Complex Geometry of Slant Submanifolds [1st ed. 2022] 9811600201, 9789811600203

Table of contents :
Preface
Contents
About the Editors
An Overview of Recent Developments in Slant Submanifolds
1 Slant submanifolds
2 Slant Submanifolds in Contact Semi-Riemannian Geometry
3 Some Generalizations
4 Slant Submanifolds and Warped Product
References
Slant Surfaces in Kaehler Manifolds
1 Introduction
2 Basic Formulas and Definitions
3 Some Examples
4 Some General Properties of Slant Surfaces
5 A Link Between Gauss Map and Slant Surfaces
6 A Link Between Regular Homotopy and Slant Surfaces
7 Mean Curvature and Gauss Curvature of Slant Surfaces
8 Spherical Slant Surfaces in C2
9 Slant Surfaces Lying in a Real Hyperplane of C2
10 Classification of Flat Slant Surfaces in C2
11 Slumbilical Surfaces in Cn
12 A Basic Inequality for Slant Surfaces
13 Special Slant Surfaces in Complex Space Forms
14 A Link Between Slant and Contact Slant Submanifolds
15 Special Slant Surfaces in Complex Hyperbolic Plane
16 The Slant Surface in CH2(-4) Satisfying the Basic Equality
17 Complex Extensors and H-Umbilical Submanifolds
18 Classification of Lagrangian H-Umbilical Submanifolds
19 A Basic Inequality for Kaehlerian Slant Submanifolds
20 An Open Problem and Three Conjectures on Slant Surfaces
References
Slant Geometry of Warped Products in Kaehler and Nearly Kaehler Manifolds
1 Introduction
2 Preliminaries
3 CR-Products in Complex Space Forms
4 Generic Products in Complex Space Forms
5 Warped Products in Real Space Forms
6 Warped Products in Complex Space Forms
7 CR-Warped Products in Kaehler Manifolds
8 CR-Twisted Products in Kaehler Manifolds
9 CR-Warped Products in Complex Space Forms
10 CR-Warped Products with Compact Holomorphic Factor
11 Bi-slant Warped Products in Kaehler Manifolds
12 Pointwise Bi-slant Warped Products in Kaehler Manifolds
13 Pointwise Semi-slant Warped Products in Kaehler Manifolds
14 Pointwise Hemi-Slant Warped Products in Kaehler Manifolds
15 Pointwise CR-Slant Warped Products in Kaehler Manifolds
16 Basics on Nearly Kaehler Manifolds
17 CR-Warped Products in Nearly Kaehler Manifolds
18 Hemi-Slant Warped Products in Nearly Kaehler Manifolds
19 Semi-slant Warped Products in Nearly Kaehler Manifolds
20 Generic Warped Products in Nearly Kaehler Manifolds
21 Bi-warped Products in Nearly Kaehler Manifolds
22 CR-Slant Warped Products in Nearly Kaehler Manifolds
23 Generalized Complex Space Forms
24 Warped Products in Generalized Complex Space Forms
25 Casorati Curvature of Bi-slant Submanifolds in Generalized Complex Space Forms
References
Slant Geometry of Riemannian Submersions from Almost Hermitian Manifolds
1 Introduction
2 Hermitian and Almost Hermitian Manifolds
3 Submanifolds of Almost Hermitian Manifolds
4 Basics on Riemannian Submersions
5 Almost Hermitian Submersions
6 Invariant Submersions
7 Anti-invariant Submersions
8 Lagrangian Submersions
9 Semi-invariant Submersions
10 Pointwise Slant Submersions
11 Bi-Slant Submersions
12 Hemi and Semi-slant Submersions
12.1 Hemi-slant Submersions
12.2 Semi-slant Submersions
13 Quasi Bi-Slant Submersions
14 Conformal Submersions
14.1 Conformal Slant Submersions
14.2 Conformal Anti-invariant Submersions
14.3 Conformal Semi-slant Submersions
References
Slant Submanifolds of the Nearly Kaehler 6-Sphere
1 Introduction
2 Preliminaries
3 Slant Submanifolds
References
Slant Submanifolds of Para Hermitian Manifolds
1 Introduction
2 Basic Formulas and Definitions
3 Characterization Results
4 Examples of Slant Submanifolds
5 Bi-Slant, Semi-slant, Hemi-slant, and CR-Submanifolds
6 Examples of Bi-Slant, Semi-slant, Hemi-slant, and CR-Submanifolds
7 Slant Submanifolds of a Para Kaehler Manifold
8 Semi-slant and Hemi-slant Submanifolds of a Para Kaehler Manifold
References
Hemi-slant and Semi-slant Submanifolds in Locally Conformal Kaehler Manifolds
1 Hemi-slant Submanifolds in Locally Conformal Kaehler Manifolds
1.1 Locally and Globally Conformal Kaehler Manifolds
1.2 Submanifolds of Riemannian Manifolds
1.3 Hemi-slant Submanifolds of an Almost Hermitian Manifold
1.4 Hemi-slant Submanifolds of a Locally Conformal Kaehler Manifold
1.5 Hemi-slant Submanifolds with Parallel Canonical Structures
2 Warped Product Hemi-slant Submanifolds in Locally Conformal Kaehler Manifolds
2.1 Warped Products
2.2 Warped Product Hemi-slant Submanifolds of a l.c.K. Manifold
2.3 An Inequality for Warped Product Mixed Geodesic Hemi-slant Submanifolds
3 Semi-slant Submanifolds of an Almost Hermitian Manifold
3.1 Semi-slant Submanifolds of a Locally Conformal Kaehler Manifold
3.2 Semi-slant Submanifolds with Parallel Canonical Structures
4 Warped Product Semi-slant Submanifolds of a l.c.K. Manifold
4.1 An Inequality for Warped Product Semi-slant Submanifolds
4.2 Warped Product Semi-slant Submanifolds in l.c.K-Space Forms
References
Slant Submanifolds and Their Warped Products in Locally Product Riemannian Manifolds
1 Introduction
2 Preliminaries
3 Slant Submanifolds of Almost Product Riemannian Manifolds
4 Warped Product Semi-slant Submanifolds
5 Warped Product Semi-slant Submanifolds of the Form MTtimesfMθ
6 Warped Product Semi-slant Submanifolds of the Form MθtimesfMT
7 Warped Product Hemi-Slant Submanifolds
8 Warped Product Pointwise Hemi-Slant Submanifolds
8.1 Pointwise Hemi-Slant Submanifolds of Locally Product Riemannian Manifolds
8.2 Warped Product Pointwise Hemi-Slant Submanifolds MperptimesfMθ
9 Generic Warped Product Submanifolds
References
Slant Submanifolds of Quaternion Kaehler and HyperKaehler Manifolds
1 Introduction
2 Ricci Curvature, Squared Mean Curvature and Shape Operator of Slant Submanifolds of Quaternion Space Forms
2.1 Ricci Curvature and Squared Mean Curvature
2.2 k-Ricci Curvature, the Squared Mean Curvature and Shape Operator of Slant Submanifolds
3 Inequalities for Slant Submanifolds of Quaternion Space Forms
3.1 B. Y. Chen Inequality for Slant Submanifolds
3.2 Inequalities for Casorati Curvature of Slant Submanifolds
4 Pointwise h-Semi-slant and Warped Product Submanifolds of Hyperkaehler Manifolds
4.1 Pointwise h-Semi-slant Submanifolds
4.2 Warped Product Submanifolds
References
Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds
1 Introduction
2 An Integral Formula of Simons' Type with a Slant Factor
2.1 Kählerian Slant Submanifolds in Complex Space Forms
2.2 Some Examples
3 Some Geometric Inequalities for Slant Submanifolds
3.1 Chen-Ricci Inequality
3.2 B. Y. Chen Inequality
3.3 Other Geometric Inequalities
4 Slant Submanifolds in Neutral Kähler Manifolds
5 Pointwise Slant Submanifolds in Almost Hermitian and Kähler Manifolds
5.1 Cohomology of Pointwise Slant Submanifolds
5.2 Some Examples
References
Lorentzian Slant Submanifolds in Indefinite Kähler Manifolds
1 Introduction
2 Preliminaries
3 Some Basic Results on Lorentzian Slant Surfaces
4 Classification Results of Lorentzian Slant Surfaces
4.1 Minimal Slant Surfaces in mathbbC12
4.2 Quasi-minimal Slant Surfaces
4.3 Slant Surfaces with Parallel Mean Curvature Vector
4.4 Pseudo-umbilical Slant Surfaces
4.5 Biharmonic and Quasi-biharmonic Slant Surfaces
References
Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds and Their Warped Product Manifolds
1 Introduction
2 Lightlike Submanifolds
3 Slant Lightlike Submanifolds
4 Totally Umbilical Slant Lightlike Submanifolds
5 Minimal Slant Lightlike Submanifolds
6 Warped Product Slant Lightlike Submanifolds
7 Pointwise Slant Lightlike Submanifolds
8 Semi-slant Lightlike Submanifolds
9 Screen Pseudo-slant Lightlike Submanifolds
References

Citation preview

Bang-Yen Chen Mohammad Hasan Shahid Falleh Al-Solamy   Editors

Complex Geometry of Slant Submanifolds

Complex Geometry of Slant Submanifolds

Bang-Yen Chen · Mohammad Hasan Shahid · Falleh Al-Solamy Editors

Complex Geometry of Slant Submanifolds

Editors Bang-Yen Chen Department of Mathematics Michigan State University East Lansing, MI, USA

Mohammad Hasan Shahid Department of Mathematics Jamia Millia Islamia New Delhi, India

Falleh Al-Solamy President, King Khalid University Abha, Saudi Arabia

ISBN 978-981-16-0020-3 ISBN 978-981-16-0021-0 (eBook) https://doi.org/10.1007/978-981-16-0021-0 Mathematics Subject Classification: 53-XX, 53C26, 53C23, 54E35, 53-02, 53A07, 53B25, 53B35 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The theory of complex submanifolds of complex manifolds began as a separate field in the nineteenth century as the theory of algebraic curves and algebraic surfaces over the complex field. In the early 1930s, E. Kaehler, J. A. Schouten and D. van Dantzig initiated the study of complex manifolds from Riemannian geometric point of view in [15, 19, 20] which led to the creation of Kaehler manifolds. In 1947, A. Weil pointed out in [21] that there exists a tensor field J of type (1,1) on the tangent bundle of a complex manifold M satisfying J 2 = −I . Then in the same year, C. Ehresmann introduced the notion of almost complex manifolds as even-dimensional manifolds which admit such a tensor field J . An almost Hermitian manifold (M, g M , J ) is an almost complex manifold (M, J ) admitting a Riemannian metric g M which is compatible with the almost complex structure J . An (immersed) submanifold of an almost Hermitian manifold (M, g M , J ) is the image of an isometric immersion φ : (N , g N ) → (M, g M , J ) from a Riemannian manifold (N , g N ) into (M, g M , J ). For a submanifold N of (M, g M , J ), there exist three important classes of submanifolds; namely the classes of complex, totally real and slant submanifolds based on the action of J on the tangent bundle T N of N defined as follows. In terms of the almost complex structure J , a submanifold N of an almost complex manifold (M, J, g) is called a complex submanifold if J (T p N ) ⊆ T p N

(1)

for any point p ∈ N . The study of complex submanifolds from differential geometric point of view with emphasis on the Riemannian structure was initiated by E. Calabi [3, 4] in the early 1950s. An excellent survey in this respect was given by K. Ogiue in [16]. v

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A submanifold N of (M, J, g) is called totally real if J maps each tangent vector of N into the corresponding normal space, that is (cf. [8, 12]) J (T p N ) ⊆ T p⊥ N

(2)

for any point p ∈ N . A totally real submanifold N in an almost Hermitian manifold M is also known as a Lagrangian submanifold if it satisfies dimR N = dimC M. For Lagrangian submanifolds from Riemannian geometric point of view, we refer to the survey article [7]. Let N be a submanifold of an almost Hermitian manifold (M, J, g). For a given vector X ∈ T N , we put J X = P X + F X,

(3)

where P X and F X denote the tangential and the normal components of J X , respectively. Then P is an endomorphism of the tangent bundle T N . For any nonzero vector X ∈ T p N at p ∈ N , the angle θ (X ) between J X and the tangent space T p N is known as the Wirtinger angle of X . In 1990, the first editor of this volume introduced in [5] the notion of slant submanifolds as follows. Definition 1 A submanifold N of an almost Hermitian manifold (M, J, g) is called slant if the Wirtinger angle θ (X ) is independent of the choice of X ∈ T p N and of p ∈ N . The Wirtinger angle of a slant submanifold is called the slant angle. A slant submanifold with slant angle θ is simply called θ-slant. Obviously, complex and totally real submanifolds are exactly θ -slant submanifolds with θ = 0 and θ = π2 , respectively. From J -action point of view, slant submanifolds (including complex and totally real submanifolds) are the simplest and the most natural submanifolds of an almost Hermitian manifold. In [11, 14], the notion of pointwise slant submanifolds of an almost Hermitian manifold was defined as a generalization of slant submanifold. The authors would like to mention that, besides complex, totally real and slant submanifolds, there is another important class of submanifolds, called CR-submanifolds introduced by A. Bejancu in [1]. For a survey on CR-submanifolds, we refer to [2, 13]. Dual to the notion of isometric immersions, there exists the notion of Riemannian submersions due to B. O’Neill [17]. By definition, a Riemannian submersion is a surjective map π : (M, g M ) → (B, g B ) from a Riemannian manifold (M, g M ) onto another Riemannian manifold (B, g B ) which preserves the scalar products of vectors normal to fibers. Dual to slant submanifolds, B. Sahin ¸ introduced in [18] the notion of slant submersions. Roughly speaking, a Riemannian submersion π : (M, g M , J ) → (B, g B )

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from an almost Hermitian manifold (M, g M , J ) onto a Riemannian manifold (B, g B ) is called a slant submersion if the vertical distribution of the submersion π is a slant distribution. The early results on slant submanifolds were collected in the book [6]. Since then, the study of slant submanifolds and slant submersions has been attracting more and more researchers, and a lot of interesting results have been achieved during the past thirty years. Given the huge amount of work on slant submanifolds published since the appearance of the last monograph [6], the editors thought it is appropriate to invite a number of specialists to contribute one or more papers to illustrate the state of the art in the theory of complex slant geometry with focuses on slant submanifolds and complex slant submersions, and many colleagues answered our call. The editors express their gratitude to all the contributors. The editors hope that the readers will find this book both as a good introduction and a useful reference of complex slant geometry to perform their research more successfully and creatively. East Lansing, MI, USA New Delhi, India Abha, Saudi Arabia

Bang-Yen Chen Mohammad Hasan Shahid Falleh Al-Solamy

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Bejancu, A.: CR submanifolds of a Kaehler manifold I. Proc. Am. Math. Soc. 69, 135–142 (1978). Bejancu, A.: Geometry of CR-Submanifolds, D. Reidel, Dordrecht (1986). Calabi, E.: Isometric embeddings of complex manifolds. Ann. of Math. 58, 1–23 (1953). Calabi, E.: Metric Riemann surfaces. Ann. of Math. Studies. 58, 1–23 (1953). Chen, B.-Y.: Slant immersions. Bull. Austral. Math. Soc. 41, 135–147 (1990). Chen, B.-Y.: Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Belgium (1990). Chen, B.-Y.: Riemannian geometry of Lagrangian submanifolds. Taiwanese J. Math. 5(4), 681–723 (2001). Chen, B.-Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications, World Scientific Publ., Hackensack, NJ (2011). Chen, B.-Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd Edition, World Scientific Publ., Hackensack, NJ (2015). Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publ., Hackensack, NJ (2017). Chen, B-Y., Garay, O.J.: Pointwise Slant submanifolds in almost Hermitian manifolds. Turk. J. Math. 36, 630–640 (2012). Chen, B.-Y., Ogiue, K.: On totally real submanifolds. Trans. Amer. Math. Soc. 193, 257–266 (1974). Dragomir, S., Shahid, M. H., Al-Solamy, F. R.: Geometry of Cauchy-Riemann submanifolds, Springer.Singapore (2016).

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14.

Etayo, F.: On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. (Debrecen) 53, 217–223 (1998). Kaehler, K.: Über eine bemerkenswerte Hermitische Metrik. Abh. Math. Sem. Univ. Hamburg 9, 173–186 (1933). Ogiue, K.: Differential geometry of Kaehler submanifolds. Adv. Math. 13, 73–114 (1974). O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966). Sahin, ¸ B.: Slant submersions from almost Hermitian manifolds. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102), 1, 93–105 (2011). Schouten, J. A., van Dantzig, D.: Über unitäre Geometrie. Math. Ann. 103 (1930), 319–346. Schouten, J. A., van Dantzig, D.: Über unitäre Geometrie konstanter Krümmung. Proc. Kon. Nederl. Akad. Amsterdam 34, 1293–1314 (1931). Weil, A.: Sur la théorie des formes différentielles attachété analytique complexe, Comm. Math. Helv. 20, 110–116 (1947).

15. 16. 17. 18. 19. 20. 21.

Contents

An Overview of Recent Developments in Slant Submanifolds . . . . . . . . . . Pablo Alegre, Joaquín Barrera, and Alfonso Carriazo

1

Slant Surfaces in Kaehler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bang-Yen Chen

19

Slant Geometry of Warped Products in Kaehler and Nearly Kaehler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bang-Yen Chen and Siraj Uddin

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Slant Geometry of Riemannian Submersions from Almost Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Bang-Yen Chen, Bogdan D. Suceavˇa, and Mohammad Hasan Shahid Slant Submanifolds of the Nearly Kaehler 6-Sphere . . . . . . . . . . . . . . . . . . . 129 Luc Vrancken Slant Submanifolds of Para Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . 139 Pablo Alegre Hemi-slant and Semi-slant Submanifolds in Locally Conformal Kaehler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Hakan Mete Ta¸stan and Sibel Gerdan Aydın Slant Submanifolds and Their Warped Products in Locally Product Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Siraj Uddin Slant Submanifolds of Quaternion Kaehler and HyperKaehler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Mohammad Hasan Shahid, Falleh Al-Solamy, and Mohammed Jamali Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Ion Mihai, Aliya Naaz Siddiqui, and Mohammad Hasan Shahid

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Lorentzian Slant Submanifolds in Indefinite Kähler Manifolds . . . . . . . . . 327 Yu Fu and Dan Yang Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds and Their Warped Product Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Rashmi Sachdeva, Garima Gupta, Rachna Rani, Rakesh Kumar, and Akhilesh Yadav

About the Editors

Bang-Yen Chen a Taiwanese-American mathematician, is University Distinguished Professor Emeritus at Michigan State University, USA, since 2012. He completed his Ph.D. degree at the University of Notre Dame, USA, in 1970, under the supervision of Prof. Tadashi Nagano. He received his M.Sc. degree from National Tsing Hua University, Hsinchu, Taiwan, in 1967, and B.Sc. degree from Tamkang University, Taipei, Taiwan, in 1965. Earlier at Michigan State University, he served as University Distinguished Professor (1990–2012), Full Professor (1976), Associate Professor (1972), and Research Associate (1970–1972). He taught at Tamkang University, Taiwan, from 1966 to 1968, and at National Tsing Hua University, Taiwan, during the academic year 1967–1968. He is responsible for the invention of δ-invariants (also known as Chen invariants), Chen inequalities, Chen conjectures, development of the theory of submanifolds of finite type, and co-developed (M+, M–)-theory. An author of 12 books and more than 500 research articles, Prof. Chen has been Visiting Professor at various universities, including the University of Notre Dame, USA; Science University of Tokyo, Japan; the University of Lyon, France; Katholieke Universiteit Leuven, Belgium; the University of Rome, Italy; National Tsing Hua University, Taiwan; and Tokyo Denki University, Japan. Mohammad Hasan Shahid is Professor at the Department of Mathematics, Jamia Millia Islamia, New Delhi, India. He earned his Ph.D. in Mathematics from Aligarh Muslim University, India, on the topic “On geometry of submanifolds” in 1988 under (Late) Prof. Izhar Husain. Earlier, he served as Associate Professor at King Abdul Aziz University, Jeddah, Saudi Arabia, from 2001 to 2006. He was a recipient of the postdoctoral fellowship from the University of Patras, Greece, from October 1997 to April 1998. He has published more than 100 research articles in various national and international journals of repute. Recently, he was awarded the Sultana Nahar Distinguished Teacher award of the Year 2017–2018 for his outstanding contribution to research. For research works and delivering talks, Prof. Shahid has visited several universities of the world: the University of Leeds, UK; the University of

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About the Editors

Montpellier, France; the University of Sevilla, Spain; Hokkaido University, Japan; Chuo University, Japan; and Manisa Celal Bayar University, Turkey. Falleh Al-Solamy is President at King Khalid University, Abha, Saudi Arabia. Earlier, he was Professor of Differential Geometry at King Abdulaziz University, Jeddah, Saudi Arabia. He studied Mathematics at King Abdulaziz University, Jeddah, Saudi Arabia, and earned his Ph.D. in Mathematics from the University of Wales Swansea, Swansea, UK, in 1998, under Prof. Edwin Beggs. His research interests concern the study of the geometry of submanifolds in Riemannian and semiRiemannian manifolds, Einstein manifolds, and applications of differential geometry in physics. Professor Al-Solamy’s research papers have been published in journals and conference proceedings of repute.

An Overview of Recent Developments in Slant Submanifolds Pablo Alegre, Joaquín Barrera, and Alfonso Carriazo

2000 AMS Mathematics Subject Classification 53C40 · 53C42 · 53C50

1 Slant submanifolds Slant submanifolds were defined in the nineties by B.-Y. Chen and the corresponding theory has had an increasing development. Actually, they present the most natural behavior of a submanifold with respect to the action of the almost complex structure of an ambient Kaehlerian manifold or the almost contact structure of an ambient Sasakian manifold. The purpose of this article is to offer a survey on the theory of slant submanifolds of a semi-Riemannian manifold, by focusing mainly on the newest papers, in fact, it could be considered as a continuation of a previous review about slant submanifolds [27]. Finally, we include an ample list of classic and new references on this topic. Let us begin by briefly remembering some basic definitions in complex Rieman g), a comnian geometry. Given an even-dimensional Riemannian manifold ( M, patible almost complex structure J is a (1, 1) tensor satisfying J 2 X = −X and  If we have such a tensor, g(J X, J Y ) = g(X, Y ), for any vector fields X, Y on M. P. Alegre (B) · A. Carriazo Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Seville, Spain e-mail: [email protected] A. Carriazo e-mail: [email protected] J. Barrera Servicio de Planes de Formación. D. G. de Formación del Profesorado e Innovación Educativa, Consejería de Educación y Deporte. Junta de Andalucía, C/ Juan Antonio de Vizarrón s/n, 41092 Seville, Spain © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_1

1

2

P. Alegre et al.

 J, g) is called an almost Hermitian manifold, which we will abbreviate then ( M,  Moreover, if the Kaehler form  given by (X, Y ) = g(X, J Y ) is simply by M.  is said to be an almost Kaehler manifold. Hermitian and Kaehler closed, then M manifolds are obtained respectively, if, in addition, the Nijenhuis torsion [J, J ] of J vanishes. On the other hand, it is well-known that an almost Hermitian manifold is a  J = 0, where ∇  denotes the Levi-Civita connection Kaehler manifold if and only if ∇  of M. In complex geometry, there are two well-known classes of submanifolds of an  namely complex submanifolds and totally real subalmost Hermitian manifold M, manifolds. They are characterized by a special behavior with respect to the ambient almost complex structure: a submanifold M is said to be complex (resp. totally real) if for any tangent vector field X , J X is also tangent (resp. is normal) to M. A more general family of submanifolds appeared in 1990 when B.-Y. Chen defined slant submanifolds in [35]. A submanifold M is said to be slant [33] if for any p ∈ M and for each nonzero vector X p ∈ T p M, the Wirtinger angle θ(X p ), 0 ≤ θ(X p ) ≤ π/2, between J p X p and T p M is a constant θ, called the slant angle of the submanifold. In particular, complex submanifolds and totally real submanifolds are slant submanifolds with angles θ = 0 and θ = π/2, respectively. On the other hand, a slant submanifold with angle θ ∈ (0, π/2) is said to be a proper slant submanifold. There is a huge number of papers dealing with slant submanifolds of an almost Hermitian manifold. There are some books and surveys about the topic: [27, 34]. On the other hand, slant submanifolds have been studied in contact Riemannian geometry. First, we recall some basic definitions in this field.  g), let us consider on M a Given an odd-dimensional Riemannian manifold ( M, (1, 1) tensor field φ, a global unit vector field ξ (called structure vector field), and a 1-form η. If φ2 X = −X + η(X )ξ, g(X, ξ) = η(X ) and g(φX, φY ) = g(X, Y ) −  then M  is said to have an almost contact η(X )η(Y ), for any vector fields X, Y on M, metric structure (φ, ξ, η, g) and it is called an almost contact metric manifold. From  φ, ξ, η, g) just by M.  now on, we will abbreviate ( M, Let  denote the fundamental 2-form given by (X, Y ) = g(X, φY ) for all vector  is said to be a contact metric manifold. Moreover, fields X, Y . If  = dη, then M X ξ = −φX , where the contact metric structure is called a K -contact structure if ∇   ∇ denotes the Levi-Civita connection of M.  is said to be normal if [φ, φ] + 2dη ⊗ ξ = 0, where [φ, φ] is The structure of M the Nijenhuis torsion of φ. A Sasakian manifold is a normal contact metric manifold. Every Sasakian manifold is a K -contact manifold and it is well-known that an almost X φ)Y = g(X, Y )ξ − contact metric manifold is a Sasakian manifold if and only if (∇ η(Y )X , for any X, Y .  is called a φ-section  a plane section π in T p M Given a Sasakian manifold M, if it is spanned by X and φX , where X is a unit tangent vector field orthogonal to ξ. The sectional curvature K (π) of a φ-section π is called φ-sectional curvature.  has constant φ-sectional curvature c, then M  is called If a Sasakian manifold M 2m+1  (c) the complete simply-connected a Sasakian space form. We denote by M Sasakian space form with dimension 2m + 1 and constant φ-sectional curvature c.  of M  2m+1 (c) is given by: The curvature tensor R

An Overview of Recent Developments in Slant Submanifolds

3

c+3  (g(Y, Z )X − g(X, Z )Y )+ R(X, Y )Z = 4

(1)

c−1 (g(X, φZ )φY − g(Y, φZ )φX + 2g(X, φY )φZ + 4 η(X )η(Z )Y − η(Y )η(Z )X + g(X, Z )η(Y )ξ − g(Y, Z )η(X )ξ), for any vector fields X, Y, Z . For more details and background, we recommend the standard Ref. [14]. Later, in [59], J. A. Oubiña introduced the notion of trans-Sasakian manifold, which were characterized by Blair and Oubiña in [17] like those manifolds that verify: X φ)Y = α(g(X, Y )ξ − η(Y )X ) + β(g(φX, Y )ξ − η(Y )φX ), (∇

(2)

 where α and β are two functions in M.  If β = 0, it is said for all X , Y in M,  is a α-Sasakian manifold. Sasakian manifolds appear as examples of αthat M  is a β-Kenmotsu manifold. Sasakian manifolds, with α = 1. If α = 0, it is said that M Kenmotsu manifolds are particular cases with β = 1. Finally, if both α and β are  is a cosymplectic manifold. Moreover, from (2) it is easy to see that the null, then M following equalities for a trans-Sasakian manifold are satisfied: X ξ = −αφX + β(X − η(X )ξ), ∇

dη = α.

On the other hand, in [6], P. Alegre, D. E. Blair and A. Carriazo introduced the notion of generalized Sasakian space form, like an almost contact manifold  φ, ξ, η, g) whose curvature tensor is given by: ( M,  R(X, Y )Z = f 1 {g(Y, Z )X − g(X, Z )Y } +

(3)

f 2 {g(X, φZ )φY − g(Y, φZ )φX + 2g(X, φY )φZ } + f 3 {η(X )η(Z )Y − η(Y )η(Z )X + g(X, Z )η(Y )ξ − g(Y, Z )η(X )ξ} ,  These manifolds are denoted where f 1 , f 2 , f 3 are differentiable functions of M.   whose by M( f 1 , f 2 , f 3 ) and generalize the notion of Sasakian space form, M(c), curvature tensor satisfy the expression (3), with f1 =

c+3 , 4

f2 = f3 =

c−1 , 4

as can be deduced immediately from (1). Now, let M be a submanifold isometrically  The definition of M as a slant immersed in an almost contact metric manifold M. submanifold was first given by Lotta in [56]. There, M is said to be slant if the angle

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between φX and T p M is a constant, which is independent of the choice of the point p ∈ M and the tangent vector X ∈ T p M, such that X and ξx are linearly independent. In particular, for θ = 0 and θ = π/2, we obtain the well-known invariant and anti is a contact invariant submanifolds, respectively. Moreover, Lotta proved that if M metric manifold, then ξ must be tangent to any non-anti-invariant slant submanifold. Slant submanifolds were studied by J. L. Cabrerizo, A. Carriazo, L. M. Fernández and M. Fernández on contact structures and mainly on K -contact and Sasakian manifolds, in [19–23]. In [20], they study and characterize slant submanifolds in K -contact and Sasakian manifolds. First, they establish a characterization of slant submanifolds through the squared of the tangential projection T of the almost contact structure on the submanifold. In fact, they show that a submanifold M of an almost contact metric  is slant if and only if there exists a constant λ ∈ [0, 1] such that T 2 = manifold M −λI + λη ⊗ ξ. Furthermore, in such case λ = cos2 θ, where θ is the slant angle of M. In addition to that, they give interesting examples of slant submanifolds. Finally, they establish two characterizations of 3-dimensional proper (i.e. neither invariant nor anti-invariant) slant submanifolds in terms of ∇T and ∇ N , respectively. In fact, these submanifolds satisfy the equality (∇ X T )Y = cos2 θ(g(X, Y )ξ − η(Y )X ), for any tangent vector fields X, Y . For an arbitrary dimension, proper slant submanifolds satisfying the above equation play an important role in contact metric geometry: they represent a similar notion to that of Kaehlerian slant submanifolds defined by Chen (see [33]). By using the Weingarten map, these authors can obtain a new result for threedimensional submanifolds of a Sasakian manifold. Indeed, such a submanifold M is slant if and only if there exists a function C : M → [0, 1] such that A N Y X = A N X Y + C(η(Y )X − η(X )Y ), for any tangent vector fields X, Y . Moreover, in this case, C = sin2 θ, where θ is the slant angle of M. Thus, it can be proved that, if M is a minimal proper slant submanifold, then (∇ X N )Y = 2η(X )N T Y + η(Y )N T X , for any tangent fields X, Y . A converse result is also given in [20], with some additional conditions. It is well-known that a Sasakian manifold induces by a natural way a Sasakian structure on every invariant submanifold. In this case, the submanifold is usually called a Sasakian submanifold. It is easy to prove that every non-anti-invariant slant submanifold admits an induced almost contact metric structure, given by sec θT , where θ  = π/2 is the slant angle. Nevertheless, it follows from Lotta’s characterization that a slant submanifold of a K -contact manifold is also K -contact if and only if it is an invariant submanifold. According to the previous result, the authors of [21] showed that it is impossible to induce a contact metric structure over a non-invariant 3-dimensional slant submanifold. But they also offered some positive results, as new characterizations of slant submanifolds, which will be useful for further works.

An Overview of Recent Developments in Slant Submanifolds

5

The same authors present in [21] the existence and uniqueness theorems for slant submanifolds of a Sasakian space form. These results are similar to those given by Chen and Vrancken for complex space forms in [39, 40]. Moreover, in [21], we can find many examples of 3-dimensional proper slant submanifolds of a Sasakian space  5 (−3) with constant φ-sectional curvature −3, as well as anti-invariant subform M manifolds of a 5-dimensional Sasakian pace form of arbitrary φ-sectional curvature c. Moreover, the first examples of proper slant submanifolds in a Sasakian manifold which is not R2m+1 with its usual Sasakian structure are found. Finally, by combining this construction with D-homothetic deformations φ∗ = φ,

ξ∗ =

1 ξ, a

η ∗ = aη,

g ∗ = ag + a(a − 1)η ⊗ η,

where a is a positive constant, it is proved that there exist 3-dimensional proper slant submanifolds in a Sasakian space form with φ-sectional curvature c, for any c < −3. There are two other papers containing results on slant submanifolds in contact geometry, written by A. Carriazo. In [24], he establishes a contact version of B.-Y. Chen’s inequality for a submanifold tangent to the structure vector field of a Sasakian  2m+1 (c) is a θ-slant immersion space form. In fact, he proves that if ϕ : M n+1 → M  2m+1 (c), then, for of a Riemannian (n + 1)-manifold into a Sasakian space form M any point p ∈ M and any plane section π ⊂ D p , we have τ − K (π) ≤

1 c+3 (n + 1)2 (n − 1) |H |2 + (n + 1)(n − 2) + 2n 2 4

+n cos2 θ +

 3(c − 1)  n cos2 θ − 2 (π) , 4 2

where τ denotes the scalar curvature of M, K (π) is the sectional curvature of π, H is the mean curvature vector of the immersion and 2 (π) = g 2 (e1 , φe2 ) is a real number in [0, 1] which is independent of the choice of the orthonormal basis {e1 , e2 } of π. In particular, if we define   (inf D K )( p) = inf K (π) : plane sections π ⊂ D p for each point p ∈ M and δ DM ( p) = τ ( p) − infD K ( p), we obtain for every 3-dimensional θ-slant submanifold M the inequality δ DM ≤

9 |H |2 + 2 cos2 θ, 4

with equality holding if and only if M is minimal.

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If we now restrict our study to T -sections, i.e., plane sections spanned by X and T X , where X is a tangent vector orthogonal to ξ, and we put (inf T K )( p) = T ( p) = τ ( p) − inf T K ( p), then the inequality inf{K (π) : T -sections π} and δ M T δM ≤

(n + 1)2 (n − 1) 1 c+3 |H |2 + (n + 1)(n − 2) + 2n 2 4 1 3(c − 1) cos2 θ +n cos2 θ + (n − 2) 2 4

holds for every (n + 1)-dimensional, non-anti-invariant θ-slant submanifold of a  2m+1 (c). Sasakian space form M By using this inequality, Carriazo presents in [26] a geometrical obstruction for the immersion of a Riemannian manifold as a minimal slant submanifold of a Sasakian space form, and he offers a non-trivial example of such a situation. On the other hand, he also obtains some topological obstructions by defining and studying a cohomology class for a proper slant submanifold of a Sasakian space form. Finally, he shows new obstructions by combining both geometrical and topological properties, in which the scalar curvature, the Ricci curvatures, and some intrinsic functions are involved. B. Y. Chen inequalities have also been studied for slant submanifolds of Kenmotsu manifolds [43], for bi-slant submanifolds in Sasakian space forms [42], Kenmotsu space forms [60] and cosymplectic space forms [46]. Later, in [54], M. A. Lone established a similar inequality but considering the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds of generalized Sasakian space forms. The notion of Casorati curvature extends the concept of the principal direction of a hypersurface of a Riemannian manifold, it is defined as C=

n+1 2m+1   1 (h r )2 . n + 1 r=n+2 i, j=1 i j

The paper [69] generalizes M. A. Lone results establishing the corresponding inequality for bi-slant submanifolds of a generalized Sasakian space form and [55] for Kenmotsu space forms. It was proved by Borrelli, Chen and Morvan [18] and independently by Ros and Urbano [63] that if M m is a Lagrangian submanifold of Cn , with mean curvature 2(m + 2) τ . The equality holds if vector H and scalar curvature τ , then H 2 ≥ 2 m (m − 1) and only if M is either totally geodesic or a (piece of a) Whitney sphere. Moreover, it was proved that M m satisfies the equality case at every point, if and only if its second fundamental form σ is given by σ(X, Y ) =

m {g(X, Y )H + g(J X, H )J Y + g(J Y, H )J X }, m+2

(4)

An Overview of Recent Developments in Slant Submanifolds

7

for any tangent vector fields X , Y . Thus, they found a simple relationship between one of the main intrinsic invariants, τ , and the main extrinsic invariant H . Remember that the Maslov form ω H as the dual form of φH , that is ω H (X ) = g(X, φH ). Well, such Maslov form, which is a closed form for a Lagrangian submanifold of C m , is a conformal form if and only if it the second fundamental form takes this form (4). At the same time, Blair and Carriazo [16] established an analogue of the above result for anti-invariant submanifolds in R2m+1 with its standard Sasakian structure. Recently, we have proved in [4] that, for a proper θ-slant submanifold, M m+1 ,  2m+1 ( f 1 , f 2 , f 3 ), the of an (α, β) trans-Sasakian generalized Sasakian space form M 2 squared mean curvature H and the scalar curvature τ verify at each point the following inequality:

H 2 ≥

 1 2(m + 2) 3 2 θ f + m f + α2 m sin2 θ . m(m + 1) f m cos τ − − 1 2 3 2 2 (m + 1)2 (m − 1)

Then, M satisfies the equality case if and only if the second fundamental form verifies  m+1 σ(X, Y ) =  m+2 (g(X, Y ) − η(X )η(Y )) H η(X ) N Y + sin12 θ g(φX, H ) − α m+2 m+1 

 η(Y ) N X , + sin12 θ g(φY, H ) − α m+2 m+1

We call ∗-slant submanifold such a submanifold verifying the equality case [4]. For a slant submanifold of a generalized Sasakian manifold, the Maslov form is not always closed. At [5] we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. However, in the case where the submanifold satisfies the equality in the above inequality, such form is not always a conformal vector field. In complex geometry, there is another environment where slant submanifolds have been studied: golden and metallic manifolds. Although the corresponding study in contact geometry has not still been developed, we include the complex case. A metallic manifold is a pair (M, J ) with M an m-dimensional manifold and J a tensor field of type (1, 1) satisfying J 2 = p J + q I for some integers p, q, where I is the identity operator on T M. If p = q = 1 one obtains the Golden structure. Moreover, if (M, g) is a Riemannian manifold endowed with a metallic structure J , such that the Riemannian metric g is J -compatible, i.e., g(J X, Y ) = g(X, J Y ), for any X, Y vector field in T M, then (M, g, J ) is a metallic Riemannian manifold. A submanifold M in a metallic (or Golden) Riemannian manifold (M, g, J ) is called slant submanifold if the angle θ(X x ) between J X x and Tx M is constant [13]. As long as we know slant submanifolds of contact metallic manifolds have not been already studied. But it is possible to consider another different generalization of the surrounding space. Thus, M.B. Hans-Uber deals with slant submanifolds of framed metric

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manifolds and S-manifolds in [49]. A (2m + r )-dimensional Riemannian manifold  g) is said to be a framed metric manifold if it is endowed with a (1, 1)-tensor ( M, field f defining an f -structure of rank 2m, s unit and mutually orthogonal vector fields ξ1 , . . . , ξs and s 1-forms η 1 , . . . , η s satisfying f 2 X = −X +



η α (X )ξα ,

g(X, ξα ) = η α (X ),

α

g( f X, f Y ) = g(X, Y ) −



η α (X )η α (Y ),

α

 and α ∈ {1, . . . , s}. In particular, almost Hermitian for any vector fields X, Y on M manifolds and almost contact metric manifolds are framed metric manifolds, with s = 0 and s = 1, respectively. Moreover, a framed metric manifold is said to be normal if  dη α ⊕ ξα = 0, [ f, f ] + 2 α

 is normal and the 2-form defined where [ f, f ] denotes the Nijenhuis torsion of f . If M by g(X, f Y ) is closed, then the manifold is called a K -manifold. Kaehler manifolds are obtained for s = 0. A normal framed metric manifold such that g(X, f Y ) = dη α (X, Y ), for any α = 1, . . . , s and any vector fields X, Y is said to be an Smanifold. Thus, Sasakian manifolds appear clearly as particular cases of S-manifolds. For more background on framed metric manifolds, we refer to [15, 77].

2 Slant Submanifolds in Contact Semi-Riemannian Geometry There have been different approaches to slant submanifolds of a semi-Riemannian manifold. Therefore, the name slant corresponds to different definitions that are more or less related to the original idea given by Chen of a constant angle. Those developments have been made in Lorentz complex, Lorentz almost contact and neutral manifolds. The study of slant submanifolds in a semi-Riemannian manifold was also initiated: B.-Y. Chen, O. Garay, and I. Mihai classified slant surfaces in Lorentzian complex space forms in [30, 31]. K. Arslan, A. Carriazo, B.-Y. Chen and C. Murathan defined slant submanifolds of a neutral Kaehler manifold in [8], while A. Carriazo and M. J. Pérez-García did it in neutral almost contact pseudo-metric manifolds in [29]. ˜ g) is said to be an almost An odd-dimensional semi-Riemannian manifold ( M, contact pseudo-metric manifold or a almost contact semi-Riemannian manifold if there exist on M˜ a (1, 1)-tensor field φ, a vector field ξ and a 1-form η such that η(ξ) =

An Overview of Recent Developments in Slant Submanifolds

9

1, φ2 (X ) = −X + η(X )ξ and the semi-Riemannian metric g on M˜ is compatible with the almost contact structure (φ, ξ, η), that is g(φX, φY ) = g(X, Y ) − εη(X )η(Y ), ˜ where ε = ±1. for any vector fields X, Y on M, 2n+1 4n+1 −→ M˜ 2n,2n+1 of a neutral (2n + 1)-manifold An isometric immersion  : Mn,n+1 into a neutral almost contact semi-Riemannian manifold is called θ-slant if there exists a real number θ, called slant angle, and an orthogonal decomposition 2n+1 = Dns ⊕ Dnt ⊕ Span{ξ}, T Mn,n+1 2n+1 of the tangent bundle T Mn,n+1 such that

(a) Dns is a space-like distribution and Dnt a time-like distribution, (b) T (Dns ) = Dnt and T (Dnt ) = Dns , (c) T 2 = sinh2 θ(I d − η ⊗ ξ), where T is the tangential part of φ. Moreover, M. A. Khan, K. Singh, and V. A. Khan first defined slant submanifolds in LP-contact manifolds in [50], after P. Alegre studied slant submanifolds of both Lorentzian Sasakian and para Sasakian manifolds in [2].  be a (2n + 1)-dimensional Lorentzian metric manifold, if it is endowed Let M with a structure (φ, ξ, η, g), where φ is a (1, 1) tensor, ξ a vector field, η a 1-form on  and g is a Lorentz metric, satisfying M φ2 X = εX + η(X )ξ, g(φX, φY ) = g(X, Y ) + η(X )η(Y ), η(ξ) = −ε, η(X ) = εg(X, ξ),

(5)

 it is called Lorentzian almost contact manifold or for any vector fields X, Y in M, Lorentzian almost para-contact manifold for ε = −1 or 1, respectively. It follows that g(φX, Y ) = εg(X, φY ) for any X, Y .  is  given by (X, Y ) = g(X, φY ), if dη = , M Let  denote the 2-form in M called normal contact Lorentzian manifold. Finally, it is called Lorentzian Sasakian (LS) or Lorentzian para Sasakian (LPS) if X φ)Y = εg(φX, φY )ξ + η(Y )φ2 X, (6) (∇ for ε = −1 or 1, respectively.  φ, ξ, η, g), A submanifold M of a Lorentzian almost (para) contact manifold, ( M, is said to be a slant submanifold if for any x ∈ M and any X ∈ Tx M, the Wirtinger’s angle, the angle between φX and T X , is a constant θ ∈ [0, 2π]. In such a case, θ it  They englobe both invariant and anti-invariant is called the slant angle of M in M. submanifold for θ = 0 and θ = π/2, respectively. A slant submanifold is called proper if it is neither invariant nor anti-invariant.

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Span{ξ} defines the time-like vector field distribution. If X is a space-like vector field, it is orthogonal to ξ, then g(φX, φX ) = g(X, X ) ≥ 0, so φX is also space-like, the same is valid for T X . For space-like vector fields the Cauchy–Schawrz inequality, g(X, Y ) ≤ |X ||Y |, is verified. Therefore the Wirtinger angle, θ, is given by: g(φX, T X ) = cos θ. |φX ||T X | It is proved that every submanifold M of a Lorentzian contact metric manifold,  normal to ξ is an anti-invariant submanifold. Also, every two-dimensional subM, manifold of a Lorentzian almost contact manifold tangent to ξ is anti-invariant. Some characterization results are given and some examples are constructed using warped products. Atceken and Hui [9], continue this line studying slant and pseudo-slant submanifolds in Lorentzian concircular structure-manifolds. A LCS-manifold is an ndimensional Lorentzian manifold admitting a unit time-like concircular vector field ξ, that is verifying (∇ X η)(Y ) = α{g(X, Y ) + η(X )η(Y )} for certain α nonzero scalar function. For α = 1 it would be a Lorentzian para-contact manifold. There has been an attempt [11], to define slant submanifolds of a (ε) para-Sasakian ˜ φ, η, g) manifold as those submanifolds M of an (ε)-para-Sasakian manifold ( M, for which the M is said to be a contact slant submanifold if the angle θ(X ) between φX and T M( p) is constant at any point p ∈ M for any X linearly independent of ξ. We think this definition is not correct as it cannot be always measured at this angle depending on the causal character of the involved vector fields. But they have been finally defined by Chanyal [41] and by Carriazo and García at [28] following the definition of slant submanifolds of a para-Hermitian manifold  g), let us given at [3]. Given and odd-dimensional semi-Riemannian manifold ( M,  a (1, 1) tensor field φ, a global unit vector field ξ, and a 1-form η. If consider on M φ2 X = X − η(X )ξ, g(X, ξ) = η(X ) and g(φX, φY ) = −g(X, Y ) + η(X )η(Y ), for  then M  is said to have an almost para-contact metric any vector fields X, Y on M, structure (φ, ξ, η, g) and it is called an almost para-contact metric manifold.  φ, ξ, η, g) is slant if the quotient A submanifold M of a para-contact manifold ( M, g(T X, T X )/g(φX, φX ) is constant, for every space-like or time-like tangent vector field X , where T is the tangent part of φ. As in the para-Hermitian case, three cases could be distinguished: type 1 type 2 type 3

if for any space-like (time-like) vector field X , T X is time-like (space-like), |T X | and > 1, |φX | if for any space-like (time-like) vector field X , T X is time-like (space-like), |T X | and < 1, |φX | if for any space-like (time-like) vector field X , T X is space-like (time-like).

An Overview of Recent Developments in Slant Submanifolds

11

In both papers appears characterization results for those submanifolds and interesting examples of each of them. There is another approach from B. Sahin definition of lightlike slant submanifolds of indefinite Kaehler manifolds [67]. At [68], B. Sahin together with C. Yildirim introduced the concept of slant submanifolds of an indefinite Sasakian manifolds. They study the existence problem and establish an interplay between slant lightlike submanifolds and contact Cauchy Riemann (CR)-lightlike submanifolds. They prove a characterization theorem and show that co-isotropic contact CR-lightlike submanifolds are slant lightlike submanifolds and give non-trivial examples.  g) is called a almost contact An odd-dimensional semi-Riemannian manifold ( M, metric manifold if there exists a (1, 1) tensor field φ, a global unit vector field ξ (called structure vector field), and a 1-form η. If φ2 X = −X + η(X )ξ, g(X, ξ) = η(X ) and  g(φX, φY ) = g(X, Y ) − η(X )η(Y ), g(ξ, ξ) = , for any vector fields X, Y on M,  is said to have an almost contact metric structure (φ, ξ, η, g) and it is called then M  φ, ξ, η, g) an almost contact metric manifold. From now on, we will abbreviate ( M,  Let  denote the fundamental 2-form given by (X, Y ) = g(X, φY ) for just by M.  is said to be a contact metric manifold. all vector fields X, Y . If  = dη, then M  The structure of M is said to be normal if [φ, φ] + 2dη ⊗ ξ = 0, where [φ, φ] is the Nijenhuis torsion of φ. A Sasakian manifold is a normal contact metric manifold, X φ)Y = −g(X, Y )ξ + η(Y )X , for any X, Y . X ξ = φX and (∇ for which ∇ m ˜ is A submanifold (M , g) immersed in a semi-Riemannian manifold ( M˜ m+n , g) called a lightlike submanifold if the metric g induced from g˜ is degenerate and the radical distribution Rad(T M) is of rank r , where 1 ≤ r ≤ m. Let S(T M) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(T M) in T M, i.e., T M = Rad(T M) ⊥ S(T M). Consider a screen transversal vector bundle S(T M ⊥ ), which is a semi-Riemannian complementary vector bundle of Rad(T M) in T M ⊥ . Since, for any local basis {Vi } of Rad(T M), there exists a local null frame {Ni } of sections with values in the orthogonal complement of S(T M ⊥ ) in [S(T M)]⊥ such that g(V ˜ i , N j ) = δi j , it follows that there exists a lightlike transversal vector bundle ltr (T M) locally spanned by {Ni }. Let tr (T M) be complementary (but not ˜ M . Let M be a q lightlike submanifold of an orthogonal) vector bundle to T M in T M| ˜ indefinite Sasakian manifold M of index 2q. Then we say that M is a slant lightlike submanifold of M˜ if the following conditions are satisfied:  (A) RadT M is a distribution on M such that φRadT M RadT M = 0. (B) For each nonzero vector field tangent to D at x ∈ U ⊂ M, the angle θ(X ) between φX and the vector space Dx is constant, that is, it is independent of the choice of x ∈ U ⊂ M and X ∈ Dx , where D is complementary distribution to φRadT M ⊕ φltr (T M) in the screen distribution S(T M).

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3 Some Generalizations The first generalization of slant immersions was given by N. Papaghiuc by introducing in [61] the notion of semi-slant submanifold of an almost Hermitian manifold, that is the tangent bundle admits a direct orthogonal decomposition T M = D1 ⊕ D2 , in a complex distribution and a slant one (the angle θ(X p ), 0 < θ(X p ) ≤ π/2, between J p X p and the distribution is a constant). They generalized CR-submanifolds. On the other hand Carriazo in [25] defined pseudo-slant submanifolds (they are also know as hemi-slant or anti-slant submanifolds) and bi-slant submanifolds. Both of them satisfy a decomposition of the tangent bundle with the following differences: for a pseudo-slant submanifold D1 is a totally real distribution and D2 is a slant distribution with angle θ2  = π/2, while a submanifold is said to be bi-slant if both distributions are slant, with angles θ1 , θ2 ∈ [0, π/2]. The notions of semi-slant and bi-slant submanifolds have also been introduced and studied in contact metric geometry by Cabrerizo, Carriazo, Fernández and Fernández in [22]. They present the definition of slant distribution in contact metric geometry, and they prove that a submanifold tangent to the structure vector field ξ of an almost contact metric manifold is slant if and only if the contact distribution D on the submanifold is a slant distribution. They also give a characterization theorem for slant distributions orthogonal to ξ. Then, a submanifold M of an almost contact metric manifold is said to be bi-slant if its tangent bundle admits an orthogonal direct decomposition T M = D1 ⊕ D2 ⊕ Span{ξ}, where D1 and D2 are two slant distributions. If D1 is an invariant distribution, i.e., φ(D1 ) = D1 and D2 is slant with nonzero angle, then M is said to be semi-slant. As in the complex case, invariant, anti-invariant, and proper slant submanifolds can be seen as particular examples of semi-slant submanifolds. On the other hand, this kind of submanifold also includes that of semi-invariant submanifolds, introduced by Bejancu and Papaghiuc in [12], which correspond in contact geometry to CRsubmanifolds. Also semi-slant and hemi-slant submanifolds of metallic manifolds have been studied [47, 48]. The definition of almost semi-invariant submanifolds have also been introduced and studied in framed metric manifolds (see [71]). We also recommend the paper [57] of M. M. Tripathi and I. Mihai, which contains many results relating to some different kinds of submanifolds. From a different point of view, another generalization of slant submanifolds has been given by F. Etayo, by defining in [44] quasi-slant submanifolds. A submanifold of an almost Hermitian manifold is said to be quasi-slant if the Wirtinger angle is pointwise constant, but it is not globally constant. The main result concerning quasi-slant submanifolds is that surfaces are always quasi-slant.

An Overview of Recent Developments in Slant Submanifolds

13

As a generalization of slant submanifolds and semi-slant submanifolds [62], K. S. Park introduced the notions of pointwise slant submanifolds and pointwise semi-slant submanifolds of an almost contact metric manifold. The submanifold M is called a pointwise slant submanifold if at each given point p ∈ M the angle θ(X ) between φX and the space M p = {X ∈ T p M, g(X, ξ) = 0} is constant for nonzero X ∈ M p , θ is called the slant function. That is, the slant angle could change for every point. But, he did not mention whether the structure vector field ξ is either tangent or normal to the submanifold. In [74], S. Uddin, A. H. Alkhaldi modified the definition of pointwise slant submanifolds of an almost contact metric manifold such that the structure vector field ξ is tangent to the submanifold, giving the following characterization: a submanifold of an almost contact metric manifold M˜ such that ξ ∈ T M, is pointwise slant if and only if T 2 = cos 2 θ(I + η ⊗ ξ) for some real valued function θ defined on the tangent bundle T M of M.

4 Slant Submanifolds and Warped Product B. Y. Chen initiated the study of warped product CR-submanifolds of a Kaehlerian manifold [36, 38], that is a submanifold M of a Kaehler manifold M˜ whose tangent vector space admits a decomposition T M = D ⊕ D ⊥ with D a holomorphic distribution and D ⊥ a totally real one. He proved that there exits warped product CR-submanifolds of type N T × f N⊥ but not of type N⊥ × f N T . He continued, together with Munteanu [37], studying PR-warped product submanifolds of a para-Kaehler manifold. After, B. Sahin studied pointwise semi-slant submanifolds and warped product pointwise semi-slant submanifolds by using the notion of pointwise slant submanifolds of a Kaehler manifold [64]. At [65], B. Sahin proved the non-existence of proper warped product semi-slant submanifolds of Kaehler manifolds on the form MT × f Mθ nor Mθ × f MT . And at [66] he proved that there are number of warped product hemi-slant submanifolds on the form M⊥ × f Mθ while on the form Mθ × f M⊥ do there exit. In [32], the authors considered the existence of warped product bi-slant submanifolds of a Kaehler manifold, that is M = M1 × f M2 with both M1 and M2 slant submanifolds. They prove that every bi-slant warped product submanifold is a Riemannian product or a warped product hemi-slant submanifold. Those ideas originated a huge amount of papers dealing with similar results, changing the submanifold: CR, semi-slant, hemi-slant... or the structure of the manifold: Kaehler, almost contact, cosymplectic, para-Kaehler, almost para contact, paracosymplectic. We summarize the results for slant submanifolds in an almost contact semi-Riemannian manifold. In [7], F. R. Al-Solamy and V. A. Khan proved that there are no warped product semi-slant submanifolds of the form MT × f Mθ in a Sasakian manifold other than contact CR-warped products, but if we assume that Mθ is a proper pointwise slant

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submanifold then I. Mihai and S. Uddin showed there exists a non-trivial class of such warped products giving non-trivial examples. S. Uddin and F. R. Solamy have studied warped product pseudo-slant immersions in Sasakian manifolds [73]. They also established an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. For almost contact metric manifolds, K. A. Khan, V. A. Khan, and S. Uddin have seen in [51] that there are no proper warped product semi-slant submanifolds in the cosymplectic case. At [53], they proved the non-existence of warped product pseudoslant submanifolds of type N⊥ × f Nθ nor Nθ × F N⊥ with the structure vector field ξ tangent to the second factor (fiber). Although non example of type Nθ × F N⊥ with ξ tangent to Nθ has been given, S. Uddin with F.R. Al-Solamy studied a relation for the squared norm of the second fundamental form of a mixed geodesic warped product pseudo-slant submanifold with the warping function. Again, if we change to pointwise semi-slant submanifolds, they do exist warped product pointwise semi-slant submanifolds MT × f Mθ in a cosymplectic manifold [1]. They give an inequality in terms of the second fundamental form and the scalar curvature using Gauss equation, study warped product pointwise semi-slant submanifolds in cosymplectic space form in various mathematical and physical terms such as Hessian, Hamiltonian, and kinetic energy. Finally, about the existence of warped product hemi-slant and semi-slant submanifold of a Kenmotsu manifold, M. Atceken proved that the warped product submanifolds of the types Nθ × f N T and Nθ × f N⊥ of a Kenmotsu manifold M˜ do not exist where the manifolds Nθ and N T (resp., N⊥ )are proper slant and φ-invariant (resp. anti-invariant) [10]. However, at [52] warped products of the types N T × f Nθ are studied and one example of type N⊥ × f Nθ is given. Srivastava, Sharma, Tiwari continued with this type of results [70]: now they deal with PR-anti-slant warped product submanifolds of a nearly paracosymplectic manifold. That is T M = D ⊥ ⊕ Dθ ⊕ < ξ >, with D ⊥ anti-invariant and Dθ a slant distribution. They prove the usual results: (i) If ξ is tangent to Nθ , there do not exist PR-anti-slant warped product submanifolds on the form N⊥ × f Nθ . (ii) If ξ is tangent to N⊥ , they present several examples of PR-anti-slant warped product submanifolds on the form N⊥ × f Nθ , characterizing when the submanifold is a PR-anti-slant product, that is the warping function is constant. (iii) There do not exist Nθ × f N⊥ and ξ ∈ (T N⊥ ), and some characterization results are given for Nθ × f F and ξ ∈ (T Nθ ). There has been still another one more generalization studying double warped product submanifolds of a trans-Sasakian manifold. Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Unal [76]. A doubly warped product manifold of N1 and N2 , denoted as f 2 N1 × f 1 N2 is endowed with a metric g defined as g = f 22 2g1 + f 12 g2 where f 1 and f 2 are positive differentiable functions on N1 and N2 respectively. But, [75] there do not exist proper doubly warped product submanifolds of a trans-Sasakian manifold. Also warped products

An Overview of Recent Developments in Slant Submanifolds

15

of the type N1 × f N2 of trans-Sasakianmanifolds do not exist if the structure vector field ξ is tangent to N2 . The usual procedures are done for N T × f Nθ and N⊥ × f Nθ with ξ tangent to the first submanifold.

References 1. Akram, A., Ozel, C.: Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications. Int. J. Geom. Methods Modern Phys. (2017) 2. Alegre, P.: Slant submanifolds of Lorentzian Sasakian and para Sasakian manifolds. Taiwan. J. Math. 17, 897–910 (2013) 3. Alegre, P., Carriazo, A.: Slant submanifolds of para-hermitian manifolds. Mediterr. J. Math. 14, 214 (2017) 4. Alegre, P., Barrera, J., Carriazo, A.: A new class of slant submanifolds in generalized Sasakian space forms. Mediterr. J. Math. 17(3), Paper No. 76, 18 pp. (2020) 5. Alegre, P., Barrera, J., Carriazo, A.: A closed form for slant submanifolds of generalized Sasakian space forms. Mathematics 7(12), 1238 (2019) 6. Alegre, P., Blair, D.E., Carriazo, A.: Generalized Sasakian-space-forms. Isr. J. Math. 141, 157–183 (2004) 7. Al-Solamy, F.R., Khan, V.A.: Warped product semi-slant submanifolds of a Sasakian manifold. Serdica Math. J. 34, 597–606 (2008) 8. Arslan, K., Carriazo, A., Chen, B.-Y., Murathan, C.: On slant submanifolds of neutral Kaehler manifolds. Taiwan. J. Math. 14(2):561–584 (2010) 9. Atceken, M., Hui, S.S.: Slant and pseudo slant submanifolds in LCS-manifolds. Czechoslov. Math. J. 63(138), 177–190 (2013) 10. Atceken, M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 34, 425–433 (2010) 11. Atceken, M., Dirik, S., Yıldırım, U.: Contact pseudo-slant submanifolds of a (ε) para Sasakian space form. Int. Math. Virtual Inst. 10(1), 59–74 (2020) 12. Bejancu, A., Papaghiuc, N.: Semi-invariant submanifolds of a Sasakian manifold. An. Stiin¸ ¸ t. Univ. Al. I. Cuza Ia¸si 27, 163–170 (1981) 13. Blaga, A.M., Hretcanu, C.E.: Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold (2018). arXiv:1803.01415 14. Blair, D.E.: Geometry of manifolds with structural group U(n) × O(s). J. Differ. Geom. 4, 155–167 (1970) 15. Blair, D.E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509. Springer, New York (1976) 16. Blair, D.E., Carriazo, A.: The contact Whitney sphere. Note di Matematica 20(2), 125–133 (2000/2001) 17. Blair, D.E., Oubiña, J.A.: Conformal and related changes of metric on the product of two almost contact metric manifolds. Publ. Mat. 34, 199–207 (1990) 18. Borrelli, V., Chen, B.-Y., Morvan, J.M.: Une caractérisation géométrique de la sphère de Whitney. C. R. Acad. Sci. Paris, Série I 321, 1485–1490 (1995) 19. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedicata 78, 183–199 (1999) 20. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42, 125–138 (2000) 21. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Structure on a slant submanifold of a contact manifold. Indian J. Pure Appl. Math. 31, 857–864 (2000) 22. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Existence and uniqueness theorem for slant immersions in Sasakian-space-forms. Publ. Math. Debrecen 58, 559–574 (2001)

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23. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Riemannian submersions and slant submanifolds. Publ. Math. Debrecen 61, 523–532 (2002) 24. Carriazo, A.: A contact version of B.-Y. Chen’s inequality and its applications to slant immersions. Kyungpook Math. J. 39, 465–476 (1999) 25. Carriazo, A.: Bi–slant immersions. In: Proceedings ICRAMS 2000, India (2000) 26. Carriazo, A.: Obstructions to slant immersions in contact manifolds. Ann. Mat. Pura Appl. 179, 459–470 (2001) 27. Carriazo, A.: New developments in slant submanifolds theory. In: Misra, J.C. (ed.) Applicable Mathematics in the Golden Age. Narosa Publishing House, New Delhi, pp. 339–356 (2002) 28. Carriazo, A., García, D.: Productos warped en variedades para-Hermíticas y para-contacto métricas. Universidad de Sevilla, Trabajo fin de Master (2020) 29. Carriazo, A., Pérez-García, M.J.: Slant submanifolds in neutral almost contact pseudo-metric manifolds. Diff. Geom. Appl. 54, 71–80 (2017) 30. Chen, B.-Y., Garay, O.: Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space E 24 . Results Math. 55, 23–38 (2009) 31. Chen, B.-Y., Mihai, I.: Classification of quasi-minimal slant surfaces in Lorentzian complex space forms. Acta Math. Hungar. 122, 307–328 (2009) 32. Chen, B.-Y., Al-Solamy, F.R., Uddin, S.: Warped product bi-slant inmersions in Kaehler manifolds. Mediterr. J. Math. 14(2) Art. 95, 11 (2017) 33. Chen, B.-Y.: Slant immersions. Bull. Austral. Math. Soc. 41, 135–147 (1990) 34. Chen, B.-Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven (1990) 35. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60, 568–578 (1993) 36. Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133, 177–195 (2001) 37. Chen, B.-Y., Munteanu, M.I.: Geometry of PR-warped products in para-Kaehler manifolds. Taiwan. J. Math. 16(4), 1293–1327 (2012) 38. Chen, B.-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6(2), 01–43 (2013) 39. Chen, B.-Y., Vrancken, L.: Existence and uniqueness theorem for slant immersions and its applications. Results Math. 31, 28–39 (1997) 40. Chen, B.-Y., Vrancken, L.: Addendum to: existence and uniqueness theorem for slant immersions and its applications. Results Math. 39, 18–22 (2001) 41. Chanyal, S.K.: Slant submanifolds in an almost paracontact metric manifold. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 67(2), 213–229 (2021) 42. Cioroboiu, D., Chen, B.-Y.: Inequalities for bi-slant submanifolds in Sasakian space forms. Demonstratio Math. 36(1), 179–187 (2003) 43. Costache, S., Chen, B.-Y.: Inequalities for slant submanifolds in Kenmotsu space forms. Bull. Transilv. Univ. Bra¸sov Ser. III 1(50), 87–92 (2008) 44. Etayo, F.: On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. Debrecen 53, 217–223 (1998) 45. Gunduzalp, Y.: Neutral slant submanifolds of a para-Kaehler manifold. Hindawi Publ. Corp. Art. 752650, 8 (2013) 46. Gupta, R.S.: B.-Y. Chen’s inequalities for bi-slant submanifolds in cosymplectic space forms. Sarajevo J. Math. 9(21)(1), 117–128 (2013) 47. Hretcanu, C.E., Blaga, A.M.: Slant and semi-slant submanifolds in metallic Riemannian manifolds. J. Funct. Spaces Art. 2864263, 13 (2018) 48. Hretcanu, C.E., Blaga, A.M.: Hemi-slant submanifolds in metallic Riemannian manifolds. Carpathian J. Math. 35(1), 59–68 (2019) 49. Hans-Uber, M.B.: Subvariedades slant en f -variedades (Tesina de Licenciatura). Universidad de Sevilla (2001) 50. Khan, K.A., Singh, K., Khan, V.A.: Slant submanifolds of LP-contact manifolds. Diff. Geo. Dyn. Syst. 12, 102–108 (2010)

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51. Khan, K.A., Khan, V.A., Uddin, S.: Warped product submanifolds of cosymplectic manifolds. Balkan J. Geom. Appl. 13, 55–65 (2008) 52. Khan, K.A., Khan, V.A., Uddin, S.: Warped product submanifolds of Kenmotsu manifold. Turk J. Math. 36, 319–330 (2012) 53. Khan, K.A., Khan, V.A., Uddin, S.: A note on warped product submanifolds of cosymplectic manifolds. Filomat 24, 95–102 (2010) 54. Lone, M.A.: Some inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in generalized Sasakian space form. Novi Sad J. Math. 47(1), 129–141 (2017) 55. Lone, M.A.: Optimal inequalities for generalized normalized δ-Casorati curvatures for bi-slant submanifolds of Kenmotsu space forms. J. Dyn. Syst. Geom. Theor. 17(1), 39–50 (2019) 56. Lotta, A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Roumanie 39, 183–198 (1996) 57. Mihai, I., Tripathi, M.M.: Submanifolds of framed metric manifolds and S-manifolds. Note Mat. 20(2), 135–164 (2000/01) 58. Mihai, I., Uddin, S.: Warped product pointwise semi-slant submanifolds of Sasakian manifolds 59. Oubiña, J.A.: New classes of almost contact metric structures. Publ. Math. Debrecen 32, 187– 193 (1985) 60. Pandey, P.K., Gupta, R.S., Sharfuddin, A.B.-Y.: Chen’s inequalities for bi-slant submanifolds in Kenmotsu space forms. Demonstratio Math. 43(4), 887–898 (2010) 61. Papaghiuc, N.: Semi-slant submanifolds of a Kaehlerian manifold. An. Stiin¸ ¸ t. Univ. Al. I. Cuza Ia¸si 40 55–61 (1994) 62. Park, K.S.: Pointwise slant and pointwise semi-slant submanifolds in almost contact metric manifolds 63. Ros, A., Urbano, F.: Lagrangian submanifolds of C n with conformal Maslov form and the Whitney sphere. J. Math. Soc. Jpn. 50(1), 203–226 (1998) 64. Sahin, B.: Warped product pointwise semi-slant submanifolds of Kaehler manifolds. Port. Math. 70, 252–268 (2013) 65. Sahin, B.: Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedicata 117, 195–202 (2006) 66. Sahin, B.: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Polon. Math. 95(3), 207–226 (2009) 67. Sahin, B.: Slant lightlike submanifolds of indefinite Hermitian manifolds. Balkan J. Geom. Appl. 13(1), 107–119 (2008) 68. Sahin, B., Yıldırım, C.: Slant lightlike submanifolds of indefinite Sasakian manifolds. Filomat 26, 277–287 (2012) 69. Siddiqui, A.N., Shahid, M.H.: A lower bound of normalized scalar curvature for bi-slant submanifoldsin generalized Sasakian space forms using Casorati curvatures. Acta Math. Univ. Comenian. 87(1), 127–140 (2018) 70. Srivastava, S.K., Sharma, A., Tiwari, S.K.: PR-anti-slant warped product submanifold of a nearly paracosymplectic manifold 71. Tripathi, M.M., Singh, K.D.: Almost semi-invariant submanifolds of an ξ-framed metric manifold. Demonstratio Math. 29, 413–426 (1996) 72. Uddin, S., Al-Solamy, F.R.: Warped product pseudo-slant submanifolds of cosymplectic manifolds. An. Stiint. Univ. Al. I. Cuza Iasi Mat Tome LXIII 3, 901–913 (2016) 73. Uddin, S., Al-Solamy, F.R.: Warped product pseudo-slant immersions in Sasakian manifolds. Publ. Math. Debrecen 91, 331–348 (2017) 74. Uddin, S., Alkhaldi, A.H.: Pointwise slant submanifolds and their warped products in Sasakian manifolds. Filomat 32(12), 1–12 (2018) 75. Uddin, S., Khan, K.A.: Warped product semi-slant submanifolds of trans-Sasakian manifolds. Differ. Geom. Dyn. Syst. 12, 260–270 (2010) 76. Unal, B.: Doubly warped products. Diff. Geom. Appl. 15, 253–263 (2001) 77. Yano, K., Kon, M.: Structures on Manifolds. Series in Pure Mathematics, vol. 3. World Scientific, Singapore (1984)

Slant Surfaces in Kaehler Manifolds Bang-Yen Chen

2000 AMS Mathematics Subject Classification 53C40 · 53C42 · 53C50

1 Introduction The theory of submanifolds of a Kaehler manifold is one of the most interesting topics in differential geometry. According to the behavior of the tangent bundle of a submanifold with respect to the action of the almost complex structure J of the ambient manifold, there are three important classes of submanifolds, namely complex submanifolds, totally real submanifolds, and slant submanifolds. The study of complex submanifolds began as a separate area of study in the nineteenth century with the investigation of algebraic curves and algebraic surfaces in classical algebraic geometry over complex field. The principal investigators are Riemann, Picard, Enriques, Castelnuovo, Severi, and Segre, among others. It was Kaehler [48], Schouten, and van Dantzig [58, 59] who first tried to study complex manifolds from the viewpoint of Riemannian geometry in the early 1930s. In their studies, a Hermitian space with the so-called symmetric unitary connection was introduced. A Hermitian space with such a connection is known today as a Kaehler manifold. It was Weil [64] who in 1947 pointed out there exists in a complex manifold a tensor field J of type (1, 1) whose square is equal to the negative of the identity transformation of the tangent bundle, that is, J 2 = −I. In the same year, C. Ehresmann introduced the notion of an almost complex manifold as an even-dimensional differentiable manifold which admits such a tensor field J of type (1, 1).

B.-Y. Chen (B) Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_2

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The study of complex submanifolds of a Kaehler manifold from the differential geometrical point of view, with emphasis on the Riemannian metric, was initiated by E. Calabi in the early of 1950s (cf. [4, 5]). In terms of the behavior of the tangent bundle T N of the submanifold N , complex submanifolds N of a Kaehler manifolds (M, g, J ) are characterized by J (T p N ) ⊆ T p N

(1)

for any point p ∈ N . Since then, it has become an active and fruitful field in modern differential geometry. Many important results on complex submanifolds in Kaehler manifolds have been obtained by many differential geometers (cf., e.g., [52, 53]). Besides complex submanifolds, there is another important class of submanifolds, called totally real submanifolds. A totally real submanifold N is a submanifold such that the almost complex structure J of the ambient manifold M carries each tangent vector of N into the corresponding normal space of N in M, i.e., J (T p N ) ⊆ T p⊥ N

(2)

for any point p ∈ N (cf. [25, 27, 36]). When dimR N = dimC M, the totally real submanifold N in M is also known as a Lagrangian submanifold. Let N be a submanifold of a Kaehler manifold (or an almost Hermitian manifold) (M, J, g). For any vector X tangent to M, we put J X = P X + F X,

(3)

where P X and F X denote the tangential and the normal components of J X , respectively. Then P is an endomorphism of the tangent bundle T N . For any nonzero vector X ∈ T p N at p ∈ N , the angle θ(X ) between J X and the tangent space T p N is called the Wirtinger angle of X . In 1990, the author [8] introduced the notion of slant submanifolds as follows. Definition 1.1 A submanifold N of an almost Hermitian manifold (M, J, g) is called a slant submanifold if the Wirtinger angle θ(X ) is independent of the choice of X ∈ T p N and of p ∈ N . The Wirtinger angle of a slant submanifold is called the slant angle. A slant submanifold with slant angle θ is simply called θ-slant. Complex submanifolds and totally real submanifolds are exactly θ-slant submanifolds with θ = 0 and θ = π2 , respectively. A slant submanifold is called proper slant if it is neither complex nor totally real. From J -action point of view, slant submanifolds are the simplest and most natural submanifolds of a Kaehler manifold. Such submanifolds arise naturally and play important roles in the theory of submanifolds. It was shown in [8] that proper slant submanifolds are characterized by the following simple condition:

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21

P 2 = λI,

(4)

where λ is a real number λ ∈ (−1, 0) and I is the identity map. It was proved in [9, p. 97] that slant submanifolds in Kaehler manifolds have the following topological property. Theorem 1.2 If N is a compact 2k-dimensional proper slant submanifold of a Kaehler manifold, then its cohomology groups satisfy H 2i (N ; R)  = {0} for each i ∈ {1, . . . , k}. Also, the following topological result was proved in [38]. Theorem 1.3 Let N be a slant submanifold in a complex Euclidean m-space Cm . If N is not totally real, then N is non-compact. In particular, there do not exist compact proper slant surfaces in any complex Euclidean m-space. Although there do not exist compact proper slant submanifolds in complex Euclidean spaces according to Theorem 1.3, there do exist compact proper slant submanifolds in complex flat tori. The first results on slant submanifolds were collected in the author’s book [9] published in 1990. Since then, there are many nice results on slant submanifolds obtained by many geometers. Since slant submanifolds (in particular, slant surfaces) of Kaehler surfaces are the most important class of slant submanifolds, the main purpose of this chapter is, thus, to present a comprehensive survey on the main results on proper slant surfaces in Kaehler manifolds (and some related results on slant submanifolds) obtained during the last three decades. Furthermore, in the last section of this chapter, we present an open problem and three conjectures on slant surfaces in complex space forms.

2 Basic Formulas and Definitions We follow the notations and definitions from [6, 9, 11, 25, 28, 50, 55, 57]. For a submanifold N of an almost Hermitian manifold M, let ∇ and ∇˜ denote the Levi-Civita connections of N and M, respectively. Then the formulas of Gauss and Weingarten of N in M are given, respectively, by ∇˜ X Y = ∇ X + σ(X, Y ), ∇˜ X ξ = −Aξ X + D X ξ,

(5) (6)

for any vector fields X, Y tangent to N and any vector field ξ normal to N , where h denotes the second fundamental form, D the normal connection, and A the Weingarten map of the submanifold N in M.

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The second fundamental form σ and the Weingarten map A of the submanifold are related by 

 Aξ X, Y = σ(X, Y ), ξ .

(7)

For any vector ξ normal to the submanifold N , we put J ξ = tξ + f ξ,

(8)

where tξ and f ξ are the tangential and the normal components of J ξ, respectively. Then f is an endomorphism of the normal bundle T ⊥ N and t is a tangent-bundlevalued 1-form on T ⊥ N . For a submanifold N in M, the mean curvature vector H is defined by H=

n 1 1 σ(ei , ei ) tr σ = n n i=1

(9)

where {e1 , . . . , en } is a local orthonormal frame of the tangent bundle T N of N . The length of H , denoted by α, is called the mean curvature. The submanifold N is said to have parallel mean curvature vector if its mean curvature vector H is parallel in the normal bundle, i.e., D H = 0 holds identically. A submanifold N in M is called totally geodesic if its second fundamental form h vanishes identically. And it is called totally umbilical if its second fundamental form satisfies σ(X, Y ) = g(X, Y )H

(10)

¯ for any X, Y tangent to N . For a submanifold N in M, the covariant derivative ∇h of its second fundamental form is defined by (∇¯ X σ)(Y, Z ) = D X (σ(Y, Z )) − σ(∇ X Y, Z ) − σ(Y, ∇ X Z )

(11)

for vector fields X, Y, Z tangent to N . The submanifold N is called a parallel sub¯ = 0 identically. manifold if it satisfies ∇σ

3 Some Examples Let Cm = (E2m , J0 ) be the complex Euclidean m-space endowed with an almost complex structure J0 on a Euclidean 2m-space E2m , where J0 is defined by J0 (a1 , . . . , am , b1 , . . . , bm ) = (−b1 , . . . , −bm , a1 , . . . , am ).

(12)

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In this section, we provide some examples of slant surfaces in complex Euclidean spaces. Example 3.1 For any positive number θ > 0, x(u, v) = (u cos θ, u sin θ, v, 0) defines a slant plane with slant angle θ in C2 . Example 3.2 For any real number k > 0, φ(u, v) = (eku cos u cos v, eku sin u cos v, eku cos u sin v, eku sin u sin v) defines a complete, non-minimal, and pseudo-umbilical proper slant surface with √ slant angle θ = cos−1 (k/ 1 + k 2 ) and with non-constant mean curvature α given √ by α = e−ku / 1 + k 2 . Example 3.3 For any real number k > 0, φ(u, v) = (u, k cos v, v, k sin v) defines a complete, non-minimal, √ and non-pseudo-umbilical proper slant flat surface with slant angle θ = cos−1 (1/ 1 + k 2 ) and constant mean curvature k/2(1 + k 2 ) and with non-parallel mean curvature vector. Example 3.4 Let k be any positive number and let γ(s) = (g(s), h(s)) be a unit speed plane curve. Then φ(u, s) = (−ks sin u, g(s), ks cos u, h(s)) √ is a non-minimal, proper slant flat surface with slant angle k/ 1 + k 2 . Example 3.5 For any nonzero real numbers p and q, we consider the following immersion form R × (0, ∞) into C2 defined by φ(u, v) = ( pv sin u, pv cos u, v sin qu, v cos qu). Then the immersion φ gives us a complete flat slant surface in C2 . An almost complex structure J acting on E2m is called compatible if (E2m , J ) is complex analytically isometric to the complex number space Cm with the standard flat Kaehlerian metric. When m is even, let J1− denote the compatible almost complex structures on E2m defined by J1− (a1 , . . . , am , b1 , . . . , bm ) = (−a2 , a1 , . . . , −am , am−1 , b2 , −b1 , . . . , bm , −bm−1 ).

(13)

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Example 3.6 Let N be a complex surface in Cm = (E2m , J0 ), m = 2n. Then, for any real number θ ∈ (0, π2 ], N is a slant submanifold in (E4n , Jθ ) with slant angle θ, where Jθ is the compatible almost complex structure on E2m defined by Jθ = (cos θ)J0 + (sin θ)J1− .

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This example shows that there exist infinitely many proper slant minimal submanifolds in C2n = (E4n , J0 ). Let N be a submanifold of an almost Hermitian manifold M. By a complex tangent point of N in M, we mean a point p ∈ N such that the tangent space T p N of N at p is invariant under the action of the almost complex structure J of M. Definition 3.7 (cf. [7, 24, 51]) A submanifold N of an almost Hermitian manifold M is called purely real if it has no complex tangent points, i.e., the Wirtinger angle θ(X )  = 0 for any unit vector X ∈ T N . The following result of the author and Y. Tazawa shows that slant surfaces in almost Hermitian manifolds exist abundantly. Theorem 3.8 ([37]) Let φ : N → M be an embedding of an oriented surface N into an almost Hermitian manifold M endowed with an almost complex structure J and an almost Hermitian metric g. If φ is purely real, then for any prescribed angle θ ∈ (0, π2 ), there exists an almost complex structure J on M satisfying the following two conditions: (a) (M, J, g) is an almost Hermitian manifold. (b) φ is a θ-slant surface with respect to (J, g).

4 Some General Properties of Slant Surfaces A submanifold N of an almost Hermitian manifold (M, J, g) is called pointwise slant if for each given point p ∈ N and each 0  = X ∈ T p N , the Wirtinger angle θ(X ) is independent of the choice of X at p (cf. [29, 45]). For a pointwise slant submanifold N , the function θ on N is called the slant function. Obviously, slant submanifolds are pointwise slant with a constant slant function. For a surface N in an almost Hermitian manifold (M, J, g), the Wirtinger angle θ = θ(X ) at any point p ∈ N is clear independent of the choice of 0  = X ∈ T p N . Hence, one has the following. Basic Fact 4.1. Every surface in an almost Hermitian manifold is pointwise slant. A proper slant submanifold N is called Kaehlerian slant if the endomorphism P of N is parallel with respect to the Riemannian connection, i.e., ∇ P = 0, where ∇ P is defined by

Slant Surfaces in Kaehler Manifolds

(∇ X P)Y = ∇ X (PY ) − P(∇ X Y ).

25

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A Kaehlerian slant submanifold is a Kaehler manifold with respect to its induced metric and the almost complex structure given by J˜ = (sec θ)P (cf. [2, 9, 17, 18]). The main purpose of this section is to present the basic properties of slant surfaces in Kaehler manifolds. An important property of slant surfaces is the following. Theorem 4.1 ([9, p. 22]) Let N be a surface in an almost Hermitian manifold M. Then the following three statements are equivalent: (1) N is a proper slant surface. (2) N is a Kaehlerian slant surface. (3) N is neither totally real nor complex in M and ∇ P = 0, i.e., P is parallel with respect to the Levi-Civita connection of N defined by (15). Another important property of slant surfaces is the following. Theorem 4.2 ([9, p. 24]) Let N be a surface in a Kaehler manifold M. Then N is slant if and only if A FY X = A F X Y for any X, Y tangent to N , where A is the shape operator of N in M. For the operator F defined by (3), define ∇ F by (∇ X F)Y = D X (FY ) − F(∇ X Y ).

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For surfaces satisfying ∇ F = 0, we have the following. Theorem 4.3 ([9, p. 27]) Let N be a surface in a Kaehler surface M. Then ∇ F = 0 if and only if either N is a complex surface, or a totally real surface, or a proper slant minimal surface of M. Next, we discuss the problem: “When is a proper slant surface in a Kaehler surface minimal?” Theorem 4.4 ([9, p. 30]) Let N be a proper slant surface in a Kaehler surface (M, J, g). If there exists a compatible complex structure J1 such that N is totally real with respect to the Kaehler manifold (M, J1 , g), then N is minimal in M. The next two results provide very simple characterizations for a slant surface in C2 to be minimal. Theorem 4.5 ([9, p. 32]) Let N be a proper slant surface in C2 . Then N is minimal if and only if there exists a compatible almost complex structure J1 on E4 such that N is totally real in (E4 , J1 ). Theorem 4.6 ([9, p. 32]) Let N be a proper slant surface in C2 . Then N is minimal if and only if there exists a compatible almost complex structure J2 on E4 such that N is a complex surface in (E4 , J2 ).

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For complete oriented slant surfaces in C2 , the following global topological result was proved by Chen and J.-M. Morvan in [34] (see [9, p. 88]). Theorem 4.7 Let N be a complete oriented proper slant surface in C2 . If the mean curvature of N is bounded below by a positive constant, then topologically N is either a circular cylinder or a 2-plane. In 1989, Ohnita [54] classified minimal slant surfaces with constant Gaussian curvature in complex space forms. In particular, he proved that totally geodesic surfaces are the only minimal slant surfaces with constant Gauss curvature in a complex hyperbolic n-space C H n . For slant surfaces in a non-flat complex space form M 2 (4c), the author and Y. Tazawa proved the following result in [39]. Theorem 4.8 Every proper slant surface in a non-flat complex space form M 2 (4c) is always non-minimal.

5 A Link Between Gauss Map and Slant Surfaces In views of Example 3.6 and Theorems 4.4–4.6, the author and Y. Tazawa asked the following natural problem in [37]. Problem 1 Let N be a surface in E4 . If there exists a compatible complex structure J on E4 such that N is slant in (E4 , J ). How many other compatible complex structures J˜ on E4 are there such that N is slant with respect to these complex structures? In order to present the solutions of [37] to this problem, we recall the geometry of the Grassmannian G(2, 4) of oriented 2-planes in E4 as follows (cf., e.g., [33]). Let {ε1 , ε2 , ε3 , ε4 } be the canonical orthonormal basis of E4 . Then 0 = ε1 ∧ · · · ∧ ε4 gives the canonical orientation of E4 and the space of 2-vectors of E4 , ∧2 E4 , is a 6-dimensional vector space equipped with the inner product  ,  given by   X 1 ∧ X 2 , Y1 ∧ Y2  = det ( X i , Y j )

(17)

and extended bilinearly. The Grassmannian G(2, 4) can be identified with the set D1 (2, 4) consisting of all unit decomposable 2-vectors in ∧2 E4 via ϕ : G(2, 4) → D1 (2, 4) given by ϕ(V ) = X 1 ∧ X 2 for any positive orthonormal basis {X 1 , X 2 } of the V ∈ G(2, 4). The Hodge star operator ∗ : ∧2 E4 → ∧2 E4 is defined by ∗ξ, η 0 = ξ ∧ η,

Slant Surfaces in Kaehler Manifolds

27

for any 2-vectors ξ, η ∈ ∧2 E4 . Hence, if we regard an oriented 2-plane V ∈ G(2, 4) as an element in D1 (2, 4) via φ, then we have ∗V = V ⊥ , where V ⊥ denotes the oriented orthogonal complement of the oriented 2-plane V in E4 . Since ∗2 = 1 and ∗ is a self-adjoint endomorphism of ∧2 E4 , we have the following orthogonal decomposition: ∧2 E4 = ∧2+ E4 ⊕ ∧2− E4

(18)

of eigenspaces of ∗ with eigenvalues 1 and −1, respectively. Let π+ and π− denote the natural projections: π± : ∧2 E4 → ∧2± E4 , respectively. For any ξ ∈ D1 (2, 4), we have π+ (ξ) =

1 1 (ξ + ∗ξ), π− (ξ) = (ξ − ∗ξ). 2 2

(19)

Let S+2 and S−2 denote the 2-spheres in ∧2+ E 4 and ∧2− E 4 centered at the origin with radius √12 , respectively. Then we have π+ : D1 (2, 4) → S+2 , π− : D1 (2, 4) → S−2

(20)

D1 (2, 4) = S+2 × S−2 .

(21)

and

For the complex Euclidean plane C2 = (E4 , J0 ), it is well-known that J0 is an orientation preserving isomorphism. We denote by J the set of all almost complex structures (or simply called complex structures) on E4 which are compatible with the inner product  , , i.e., J ={J : E4 → E4 : J is linear, J 2 = −I, and J X, J Y  = X, Y  for any X, Y ∈ E4 }.

(22)

An orthonormal basis {e1 , e2 , e3 , e4 } on E4 is called a J -basis if J e1 = e2 and J e3 = e4 . Any two J -bases of the same J have the same orientation. Using the canonical orientation on E4 , we divide J into two disjoint subsets of J : J + = {J ∈ J : J −bases are positive}, J − = {J ∈ J : J −bases are negative}. For any J ∈ J , there exists a unique 2-vector ζ J ∈ ∧2 E4 defined by ζ J , X ∧ Y  = − J (X, Y ) =: − X, J Y  ,

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B.-Y. Chen

for X, Y ∈ E4 . In other words, ζ J is nothing but the metrical dual of − J , where  J is the Kaehler form associated with J . For any V ∈ G(2, 4) and any J ∈ J , we choose a positive orthonormal basis {e1 , e2 } of V . We put θ J (V ) = cos−1 (J e1 , e2 ).

(24)

Then α J (V ) ∈ [0, π] and θ(X ) = min {θ J (V ), π − θ J (V )}. A 2-plane V ∈ G(2, 4) is said to be θ-slant if θ J (V ) = θ, identically. If N is an oriented surface in the complex Euclidean space C2 , then N has a unique complex structure determined by its orientation and its induced metric. With respect to the angle θ J (J ∈ J ), we have N is holomorphic ⇐⇒ θ J (T N ) ≡ 0, N is anti − holomorphic ⇐⇒ θ J (T N ) ≡ π, N is totally real ⇐⇒ θ J (T N ) ≡

π . 2

The following proposition from [37] establishes fundamental relations between the slant angle and the projections π+ and π− via the Gauss map. Proposition 5.1 We have the following: (i) If J ∈ J + , then θ J (V ) is the angle between π+ (V ) and ζ J . (ii) If J ∈ J − , then θ J (V ) is the angle between π− (V ) and ζ J . By applying Proposition 5.1, the following three results were obtained in [37] by Chen and Tazawa. Theorem 5.2 We have the following: (1) Let φ : N → E4 be a minimal immersion. If there exists a compatible complex structure Jˆ ∈ J + (respectively, Jˆ ∈ J − ) such that the immersion φ is slant with respect to Jˆ, then (1a) For any θ ∈ [0, π], there is a compatible complex structure Jθ ∈ J + (respectively, Jθ ∈ J − ) such that φ is θ-slant with respect to the complex structure Jθ . (1b) The immersion φ is slant with respect to any complex structure J ∈ J + (respectively, J ∈ J − ). (2) If φ : N → E4 is a non-minimal immersion, then there exist at most two complex structures ±J + ∈ J + and at most two complex structures ±J − ∈ J − such that the immersion φ is slant with respect to them.

Slant Surfaces in Kaehler Manifolds

29

Theorem 5.3 We have the following: (a) If φ : N → C2 = (E4 , J0 ) is holomorphic, then the immersion φ is slant with respect to every complex structure J ∈ J + . (b) If φ : N → C2 = (E4 , J0 ) is anti-holomorphic, then the immersion φ is slant with respect to every complex structure J ∈ J − . Theorem 5.4 For any immersion φ : N → E4 , exactly one of the following four cases occurs: (1) (2) (3) (4)

φ is not slant with respect to every compatible complex structure on E4 . φ is slant with respect to infinitely many compatible complex structures on E4 . φ is slant with respect to exactly two compatible complex structures on E4 . φ is slant with respect to exactly four compatible complex structures on E4 .

Definition 5.5 An immersion φ : N → E4 is called doubly slant if it is slant with respect to some complex structure J + ∈ J + and at the same time it is slant with respect to another complex structure J − ∈ J − . Remark 5.6 Examples 3.1–3.5 are examples of doubly slant surfaces. For doubly slant surfaces, we have the following results from [37]. Theorem 5.7 Every non-minimal immersion φ : N → E4 which is slant with respect to more than two complex structures is doubly slant. Theorem 5.8 If φ : N → E4 is a doubly slant immersion, then both the Gauss curvature G and the normal curvature G D vanish, i.e., G = G D = 0. Example 5.9 Let φ : E3 → E4 be the map from E3 into E4 defined by φ(x0 , x1 , x2 ) = (x1 , x2 , 2x0 x1 , 2x0 x2 ). Then φ induces an immersion φˆ : S 2 → E4 from the unit 2-sphere S 2 into E4 , known ˆ as the Whitney immersion which has a unique self-intersection point at φ(−1, 0, 0) = ˆ φ(1, 0, 0). It is known that this immersion φˆ : S 2 → E4 is a totally real immersion with respect to two suitable compatible complex structures on E4 . Moreover, since the surface is non-flat, φˆ is a slant immersion with respect to only two compatible complex structures. Example 5.10 Let N be the surface in E4 defined by x(u, v) = (u, v, k cos v, k sin v). Then N is the Riemannian product of a line and a circular helix in a hyperplane E3 of E4 . Let J1 , J2 be the compatible complex structures on E4 Defined, respectively, by

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B.-Y. Chen

J1 (a, b, c, d) = (−b, a, −d, c), J2 (a, b, c, d) = (b, −a, −d, c). Then J1 ∈ J + and J2 ∈ J − . Moreover, a direct computation shows that N is slant with respect to the following four complex structures: J1 , −J1 , J2 , −J2 , with slant angles given by     1 −1 cos−1 √ , cos−1 √ 1 + k2 1 + k2     −1 1 −1 −1 cos , cos , √ √ 1 + k2 1 + k2 respectively. For slant surfaces in C2 whose rank of its Gauss map is less than 2, Chen and Tazawa proved the following classification theorem. Theorem 5.11 ([38]) If φ : N → C2 = (E 4 , J1 ) is a slant immersion such that the rank of its Gauss map is less than 2, then the image φ(N ) is a union of some flat ruled surfaces in E4 . Therefore, locally, φ(N ) is a cylinder, a cone, or a tangential developable surface in C2 . Furthermore, we have the following: (1) A cylinder in C2 is a slant surface if and only if it is of the form {c(s) + te}, where e is a fixed unit vector and c(s) is a generalized helix with axis J1 e contained in a hyperplane of E4 and with e as its hyperplane normal. (2) A cone in C2 is a slant surface if and only if, up to translations, it is of the form {tc(s)}, where (φ ◦ c)(s) is a generalized helix in S 3 (1) with axis X˜ 1 . (3) A tangential developable surface {c(s) + (t − s)c (s)} in C2 is a slant surface if and only if, up to rigid motions, (φ ◦ c )(s) is a generalized helix in S 3 (1) with axis X˜ 1 .

6 A Link Between Regular Homotopy and Slant Surfaces In this section, we present a link between slant surfaces and regular homotopy established by Chen and J.-M. Morvan in [35]. First, we give the following. Definition 6.1 A regular homotopy is a family of immersions ψt , t ∈ [0, 1], from a manifold into another such that ψt and its derivatives depend continuously on the parameter t. Immersions ψ0 and ψ1 are said to be regularly homotopic if there exists a regular homotopy connecting ψ0 to ψ1 . It is known from M. Gromov’s article [46] that every purely real immersion of a surface N in C2 is regularly homotopic to some Lagrangian immersion of N .

Slant Surfaces in Kaehler Manifolds

31

Moreover, according to M. Aubin’s article [3], all purely real immersions of a surface in C2 are regularly homotopic to each other. In particular, Aubin’s result shows that there exist no regularly homotopic invariants for purely real surfaces in C2 . For an oriented Riemannian surface N with volume form ∗1, there is a canonical endomorphism j : T N → T N defined by  j X, Y  = 2(∗1)(X, Y ), X, Y ∈ T N .

(25)

This endomorphism j is the canonical almost complex structure of the surface N . In particular, if {e1 , e2 } is a positive orthonormal frame field of N , we have je1 = e2 , je2 = −e1 . A Kaehler surface M admits a canonical symplectic structure  given by (X, Y ) = X, J Y  ,

X, Y ∈ T M.

(26)

For an oriented purely real surface N immersed in a Kaehler surface M, Chen and Morvan introduced in [35] a 1-form  on N defined by (X ) =

1 {2(H, X ) + sin θ(dθ ◦ j)(X )}, 2π sin2 θ

X ∈ T N,

(27)

where θ is the Wintinger function on N and H is the mean curvature vector field. The following result was proved in [35]. Theorem 6.2 Let N be an oriented purely real surface in a Kaehler surface M. If N is flat, then d = 0, and hence,  defines a cohomology class: [] ∈ H 1 (M; R). Remark 6.3 The cohomology class [] ∈ H 1 (M; R) via (27) reduces to the Moslov class m(L) when the purely real surface N is a Lagrangian surface L in C2 . An easy application of Theorem 6.2 is the following. Theorem 6.4 Let N be a purely real minimal surface in a flat Kaehler surface M. If N is compact, then N is a slant surface. In particular, we have  = 0. For purely real surfaces in C2 , we also have the following two results from [35]. Theorem 6.5 Let M be an oriented purely real surface in C2 . Then the cohomology class [] of M is an integral class, i.e., [] ∈ H 1 (M; Z ). Theorem 6.6 Let M be an oriented purely real surface in C2 . Then the cohomology class [] is an invariant through totally real regular homotopy. Via Theorem 6.6, we have the following link between regular homotopy and minimal proper slant surfaces. Theorem 6.7 An oriented purely real surface in C2 has trivial cohomology class [] if it is regularly homotopic to a minimal proper slant surface in C2 through totally real homotopy.

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7 Mean Curvature and Gauss Curvature of Slant Surfaces Slant surfaces in the complex Euclidean plane C2 form the most fundamental family of slant surfaces. For slant surfaces with parallel mean curvature vector in C2 , we have the following classification. Theorem 7.1 ([8]) Let N be a surface in the complex Euclidean plane C2 . Then N is a slant surface with parallel mean curvature vector if and only if it is one of the following surfaces: (1) An open portion of the product surface of two plane circles. (2) An open portion of a circular cylinder which is contained in a hyperplane of C2 . (3) A minimal slant surface in C2 . Moreover, if either case (a) or case (b) occurs, then N is totally real in the complex Euclidean plane C2 . By applying Theorem 7.1, we have the following two corollaries. Corollary 7.2 Let N be a surface in the complex Euclidean plane C2 . Then N is a slant surface in C2 with parallel second fundamental form if and only if N is one of the following surfaces: (a) An open portion of the product surface of two plane circles. (b) An open portion of a circular cylinder which is contained in a hyperplane of C2 . (c) An open portion of a plane in C2 . Moreover, if either case (a) or case (b) occurs, then N is totally real in the complex Euclidean plane C2 . Obviously, slant surfaces with parallel mean curvature vector or with parallel second fundamental form have constant mean curvature α automatically. For slant surfaces with constant mean curvature in C2 , we have the following. Theorem 7.3 Let N be a slant surface in C2 with constant mean curvature. Then N is spherical if and only if it is an open portion of the product surface of two plane circles. Theorem 7.4 Let N be a slant surface in C2 with constant mean curvature. Then N lies in a hyperplane of C2 if and only if it is either an open portion of a 2-plane or an open portion of a circular cylinder. For proper slant surfaces with constant mean curvature, we also have the next two results of J. Yang. Theorem 7.5 ([60]) A flat proper slant surface with nonzero constant mean curvature in C2 is an open portion of a helical cylinder.

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33

Theorem 7.6 ([60]) There do not exist proper slant surfaces with nonzero constant mean curvature and nonzero constant Gaussian curvature in the complex Euclidean plane C2 . The author and L. Vrancken proved the following three existence theorems for slant surfaces with either prescribed Gaussian curvature or prescribed mean curvature. Theorem 7.7 ([40]) For any given constant θ ∈ (0, π2 ] and a given function λ, there exist infinitely many θ-slant surfaces in C2 with λ as the prescribed mean curvature function. Theorem 7.8 ([40]) For any given constant θ ∈ (0, π2 ] and a given function K , there exist infinitely many θ-slant surfaces in C2 with K as the prescribed Gaussian curvature function. Theorem 7.9 ([42]) Locally, for any given real number θ ∈ (0, π2 ] and any given function K = K (x), there exist infinitely many θ-slant surfaces in the complex projective plane C P 2 (4) and in the complex hyperbolic plane C H 2 (−4) with K as the prescribed Gaussian curvature. The proof of these theorems is based on the following Existence and Uniqueness Theorems for slant submanifolds obtained in [40, 41] by Chen and Vrancken. Theorem 7.10 (Existence Theorem) Let c, θ be two constants with 0 < θ ≤ π2 and N a simply connected Riemannian n-manifold with inner product  , . Suppose there exist an endomorphism P of the tangent bundle T N and a symmetric bilinear T N -valued form α on N such that for X, Y, Z , W ∈ T M, we have P 2 = −(cos2 θ)I, P X, Y  + X, PY  = 0,

(28) (29)

(∇ X P)Y, Z  = α(X, Y ), Z  − α(X, Z ), Y  ,

(30)

R(X, Y ; Z , W ) = csc θ{α(X, W ), α(Y, Z ) − α(X, Z ), α(Y, W )} + c{X, W  Y, Z  − X, Z  Y, W  + P X, W  PY, Z  (31) − P X, Z  PY, W  + 2 X, PY  P Z , W }, 2

(∇ X α)(Y, Z ) + csc2 θ{Pα(X, α(Y, Z )) + α(X, Pα(Y, Z ))}

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+ (sin2 θ)c{X, P Z  Y + X, PY  Z } is totally symmetric. Then there exists a θ-slant isometric immersion from M into a simply connected complete Kaehler n-manifold M n (4c) with constant holomorphic curvature 4c whose second fundamental form h is given by h(X, Y ) = csc2 θ(Pα(X, Y ) − J α(X, Y )).

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Theorem 7.11 (Uniqueness Theorem) Let φ1 , φ2 : N → M n (4c) be two θ-slant (0 < θ ≤ π2 ) isometric immersions of a connected Riemannian n-manifold N into a simply connected complete Kaehler n-manifold M n (4c) with second fundamental form σ 1 and σ 2 . If c  = 0 and 

   σ 1 (X, Y ), J φ1 Z = σ 2 (X, Y ), J φ2 Z

(34)

for all vector fields X, Y, Z tangent to M, then P1 = P2 and there exists an isometry  of M n (4c) such that φ1 = (φ2 ). Remark 7.12 The corresponding Existence and Uniqueness Theorems for pointwise slant immersions of Riemannian n-manifolds into a complex space form M˜ n (c) of constant holomorphic sectional curvature c have been obtained recently in [1].

8 Spherical Slant Surfaces in C2 Let S 3 (1) denote the unit hypersphere in C2 centered at the origin. It is known that S 3 (1) is the Lie group consisting of all unit quaternions Q1 = {u = a + ib + jc + kd : |u| = 1}, which can also be regarded as a subgroup of the orthogonal group O(4) in a natural way. Let 1 denote the identity element of the Lie group S 3 (1) given by 1 = (1, 0, 0, 0) ∈ S 3 (1) ⊂ E4 .

(35)

Let us put X 1 = (0, 1, 0, 0), X 2 = (0, 0, 1, 0), X 3 = (0, 0, 0, 1) ∈ T1 S 3 (1).

(36)

Let X˜ i denote the left-invariant vector fields obtained from the extensions of X i on S 3 (1) for i ∈ {1, 2, 3}, and let φ : S 3 (1) → S 3 (1) denote the orientation-reversing isometry defined by ψ(a, b, c, d) = (a, b, d, c).

(37)

Then the left-translation L p and the right-translation R p on S 3 (1) are isometries which are analogous to the parallel translations on E3 and they are given by

Slant Surfaces in Kaehler Manifolds

35

⎛

a ⎜b T (L p q) = ⎜ ⎝c d ⎛ a ⎜b T (R p q) = ⎜ ⎝c d

−b a d −c

−c −d a b

−b a −d c

−c d a −b

⎞⎛ ⎞ −d x ⎜y⎟ c ⎟ ⎟⎜ ⎟, −b ⎠ ⎝ z ⎠ a w ⎞⎛ ⎞ −d x ⎜y⎟ −c ⎟ ⎟⎜ ⎟ b ⎠⎝z⎠ a w

(38)

(39)

for p = (a, b, c, d), q = (x, y, z, w) ∈ S 3 (1) ⊂ E4 , where T A is the transpose of A. If η denotes the unit outer normal of S 3 (1) ⊂ E4 and J1 and J1− denote the complex structures on E4 defined, respectively, by J1 (a, b, c, d) = (−b, a, −d, c), J1− (a, b, c, d) = (−b, a, d, −c),

(40) (41)

then for any q ∈ S 3 (1), we have (cf. [38]) (J1 η)(q) = Rq∗ X 1 , (J1− η)(q)

= L q∗ X 1 = X˜ 1 (q).

(42) (43)

Hence, J1 η and J1− η are right-invariant and left-invariant vector fields on S 3 (1), respectively. Further, we also have the following result from [38]. Proposition 8.1 Let W ∈ G(3, 4) and V ∈ G(2, 4) such that V ⊂ W . Then V is θ-slant with respect to a complex structure J ∈ J + (resp., J ∈ J − ) if and only if ξV , J ηW  = − cos θ

(r esp., ξV , J ηW  = cos θ),

(44)

where ξV and ηW are positive unit normal vectors of V in W and of W in E4 , respectively. Let φ : N → S 3 (1) ⊂ E4 be a spherical immersion of an oriented surface N into S (1) and let ξ be the positive unit normal of φ(N ) in S 3 (1). Since every spherical surface is non-minimal in C2 , every spherical surface in C2 cannot be a complex surface of C2 . The next result characterizes slant spherical surfaces in C2 . 3

Proposition 8.2 Let φ : N → S 3 (1) ⊂ E4 be an immersion of an oriented surface N into the hypersphere S 3 (1) of E4 . Then we have the following: (a) The immersion φ is θ-slant with respect to the complex structure J1 if and only if

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B.-Y. Chen

ξ( p), J1 η(φ( p)) = − cos θ

f or any p ∈ N .

(45)

(b) The immersion φ is α-slant with respect to the complex structure J1− if and only if ξ( p), X˜ 1 (φ( p)) = cos θ

f or any p ∈ M.

(46)

(c) The immersion φ is θ-slant with respect to the complex structure J1 if and only if the composition ψ ◦ φ is θ-slant with respect to the complex structure J1− , where ψ : S 3 (1) → S 3 (1) is defined by (37). Let g+ and g− denote the two maps from N into the unit sphere S 2 (1) in the unit T1 S 3 (1) = {X ∈ T1 S 3 (1) : |X | = 1} by g+ ( p) = (L φ( f ( p))∗ )−1 (φ∗ ξ( p)), g− ( p) = (L f ( p)∗ )−1 (ξ( p))

(47)

for p ∈ N . In fact, g+ and g− are the analogues of the classical Gauss map of a surface in E3 in which the parallel translations in E3 are replaced by the left-translations L q on S 3 (1). We also define a circle Sθ1 for θ ∈ [0, π] on the unit sphere S 2 (1) in T1 S 3 (1) by Sθ1 = {X ∈ T1 S 3 (1) : |X | = 1, X, X 1  = − cos θ}.

(48)

The next result from [38] also characterizes spherical slant surfaces in the complex Euclidean plane C2 . Proposition 8.3 Let φ : N → S 3 (1) ⊂ E4 be an immersion of an oriented surface N . Then we have the following: (1) φ is θ-slant with respect to the complex structure J1 if and only if g+ (N ) ⊂ Sθ1 ⊂ T1 S 3 (1).

(49)

(2) φ is θ-slant with respect to the complex structure J1− if and only if 1 g− (N ) ⊂ Sπ−θ ⊂ T1 S 3 (1).

(50)

The following are two examples for g+ and g− . Example 8.4 If N = S 1 × S 1 is the flat torus in E 4 defined by 1 φ(u, v) = √ (cos u, sin u, cos v, sin v), 2

(51)

then the images of N under the spherical Gauss maps g+ and g− are the great circles perpendicular to X 1 = (0, 1, 0, 0).

Slant Surfaces in Kaehler Manifolds

37

Example 8.5 If N = S 2 is the totally geodesic 2-sphere of S 3 (1) parametrized by φ(u, v) = (cos u cos v, sin u cos v, sin v, 0), then g+ (u, v) = (0, − sin v, − cos u cos v, sin u cos v), g− (u, v) = (0, sin v, sin u cos v, − cos u cos v). Hence, both g+ and g− are isometries. Now, we provide the following. Definition 8.6 Let γ(s) be a curve in S 3 (1) parametrized by arclength s and assume γ  (s) is given by γ  (s) =

3 

f i (s) X˜ i (c(s))

(52)

i=1

for some functions f 1 , f 2 , f 3 . Then we call the curve γ(s) a helix in S 3 (1) with axis vector field X˜ 1 if f 1 (s) = b, f 2 (s) = a cos(ks + s0 ), f 3 (s) = a sin(ks + s0 )

(53)

for some constants a, b, k, and s0 satisfying a 2 + b2 = 1. We call the curve γ(s) a generalized helix in S 3 (1) with axis vector field X˜ 1 if it satisfies γ  (s), X˜ 1 (γ(s)) = constant. Remark 8.7 Helices in S 3 (1) defined above are the analogues of Euclidean helices in a Euclidean 3-space E3 . Definition 8.8 An immersion φ : D → S 3 (1) of a domain D around the origin (0, 0) from a Cartesian plane R2 into S 3 (1) is called a helical cylinder in S 3 (1) if φ(s, t) = β(t) · γ(s)

(54)

for some helix γ(s) in S 3 (1) with axis X˜ 1 satisfying k = − b2 with ab < 0, and for some curve β(t) in S 3 (1) which is either a geodesic or a curve of constant torsion 1 parametrized by arclength such that (i) c(0) = γ(0), and (ii) the osculating planes of γ(s) and of β(t) coincide at t = s = 0. Note that the binormal of γ(s) is normal to φ(D) in S 3 (1). Here, we orient the curve γ in such a way that the binormal of γ(s) is the positive unit normal of φ(D). The following result of Chen and Tazawa in [38] classifies all spherical slant surfaces in C2 .

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B.-Y. Chen

Theorem 8.9 Let φ : N → S 3 (1) ⊂ C2 = (E4 , J1 ) be a spherical immersion of an oriented surface N into the complex 2-plane C2 = (E4 , J1 ). Then φ is proper slant if and only if φ(N ) is locally of the form {ψ(β(t) · γ(s))} where ψ is the isometry on S 3 (1) defined by (37) and β(t) · γ(s) is a helical cylinder in S 3 (1).

9 Slant Surfaces Lying in a Real Hyperplane of C2 In this section, we consider proper slant surfaces N in the complex Euclidean plane C2 such that N lying in a real hyperplane of C2 . For such slant surfaces, the author and Tazawa obtained the following result in [38]. Theorem 9.1 Let φ : N → C2 be a proper slant immersion of an oriented surface N into C2 . If φ(N ) is contained in a hyperplane W of E4 , then φ is a doubly slant immersion and φ(N ) is a union of some flat ruled surfaces in W . Therefore, locally, φ(N ) is a cylinder, a cone, or a tangential developable surface in W . Furthermore, we have the following: (1) A cylinder in W is a proper slant surface with respect to the complex structure J1 on E4 if and only if it is a portion of a 2-plane. (2) A cone in W is a proper slant surface with respect to the complex structure J1 on E4 if and only if it is a circular cone. (3) A tangential developable surface in W is a proper slant surface with respect to a complex structure J1 on E4 if and only if it is a tangential developable surface obtained from a generalized helix in W .

10 Classification of Flat Slant Surfaces in C2 As before, let S 3 (1) denote the unit hypersphere in C2 centered at the origin. Then S 3 (1) admits a canonical Sasakian structure with the structure vector field given by ξ = iz, z ∈ S 3 (1). The purpose of this section is to present the classification theorem of flat slant surfaces in the complex Euclidean plane C2 . In order to do so, we recall the following definition from [21]. Definition 10.1 A unit speed curve z : I → S 3 (1) ⊂ C2 defined over an open interval I is called a θ-Legendre curve if it satisfies  iz(s), z  (s) = cos θ



for some constant angle θ ∈ (0, π2 ].

(55)

Slant Surfaces in Kaehler Manifolds

39

Remark 10.2 A θ-Legendre curve with θ = π2 is also known as Legendre curve. Also, a θ-Legendre curve in S 3 (1) is also known as a generalized helix in S 3 (1) in [38]. The following result shows that θ-Legendre curves in S 3 (1) can be simply characterized by an ordinary differential equation. Proposition 10.3 ([21]) A unit speed curve z : I → S 3 (1) ⊂ C2 is a θ-Legendre curve with nonzero curvature in S 3 (1) if and only if it satisfies the second-order differential equation: z  (s) − (csc θ)λ(s)iz  (s) + {1 − (cot θ)λ(s)}z(s) = 0 for some nonzero function λ over I . Remark 10.4 Proposition 10.3 extends a result of [13] for Legendre curves in S 3 (1). Next, we provide some examples of θ-Legendre curves in S 3 (1). Example 10.5 For any given θ ∈ (0, π2 ], there exist many θ-Legendre curves with nonzero curvature in S 3 (1). For example, for a given constant λ  = 0, the map z(s) = e(iλ csc θ) 2 s

 cos

s 

λ2 + (2 − λ cot θ)2



2

s   (2 cos θ − λ csc θ) sin λ2 + (2 − λ cot θ)2 , + i 2 λ2 + (2 − λ cot θ)2

s   2 sin θ 2 2  sin λ + (2 − λ cot θ) 2 λ2 + (2 − λ cot θ)2

defines a unit speed θ-Legendre curve with nonzero curvature in S 3 (1). The next result from [21] classifies all flat slant surfaces in C2 . Theorem 10.6 We have the following. (a) Let θ ∈ (0, π2 ]. Then, for any θ-Legendre curve z : I → S 3 (1) ⊂ C2 defined on an open interval I and any function β of one variable defined I , the map 

y

φ(x, y) = z(y)x +

β(u)z  (u)du

0

defines a flat θ-slant surface in C2 . (b) Let θ ∈ (0, π2 ]. Then, for any given nonzero function ϕ = ϕ(y) of one variable defined on an open interval I containing 0, the map   φ(x, y) = x + i cos θ 0

y

dt , sin θ ϕ(t)



y 0

ei(csc θ)t dt ϕ(t)



40

B.-Y. Chen

defines a flat θ-slant surface in C2 . (c) If μ(x)  = 0 and k(y) are two functions of one variable  x defined on some open intervals containing 0 such that ψ = k(y) + cot θ 0 dμx  = 0 for some θ ∈ (0, π2 ], then the map   φ(x, y) = i

y

k(y)eiy csc θ dy + eiy csc θ cos θ

0

 sin θ 0

x

ix csc θ

e dx μ(x)



 0

x

dx , μ(x)

defines a flat θ-slant surface in C2 . (d) If θ ∈ (0, π2 ) and u(x) and v(y) are two functions of one variable defined on some open intervals containing 0 with v(0)  = 0 and v  (y)  = 0, then 

x

φ(x, y) = 0

u(x)ei x csc θ d x − iv(y)ei x csc θ sin θ + iv(0) sin θ,   y  iy csc θ v (y)e dy sin θ tan θ 0

defines a flat θ-slant surface in C2 . (e) Suppose f is a nonzero function of one variable defined on an open interval containing 0 such that f  = cot θ for some θ ∈ (0, π2 ] and suppose ρ = ρ(x, y) with ∂ρ/∂ y  = 0 is a solution of the wave equation: ρx y −

  f  (x − y) ρ y + (cot θ) f (x − y) − f 2 (x − y) ρ = 0 f (x − y)

(56)

and K is a C2 -valued solution of the ordinary differential equation: 

 f  (u) K (u) + i csc θ− K  (u) + f (u)( f (u)−cot θ)K (u) = 0 f (u)−cot θ (57) 

satisfying |K |2 = 1 and |K  |2 = ( f − cot θ)2 , then   x ρ y K  (x − y) ix csc θ ix csc θ − (ρK (x − y)) y e d x dy φ(x, y) = e f (cot θ− f ) 0 0  x ρK (x − y)eix csc θ d x, f = f (x − y) + 

y



0

defines a flat θ-slant surface in C2 .

(58)

Slant Surfaces in Kaehler Manifolds

41

Conversely, locally every flat slant surface in C2 is either an open part of a slant plane or, up to rigid motions of C2 , a surface given by one of the five classes of slant surfaces given above. Remark 10.7 For any nonzero function f of one variable and any θ ∈ (0, π2 ], the wave equation (56) admits infinitely many solutions. For example, every linear combination of  x−y  f (t)dt − x cot θ , ρ1 = sin 0  x−y  ρ2 = cos f (t)dt − x cot θ , 0  x−y  ρ3 = f (u) cos(u cot θ)du ρ1 0  x−y    x−y  u + f (u) sin(u cot θ)du − 2 f (t)dt − x cot θ du ρ2 , 0 0 0  x−y  ρ4 = f (u) cos(u cot θ)du ρ2 0  x−y    x−y  u − f (u) sin(u cot θ)du − 2 f (t)dt − x cot θ du ρ1 0

0

0

is a solution of the wave equation (56). Concerning the second order differential equation (56), we have the following existence theorem from [21]. Theorem 10.8 For any given nonzero function f of one variable defined on an open interval I and for any θ ∈ (0, π2 ] such that f = cot θ, there exists a C2 -valued solution K = K (u) of the second-order ordinary differential equation:  K  (u)+ i csc θ −

f  (u) f (u) − cot θ



K  (u) + f (u) ( f (u) − cot θ) K (u) = 0, (59)

that also satisfies the two conditions: |K |2 = 1, |K  |2 = ( f − cot θ)2 .

(60)

Remark 10.9 The conditions |K | = 1 and |K  | = ( f − cot θ)2 in statement (e) of Theorem 10.6 are necessary. For instance, although

u u u i K (u) = e− 2 u csc θ cos + i(csc θ + cos θ cot θ) sin , cos θ sin 2 2 2 is a C2 -valued solution of (57) associated with f = 21 cot θ, the map (58) with ρ = 1 e 2 (x−y) cot θ does not define a flat θ-slant surface in C2 .

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The next theorem shows the existence of ample θ-Legendre curves in S 3 (1) which implies that class (a) of flat slant surfaces in C2 is quite large too. Theorem 10.10 For any θ ∈ (0, π2 ] and any nonzero function λ(s) defined on an open interval I , there is a unit speed θ-Legendre curve in S 3 (1) whose curvature in S 3 (1) is given by κ = λ csc θ. Remark 10.11 For visualization of some of the flat slant surfaces in C2 given in Theorem 10.6, see Y. Tazawa’s articles [61–63].

11 Slumbilical Surfaces in Cn A non-minimal submanifold N of a Riemannian manifold is called pseudo-umbilical if its shape operator A H with respect to the mean curvature vector H is proportional to the identity map I , i.e., A H = μI

(61)

for some function μ on N (cf. [6]). It is known from [20] that the shape operator of every Kaehlerian slant submanifold in a Kaehler manifold satisfies another condition: A F X Y = A FY X

(62)

for any vectors X, Y tangent to N . Based on these facts, the author introduced the notion of slumbilical submanifolds in [22] as those slant submanifolds which satisfy both Condition (61) and Condition (62). Note that Theorem 4.2 implies that Condition (62) holds automatically for every slant surface in a Kaehler manifold. From a shape operator point of view, slumbilical submanifolds of a Kaehler manifold are the simplest slant submanifolds besides totally geodesic ones. In [22], the author classified all slumbilical submanifolds in complex space forms. In particular, for slumbilical surfaces in complex Euclidean spaces, the author proved the following classification theorem. Theorem 11.1 Let φ : N → Cm be a θ-slant immersion of a surface N into Cn with slant angle θ ∈ (0, π2 ]. Then the immersion is slumbilical if and only if one of the following five cases occurs: (1) N is an open portion of the Euclidean plane E2 and N is immersed as an open portion of a slant plane in Cm . (2) N is an open portion of E2 equipped with the flat metric   g = e−2y cot θ d x 2 + (ax + b)2 dy 2

Slant Surfaces in Kaehler Manifolds

43

for some real numbers a, b with a  = 0. And, up to rigid motions of Cm , the immersion is given by  −1 

 (ax + b)1+ia csc θ −y cot θ 1 + a2 y e cos a + i csc θ 

  a sin θ + i

  a cos θ sin 1 + a2 y , √ sin 1 + a 2 y , 0, . . . , 0 . + i√ 1 + a2 1 + a2

φ(x, y) =

(3) N is an open portion of E2 with the flat metric   g = e−2y cot θ d x 2 + b2 dy 2 for some positive number b. Moreover, up to rigid motions of Cm , the immersion is given by φ(x, y) = b sin θeib

−1

x csc θ−y cot θ



 cos y, sin y, 0, . . . , 0 .

(4) θ = π2 and N is an open portion of the warped product of a line and an open interval I with the warped metric g = dx2 +

(ax + b)2 2 dy 1 + a2

for some real numbers a, b with a  = 0. Moreover, up to rigid motions of Cm , the immersion is given by −1

φ(x, y) =

(ax + b)1+ia (cos y, sin y, 0, . . . , 0). a+i

(5) θ = π2 and N is an open portion Of the direct product R × I of a line and an open interval I . Moreover, up to rigid motions of Cm , the immersion is given by φ(x, y) = beib

−1

x

(cos y, sin y, 0, . . . , 0).

12 A Basic Inequality for Slant Surfaces Let N be a proper θ-slant surface N in a Kaehler surface M. If e1 is a unit vector field tangent to N , we may choose a canonical orthonormal frame field {e1 , e2 , e3 , e4 } such that e2 = (sec θ)Pe1 , e3 = (csc θ)Fe1 , e4 = (csc θ)Fe2 . We simply call such an orthonormal frame field an adapted orthonormal frame for the slant surface.

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B.-Y. Chen

For slant surfaces in a complex space form, the author proved in [15] the following result. Theorem 12.1 Let N be a proper θ-slant surface in a complex space form M 2 (4c) with constant holomorphic sectional curvature c. Then the squared mean curvature H 2 and the Gauss curvature G of N satisfy H 2 ( p) ≥ 2G( p) − 2(1 + 3 cos2 θ)c,

(63)

at each point p ∈ N . The equality sign of (63) holds at a point p ∈ N if and only if, with respect to a suitable adapted orthonormal basis {e1 , e2 , e3 , e4 } at p, the shape operators of N at p take the following form: 

   3λ 0 0λ , Ae4 = . Ae3 = 0 λ λ0

(64)

The next proposition from [15] shows that, for each θ ∈ (0, π2 ), the inequality (63) is sharp for θ-slant surfaces at some point in C2 . Proposition 12.2 For each θ ∈ (0, π2 ), there exists a non-totally geodesic θ-slant surface N in C2 which satisfies the equality sign of (63) at some points in M. For slant surfaces in complex space form M 2 (4c) with c  = 0, we also have the following two results from [15]. Theorem 12.3 For any θ ∈ (0, π2 ), there exists a non-minimal special θ-slant surface with constant Gauss curvature G = −4 cos2 θ < 0 in the complex hyperbolic plane C H 2 (−4). Theorem 12.4 Let N be a proper slant surface in complex space form M 2 (4c) which satisfies the equality sign of (63) identically, then either (1) N is a totally geodesic slant surface in flat Kaehler surface (c = 0) or (2) c < 0, N has constant Gauss curvature G = 23 c, and N is a slant surface with   slant angle θ = cos−1 13 . The following corollary is an immediate consequence of Theorem 12.4. Corollary 12.5 There are no proper slant surfaces in C P 2 (4) which satisfy the equality sign of (63) identically. Remark 12.6 Theorem 12.4 shows that inequality (63) is also sharp for c < 0. Also, Theorem 12.4 implies that Proposition 12.2 fails globally.

Slant Surfaces in Kaehler Manifolds

45

13 Special Slant Surfaces in Complex Space Forms In view of (64), the author introduced the following definition in [14, 23]. Definition 13.1 A slant surface N in a Kaehler surface is called special slant if, with respect to some suitable adapted orthonormal frame {e1 , e2 , e3 , e4 }, the shape operators of N take the following special form: 

   bλ 0 0λ , Ae4 = , b ∈ R. Ae3 = 0 λ λ0

(65)

For special slant surfaces, the author proved the following. Proposition 13.2 ([15]) Every proper slant minimal surface N in a Kaehler surface is special slant which satisfies (65) with b = −1. The next theorem completely classifies special proper slant surfaces in the complex Euclidean plane C2 . Theorem 13.3 ([15]) A proper slant surface M in the complex Euclidean plane C2 is special slant if and only if it is a slant minimal surface. Furthermore, the author proved in [14] the following results for special slant surfaces in a complex hyperbolic plane. Theorem 13.4 For each b ∈ (2, 5), there exists a special slant immersion φb : M

4(2−b) 3(b−1)



→ C H 2 (−4)

of a simply connected surface with constant Gauss curvature G = C H 2 (−4) with slant angle θ = cos

−1

4(2−b) 3(b−1)

< 0 into

   1 (5 − b)(b − 2) 3 b−1

whose shape operator satisfies     5−b bλ 0 0λ Ae3 = , Ae4 = , λ= 0 λ λ0 3(b − 1) 

with respect to some adapted orthonormal frame field. Put R2 = {(x, y) : x, y ∈ R} and let U be a simply connected domain of the Cartesian 2-plane R2 . Let E = E(x, y) a positive function on U satisfying the following conditions:

46

B.-Y. Chen

∂ ∂x



1 ∂ E ∂x



1 ∂E E ∂y

 = 0,

∂E  = 0. ∂y

(66)

For a given θ ∈ (0, π2 ), we put G=

sec θ E y , 2 E

Ey =

∂E . ∂y

(67)

Let us denote by N (θ, E) the Riemannian surface (U, g) equipped with the metric tensor g = E 2 d x 2 + G 2 dy 2 ,

(68)

where E and G are given as above. There exist many functions E = E(x, y) on U which satisfy condition (66). For example, the product of any positive function f = f (x) and any non-constant positive function  = (y) on U satisfies condition (66) automatically, and (x + y)−2 is a non-product function satisfying (66). The author proved the following theorems in [14]. Theorem 13.5 For each N (θ, E) with θ ∈ (0, π2 ), there exists a special θ-slant immersion ψθ,E : N (θ, E) → C H 2 (−4) whose shape operators take the forms:  Ae3 =

   −2 sin θ 0 0 − sin θ , Ae4 = 0 − sin θ − sin θ 0

with respect to the adapted orthonormal frame field given by e1 = 1 ∂ . G ∂y

1 ∂ E ∂x

and e2 =

Theorem 13.6 Let φ : N → M 2 (4c) be a proper slant surface in a complete simply connected complex space form M 2 (4c) satisfying the equality sign of (61) identically, then either (a) c = 0 and N is a totally geodesic slant surface in C2 , or (b) c < 0 and N is a surface of constant Gauss curvature G = 23 c. Moreover, up to rigid motions of C H 2 (4c), φ is given by the immersion φ3 : N ( 23 c) → C H 2 (4c)   with slant angle θ = cos−1 13 . Theorem 13.7 Let φ : N → M 2 (4c) be a non-minimal slant surface in a complete simply connected complex space form with slant angle θ ∈ (0, π2 ). If φ is special slant satisfying (65) with b = 2, then c < 0. Moreover, if c = −1, then N = N (θ, E) for some function E satisfying (66). Furthermore, up to rigid motions, φ is given by ψθ,E : N (θ, E) → C H 2 (−4) (cf. Theorem 13.5). Theorem 13.8 Let φ : N → M 2 (4c) be a non-minimal proper θ-slant surface with constant mean curvature. If φ is special slant, i.e., it satisfies (66) for some constant b, then c < 0. Moreover, if c = −1, then we have either

Slant Surfaces in Kaehler Manifolds

47

(1) b ∈ (2, 5) and N is a surface of constant curvature G = rigid motions, φ is given by the immersion  φb : N

4(2 − b) 3(b − 1)

4(2−b) . 3(b−1)

Moreover, up to

 → C H 2 (−4)

  , or with slant angle θ = cos−1 13 (5−b)(b−2) b−1 (2) b = 2 and N = N (θ, E) for some function E satisfying (66). Moreover, up to rigid motions, φ is given by ψθ,E : N (θ, E) → C H 2 (−4).

14 A Link Between Slant and Contact Slant Submanifolds Now, we present a general construction procedure of slant submanifolds in C H m (−4) via Hopf’s fibration π : H12m+1 (−1) → C H m (−4) described in [39] . denote the complex number Let Cm+1 1  (m + 1)-space equipped with the pseudoEuclidean metric g0 = −dz 0 d z¯ 0 + mj=1 dz j d z¯ j . Put   H12m+1 (−1) = z = (z 0 , z 1 , . . . , z m ) : z, z = −1 , where  ,  denotes the inner product on Cm+1 induced from g0 . Then we have a 1 C∗ -action on H12m+1 (−1) defined by z  → λz. At a point z ∈ H12m+1 , iz is tangent to t = iz t which lies in the the flow of the action. The orbit is given by z t = eit z with dz dt negative-definite plane spanned by z and iz. The quotient space H12m+1 (−1)/ ∼ is the complex hyperbolic space C H m (−4). The almost complex structure J on C H m (−4) via the totally geodesic is induced from the canonical complex structure J on Cm+1 1 fibration: π : H12m+1 (−1) → C H m (−4). Let φ : N → C H m (−4) be an isometric immersion. Then the pre-image Nˆ = −1 π (N ) is a principal circle bundle over N with totally geodesic fibers and the lift φˆ : Nˆ → H12m+1 (−1) of φ is an isometric immersion such that the diagram: ˆ

φ Nˆ −−−−→ H12m+1 (−1) ⏐ ⏐ ⏐ ⏐π π  φ

N −−−−→ C H m (−4) commutes. Conversely, if ψ : Nˆ → H12m+1 (−1) is an isometric immersion which is invariant under the action of C∗ = C \ {0}, then there exists a unique isometric immersion ψπ : π( Nˆ ) → C H m (−4), projection of ψ, such that the associated diagram commutes.

48

B.-Y. Chen

On H12m+1 (−1) ⊂ Cm+1 , let us consider the almost contact metric structure 1 onto the tangent bundle of H12m+1 (−1) obtained from the projection of J of Cm+1 1 and with the structure vector field given by ξ = V = J z. Analogous to the definition of slant submanifolds in almost Hermitian manifold, we have the following definition of contact slant submanifolds in almost contact metric manifolds (cf. [39]). Definition 14.1 Let (C 2m+1 , g, ϕ, ξ) be a (2m + 1)-dimensional almost contact metric manifold with Riemannian (or a pseudo-Riemannian) metric g, the almost contact (1, 1)-tensor ϕ, and the structure vector field ξ. An immersion ψ : N → C 2m+1 of a manifold N into C 2m+1 is called contact θ-slant if it satisfies (i) the structure vector field ξ of C 2m+1 is tangent to ψ∗ (T N ) and (ii) for each nonzero vector X tangent to ψ∗ (T p N ) and perpendicular to ξ, the angle θ(X ) between ϕ(X ) and ψ∗ (T p N ) is independent of the choice of X . The following result provides a simple link between slant submanifolds of C H m (−4) and contact slant submanifolds in H12m+1 (−1). Theorem 14.2 The isometric immersion φ : N → C H m (−4) is θ-slant if and only if φˆ : Mˆ → H12m+1 is contact θ-slant. Let us denote by ∇ˆ and ∇˜ the Levi-Civita connections of H12m+1 (−1) and C H m (−4), Q ∗ the horizontal lift of Q, and by h and hˆ the second fundamental ˆ respectively. Then h and hˆ are related by forms of φ and φ, ˆ ∗ , V ) = (F X )∗ , h(V, ˆ ˆ ∗ , Y ∗ ) = (h(X, Y ))∗ , h(X V) = 0 h(X

(69)

for vector fields X, Y tangent to M, where F X is the normal component of J X in C H m (−4). If follows from Theorem 14.2 that, in order to get the explicit expression of a desired θ-slant submanifold in C H m (−4), it is sufficient to construct a contact θslant submanifold in H12m+1 (−1) ⊂ Cm+1 whose second fundamental form satisfies π∗ hˆ = h, and vice versa. Let N be a θ-slant submanifold of C H m (−4) and let ι : H12m+1 (−1) → Cm+1 1 be the standard inclusion map. We denote by ∇˘ the Levi-Civita connection of Cm+1 . 1 Then we have (cf. [32]) ∇˘ X ∗ Y ∗ = (∇ X Y )∗ + (h(X, Y ))∗ + J X, Y  iz + X, Y  z, ∇˘ X ∗ V = ∇˘ V X ∗ = (J X )∗ ,

(70)

∇˘ V V = −z,

(72)

for X, Y tangent to N .

(71)

Slant Surfaces in Kaehler Manifolds

49

In principle, we obtain the representation of a desired θ-slant submanifold of C H m (−4) by solving the system (70)–(72) of partial differential equations. The general construction procedure goes as follows: First, we need to determine both the intrinsic and extrinsic structures of the θ-slant submanifold N in order to obtain the precise form of the differential system (70)– (72). Next, we need to construct a coordinate system on the associated contact θ-slant submanifold Nˆ = π −1 (N ). After that, we shall solve the differential system via the coordinate system on Nˆ to obtain the solution of the partial differential equations ˆ of the system gives rise to the explicit expression of the system. The solution, say φ, associated contact θ-slant submanifold Nˆ in H12m+1 (−1) which in turn provides the representation of the desired θ-slant submanifold via the Hopf fibration π.

15 Special Slant Surfaces in Complex Hyperbolic Plane By applying Theorem 14.2 and the construction method given in Sect. 14, Chen and Tazawa obtained the next result for special slant surfaces in C H 2 (−4) whose shape operator satisfies (65) with b  = 2. Theorem 15.1 ([39]) For each given a ∈ (0, 1), let ψa : R3 → C31 denote the map of R3 into C31 defined by √  1−a 2 (1 + e−av ) a 2 e−av u 2 cosh av−1 ψa (u, v, t) = eit 1 + + , − iu √ a2 2(4 − 3a 2 ) 4 − 3a 2 √ 1−a 2 e−av ((4−3a 2 )(eav −1)(a 2 −1+eav )+a 4 u 2 ) 2−2a 2 +(2−a 2 )e−av u +i , 4 − 3a 2 a 2 (4 − 3a 2 )3/2 √  (4−3a 2 )(2−2a 2 + (2a 2 −3)e−av +eav ) + a 4 e−av u 2 a 1−a 2 (1−e−av ) u +i . 4−3a 2 2a(4 − 3a 2 )3/2 Then we have the following: (1) ψa defines an immersion of R3 into the anti-de Sitter space time H15 (−1). → H15 (−1) (2) With respect to the induced metric on R3 , ψa : R3  is a contact θ-slant  2 1−a isometric immersion with slant angle θ = cos−1 a 4−3a . 2 (3) The image ψ(R3 ) in H15 (−1) is invariant under the action of C∗ . (4) The projection φb : π(R3 ) → C H 2 (−4) of ψa : R3 → H15 (−1) via the Hopf fibration π : H15 (−1) → C H 2 (−4) is a special θ-slant isometric immersion then θ given by (2), and whose shape operator satisfies

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   bλ 0 0λ , Ae4 = , 0 λ λ0  8 − 3a 2 b= ∈ (2, 5), λ = 1 − a2, 4 − 3a 2 Ae3 =

with respect to an adapted orthonormal frame field {e1 , e2 , e3 , e4 }. (5) φb : π(R3 ) → C H 2 (−4) has nonzero constant mean curvature and constant Gauss curvature. Conversely, up to rigid motions, every proper slant surface with constant mean curvature in C H 2 (−4) is obtained in the above way if the shape operator satisfies (63) with b  = 2. For special slant surfaces in C H 2 (−4) whose shape operator satisfies (63) with b = 2, we have the next two results from [39]. Theorem 15.2 We have the following: (1) For each given θ ∈ (0, π2 ), the map

ψ(u, v, t) = eit sec2 θ cosh(v cos θ) − tan2 θ + iuev cos θ , uev cos θ + i(ev cos θ + tan2 θ − sec2 θ cosh(v cos θ)),  i(ev cos θ − 1) tan θ

(73)

defines a contact θ-slant immersion into the anti-de Sitter space time H15 (−1). (2) The projection of the contact θ-slant immersion given by (73) via the Hopf fibration π is a special θ-slant immersion which satisfies (65) with b = 2. Theorem 15.3 Up to rigid motions of C H 2 (−4), for each θ ∈ (0, π2 ), there exist more than one special θ-slant isometric immersions of a surface of constant negative Gauss curvature G = −4 cos2 θ into C H 2 (−4) whose shape operators satisfy (65) with b = 2.

16 The Slant Surface in C H 2 (−4) Satisfying the Basic Equality In this section, we discuss the equality case of the basic inequality (61) for proper θ-slant surfaces in C H 2 (−4). Also, by using the construction method described in Sect. 13, Chen and Tazawa obtained the following theorem. Theorem 16.1 ([39]) If N is a proper θ-slant surface in C H 2 (−4), then the squared mean curvature H 2 and the Gauss curvature G of N satisfy

Slant Surfaces in Kaehler Manifolds

H 2 ≥ 2G + 2(1 + 3 cos2 θ)

51

(74)

with the equality sign holding identically if and only if, up to rigid motions, N is given by the projection φ3 of the immersion ψ : R3 → C31 defined by 

i√ 1 3 6u(1 + e−av ) − , cosh av + 16 u 2 e−av − 2 6 2     √ 1 1 1 1 1 (1 + 2e−av )u + i 6 eav + e−av + u2 − , 3 4 12 18 3 √    √ 1 1 av 2 5 1 2 −av −av (1 − e )u + i 3 + e +e u − , 6 6 4 18 12

ψ(u, v, t) = e

with a =



2 3

it

(75)

via the hyperbolic Hopf fibration π : H15 (−1) → C H 2 (−4).

In [49], K. Kenmotsu and D. Zhou studied surfaces with parallel mean curvature vector in a complex space form M 2 (4c) with c  = 0. In particular, they proved the following. Theorem 16.2 Let ψ : N → C H 2 (−4) be a surface with parallel mean curvature vector in C H 2 (−4). If N has constant mean curvature α = √23 and constant Gauss curvature, then N is proper slant. Moreover, it is locally congruent to Chen’s surface described by (75) in Theorem 15.1.

17 Complex Extensors and H-Umbilical Submanifolds The notion of H-umbilical Lagrangian submanifolds in a Kaehler manifold was introduced by the author in [12, 19] as follows. Definition 17.1 A non-totally geodesic Lagrangian submanifold of dimension n in a Kaehler manifold is said to be Lagrangian H -umbilical if its second fundamental form σ takes the following simple form: σ(e1 , e1 ) = λJ e1 , σ(e2 , e2 ) = · · · = σ(en , en ) = μJ e1 , σ(e1 , e j ) = μJ e j , σ(e j , ek ) = 0, 2 ≤ j  = k ≤ n,

(76)

for some suitable functions λ and μ with respect to some suitable orthonormal local frame field and some orthonormal frame field {e1 , . . . , en } on the submanifold. It is direct to verify that every non-minimal Lagrangian H -umbilical submanifold N satisfies the following two conditions: (a) The vector J H is an eigenvector of the shape operator A H and

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(b) the restriction of A H to (J H )⊥ is proportional to the identity map, where (J Hx )⊥ denotes the orthogonal complement to J Hx in Tx N at each given point x ∈ N . It is easy to verify that Lagrangian H-umbilical submanifolds are the simplest Lagrangian submanifolds satisfying both conditions (a) and (b). Thus, Lagrangian H -umbilical submanifolds can be considered as the “simplest” Lagrangian submanifolds next to the totally geodesic ones. Next, we mention the following Existence Theorem for Lagrangian H-umbilical submanifold in Cn . ˜ Theorem 17.2 ([12]) For any given function λ˜ = λ(s) and an integer n ≥ 2, there n ˜ exists a Lagrangian H-umbilical submanifold of C which satisfies (76) with λ = λ. The simplest examples of Lagrangian H-umbilical submanifolds in a complex Euclidean space are obtained from the notion of complex extensors introduced in [12] which is defined as follows. Definition 17.3 Let G : N n−1 → Em denote an isometric immersion of a Riemannian (n − 1)-manifold into Euclidean m-space Em and let F : I → C be a unit speed curve in the complex plane. Then we can extend the immersion G : N n−1 → Em to an immersion of I × N n−1 into a complex Euclidean m-space Cm given by φ = F ⊗ G : I × N n−1 → C ⊗ Em = Cm , where F ⊗ G denotes the tensor product immersion of F and G defined by (F ⊗ G)(s, p) = F(s) ⊗ G( p), s ∈ I, p ∈ N n−1 . We simply call such an extension F ⊗ G of the immersion G a complex extensor of G (or of submanifold N n−1 ) via the unit speed curve F : I → C. The following theorem provides a simple link between complex extensors and Lagrangian H-umbilical submanifolds. Theorem 17.4 ([12]) Let ι : S n−1 → En be the inclusion of the unit hypersphere of Em (centered at the origin). Then every complex extensor of ι via a unit speed curve F : I → C is a Lagrangian H-umbilical submanifold of Cn unless F(s) = (s + a)c for some real number a and some unit complex number c. In this case, F is an open part of a line. Now, we provide some examples of complex extensors (see [12]). Example 17.5 (Whitney’s sphere). Let w : S n → Cn be the map defined by w(y0 , y1 , . . . , yn ) =

1 + i y0 (y1 , . . . , yn ), 1 + y02

y02 + y12 + · · · + yn2 = 1.

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53

Then w is a Lagrangian immersion of the n-sphere into Cn which is called the Whitney n-sphere. The Whitney n-sphere is a complex extensor φ = F ⊗ ι of ι : S n−1 → En via F, where F = F(s) is an arclength reparametrization of the curve f : I → C given by sin ϕ + i sin ϕ cos ϕ f (ϕ) = . 1 + cos2 ϕ Whitney’s n-sphere is a Lagrangian H-umbilical submanifold that satisfies (76) with λ = 3μ. In fact, up to homothetic transformations, Whitney’s n-sphere is the only Lagrangian H-umbilical submanifold in Cn with λ = 3μ. Example 17.6 (Lagrangian pseudo-spheres) For a real number b > 0, let F : R → C be the unit speed curve given by F(s) =

e2bsi + 1 . 2bi

With respect to the induced metric, the complex extensor φ = F ⊗ ι of the unit hypersphere of En via F : R → C is a Lagrangian isometric immersion of an open portion of an n-sphere S n (b2 ) of sectional curvature b2 into Cn which is simply called a Lagrangian pseudo-sphere. A Lagrangian pseudo-sphere is a Lagrangian H-umbilical submanifold satisfying (1.1) with λ = 2μ. Conversely, Lagrangian pseudo-spheres are the only Lagrangian H-umbilical submanifolds of Cn which satisfy (76) with λ = 2μ. Example 17.7 For a nonzero real number a, let  F(s) =

s

e−ia ln t dt,

s f (t)dt denotes an anti-derivative of f (s). Then the complex extensor of where the unit hypersphere of En via F : R → C is a Lagrangian H-umbilical submanifold of Cn satisfying (76) with λ = μ. A Lagrangian H-umbilical submanifold with λ = μ is simply called a Lagrangianumbilical submanifold. / R. Then the Example 17.8 Let a ∈ C and θ be a real number such that ae−iθ ∈ complex extensor of the unit hypersphere via F(s) = a + eiθ s is a Lagrangian Humbilical submanifold satisfying (76) with λ = 0. The following theorem provides a simple characterization of Lagrangian pseudospheres. Theorem 17.9 ([12]) Let N be a Lagrangian submanifold of Cn with n ≥ 3. Then, up to rigid motions of Cn , N is a Lagrangian pseudo-sphere if and only if N is a Lagrangian H-umbilical submanifold with nonzero constant sectional curvature.

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18 Classification of Lagrangian H-Umbilical Submanifolds In this section, we present the following classification theorems for Lagrangian Humbilical submanifolds. First, we have the following theorems. Theorem 18.1 Let n ≥ 3 and let L : N → Cn be a Lagrangian H-umbilical isometric immersion. Then we have the following: (i) If N is of constant sectional curvature, then either N is flat or, up to rigid motions of Cn , L is a Lagrangian pseudo-sphere. (ii) If N contains no open subset of constant sectional curvature, then, up to rigid motions of Cn , L is a complex extensor of the unit hypersphere of En . For Lagrangian H-umbilical surfaces in the complex Euclidean plane C2 , we have the following. Theorem 18.2 ([12]) For a Lagrangian H-umbilical surface N of C2 , we have the following: (i) If N is a minimal Lagrangian surface of C2 without totally geodesic points, then N is a Lagrangian H-umbilical surface of C2 . (ii) Let L : M → C2 be a Lagrangian H-umbilical surface satisfying σ(e1 , e1 ) = λJ e1 , σ(e1 , e2 ) = μJ e2 , σ(e2 , e2 ) = μJ e1 such that the integral curves of e1 are geodesics in N . Then we have the following: (ii.1) If N is of constant sectional curvature, then either N is flat or, up to rigid motions of C2 , L is a Lagrangian pseudo-sphere. (ii.2) If N contains no open subset of constant sectional curvature, then, up to rigid motions of C2 , L is a complex extensor of the unit circle of E2 . For flat Lagrangian H-umbilical submanifolds in Cn , we have the following. Theorem 18.3 ([12]) (a) Let N be a simply connected open portion of the twisted product manifold f R × En−1 with twisted product metric g= f

2

d x12

+

n 

d x 2j ,

(77)

α j (x1 )x j

(78)

j=2

where the twisted function is of the form: f = β(x1 ) +

n  j=2

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55

for some functions β, α2 , . . . , αn of x1 . Then, up to rigid motions of Cn , there is a unique Lagrangian isometric immersion L f : N → Cn without totally geodesic points whose second fundamental form satisfies σ(e1 , e1 ) = λJ e1 , σ(e1 , e j ) = σ(e j , ek ) = 0, 2 ≤ j, k ≤ n, where λ = f −1 , e1 = λ

∂ ∂ ∂ , e2 = , . . . , en = . ∂x1 ∂x2 ∂xn

(b) If n ≥ 3 and L : N → Cn is a Lagrangian H-umbilical isometric immersion of a flat manifold into Cn without totally geodesic points, then M is an open portion of a twisted product manifold f R × En−1 with twisted product metric given by (77) and twisted function f given by (78) for some functions β, α2 , . . . , αn . Moreover, up to rigid motions of C2 , L is given by the unique Lagrangian immersion L f given in Statement (a). (c) If L : N → C2 is a Lagrangian H-umbilical isometric immersion of a flat surface into C2 without totally geodesic points, then one of the following two cases occurs. (c1) N is an open portion of a twisted product surface f R × E1 with twisted product metric given by (77) and twisted function f given by (78) for some functions β, α2 . Moreover, up to rigid motions of Cn , L is given by the unique Lagrangian immersion L f mentioned in Statement (a). (c2) L is the complex extensor φ = F ⊗ G of a circle of radius, say r , in E2 via F, where F is the unit speed curve in C given by e(i+b)s , b ∈ R. F(s) = √ 1 + b2 Remark 18.4 For explicit description of flat Lagrangian H-umbilical submanifolds in Cn , see [16]. Also, for the classification of Lagrangian H-umbilical submanifolds in Cn , see [13].

19 A Basic Inequality for Kaehlerian Slant Submanifolds The notion of Lagrangian H-umbilical submanifolds in Cn was extended to proper slant submanifolds in [20] as follows. Definition 19.1 An n-dimensional proper slant submanifold N in a Kaehler manifold M˜ is called H -umbilical if its second fundamental form σ takes the following simple form:

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B.-Y. Chen

σ(e1 , e1 ) = λe1∗ , σ(e2 , e2 ) = · · · = σ(en , en ) = μe1∗ , σ(e1 , e j ) = μe j ∗ , σ(e j , ek ) = 0, j  = k, 2 ≤ j, k ≤ n

(79)

for some suitable functions λ and μ with respect to some suitable orthonormal local frame field e1 , . . . , en , where e1∗ , . . . , en ∗ are unit vectors in the directions of Fe1 , . . . , Fen , respectively. An H -umbilical slant submanifold with μ = 0 is simply called cylindrical slant. Remark 19.2 All H -umbilical slant submanifolds of a Kaehlerian manifold are Kaehlerian slant. And H -umbilical slant submanifolds of a Kaehlerian manifold are the simplest slant submanifolds next to totally geodesic ones (if they exist). The next result classifies n-dimensional H-umbilical proper slant submanifolds in complex space form M˜ n (4c). Theorem 19.3 ([20]) Let N be an n-dimensional H-umbilical proper slant submanifold of a complete simply connected complex space form M˜ n (4c) with n > 2. Then one of the following three statements must hold: (1) N is flat and it is immersed as an open part of a slant n-plane in the complex Euclidean n-space Cn . (2) N is flat and is is immersed as a cylindrical slant submanifold in Cn . (3) c < 0, N is a slant space form of constant slant sectional curvature 4c cos2 θ, and N is immersed as an H-umbilical submanifold in √ the complex hyperbolic n-space C H n (4c) satisfying (76) with λ = 2μ = ±2 −c sin θ. For Kaehlerian proper slant submanifolds, we have the following result from [20] which extends Theorem 12.1 for slant surfaces. Theorem 19.4 Let φ : N → M˜ n (4c), c ∈ {−1, 0, 1}, be a Kaehlerian θ-slant submanifold of dimension n into a complete simply connected complex space form M˜ n (4c). Then we have H2 ≥

  2(n + 2) n+2 3 cos2 θ τ − 1 + c, n 2 (n − 1) n n−1

(80)

where H 2 is the squared mean curvature and τ is the scalar curvature of N . Furthermore, the equality sign of (80) holds identically if and only if either (1) θ = 0 and N is a totally geodesic complex submanifold of M˜ n (4c), or n (2) c = 0 and N is a totally geodesic   θ-slant submanifold in C , or (3) c = −1, n = 2, θ = cos−1 13 , and N is a surface of constant curvature − 23 . Moreover, up to rigid motions, φ = π ◦ z where z : R3 → H15 (−4) ⊂ C31 is the immersion

Slant Surfaces in Kaehler Manifolds

57

√ z(u, v, t) = eit − 21 + 23 cosh av + 16 u 2 e−av − 6i 6u(1 + e−av ), √  1 1 av  −av −av 1 1 1 2 , (1 + 2e )u + i 6 − + e + e ( + u 3 3 4 12 18  √ √  5   2 1 2 , (1 − e−av )u + i 3 16 + 41 eav + e−av − 12 + 18 u 6 where a = C H 2 (−4).

√ 2/3 and π denotes the hyperbolic Hopf fibration π : H15 →

20 An Open Problem and Three Conjectures on Slant Surfaces In view of the results on the mean curvature of slant surfaces given in this chapter, the following problem seems to be quite interesting. Problem 2 Classify slant surfaces with constant mean curvature in complex space forms M n (4c). In particular, classify slant surfaces with constant mean curvature in C2 . A submanifold M of a Euclidean m-space Em is called biharmonic if its position vector field x satisfies 2 x = 0,

(81)

i.e., each Euclidean coordinate function of M is a biharmonic function on (M, g), where g is the induced metric on the submanifold (cf. [10, 26, 27, 30, 31, 56]). It follows from Hopf’s lemma that every biharmonic submanifold in a Euclidean space is non-compact. Further, it was known from [43, 44] that biharmonic submanifolds of a Euclidean space are of infinite type (in the sense of [27]) unless they are minimal. The study of biharmonic submanifolds was initiated by the author in the middle of the 1980s in his program of understanding submanifolds of finite type. Independently, biharmonic submanifolds as biharmonic maps were also investigated by G.-Y. Jiang in [47] for his study of Euler–Lagrange’s equation of bienergy functional. It was shown independently by the author and Jiang in [47] that there are no biharmonic surfaces in E3 except the minimal ones. This non-existence result was generalized by I. Dimitric in his doctoral thesis [43] at Michigan State University (see also [44]). Later, biharmonic submanifolds have been investigated by many mathematicians (see the book [56] for the most recent comprehensive survey). Obvious, minimal submanifolds of Em are trivially biharmonic. On the other hand, the following Chen’s biharmonic conjecture is well-known. Conjecture 1 ([10]) Minimal submanifolds are the only biharmonic submanifolds in Euclidean spaces.

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The study of biharmonic submanifolds is nowadays a very active research subject (cf., e.g., [26, 27, 56]). In particular, since 2000, biharmonic submanifolds have been receiving a growing attention and have become a popular research subject of study with many important progresses. For slant submanifolds, Conjecture 1 became Conjecture 2 Minimal submanifolds are the only biharmonic slant submanifolds in complex Euclidean spaces. In particular, we have the following. Conjecture 3 Minimal surfaces are the only biharmonic slant surfaces in complex Euclidean spaces.

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47. Jiang, G.-Y.: 2-harmonic isometric immersions between Riemannian manifolds. Chin. Ann. Math. Ser. A 7, 130–144 (1986) 48. Kaehler, K.: Über eine bemerkenswerte Hermitische Metrik. Abh. Math. Sem. Univ. Hamburg 9, 173–186 (1933) 49. Kenmotsu, K., Zhou, D.: The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms. Amer. J. Math. 122, 295–317 (2000) 50. Matsumoto, K., Mihai, I., Tazawa, Y.: Ricci tensor of slant submanifolds in complex space forms. Kodai Math. J. 26, 85–94 (2003) 51. Mihai, I.: Slant submanifolds in complex space forms. In: Topics in Differential Geometry, pp. 157–182. Ed. Acad. Române, Bucharest (2008) 52. Ogiue, K.: Differential geometry of Kaehler submanifolds. Adv. Math. 13, 73–114 (1974) 53. Ogiue, K.: Recent topics in submanifold theory. Sugaku 39, 305–319 (1987) 54. Ohnita, Y.: Minimal surfaces with constant curvature and Kaehler angle in complex space forms. Tsukuba J. Math. 13, 191–207 (1989) 55. Oiaga, A., Mihai, I., Chen, B.Y.: Inequalities for slant submanifolds in complex space forms. Demonstratio Math. 32, 835–846 (1999) 56. Ou, Y.-L., Chen, B.-Y.: Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry. World Scientific Publ, Hackensack (2020) 57. Sahin, B.: Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedicata 117, 195–202 (2006) 58. Schouten, J.A., van Dantzig, D.: Über unitäre Geometrie. Math. Ann. 103, 319–346 (1930) 59. Schouten, J.A., van Dantzig, D.: Über unitäre Geometrie konstanter Krümmung. Proc. Kon. Nederl. Akad. Amsterdam 34, 1293–1314 (1931) 60. Yang, J.: On slant surfaces with constant mean curvature in C 2 . J. Geom. 59, 184–201 (1997) 61. Tazawa, Y.: Construction of slant immersions. Bull. Inst. Math. Acad. Sinica 22, 153–166 (1994) 62. Tazawa, Y.: Construction of slant immersions II. Bull. Belg. Math. Soc. Simon Stevin 1, 569– 576 (1994) 63. Tazawa, Y.: Visualization of flat slant surfaces in C2 . Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Bilbao, 2000), pp. 433–436, Contemporary Mathematics, vol. 288. American Mathematical Society, Providence (2001) 64. Weil, A.: Sur la théorie des formes différentielles attachété analytique complexe. Comm. Math. Helv. 20, 110–116 (1947)

Slant Geometry of Warped Products in Kaehler and Nearly Kaehler Manifolds Bang-Yen Chen and Siraj Uddin

2000 AMS Mathematics Subject Classification 53C40 · 53C42 · 53C50

1 Introduction ˜ g, J ) endowed with a Among submanifolds of an almost Hermitian manifold ( M, Riemannian metric g and a compatible almost complex structure J , a holomorphic submanifold N of M˜ is characterized by J (T p N ) ⊆ T p N , for all p ∈ N .

(1)

In other words, N is a holomorphic submanifold of M˜ if and only if, for any nonzero vector X tangent to N at any point p ∈ N , the Wintinger angle between J X and the tangent plane T p N is equal to zero. Besides holomorphic (or complex) submanifolds, there exists another important class of submanifolds, called totally real submanifolds. A totally real submanifold N of an almost Hermitian manifold M˜ or, in particular, of a Kaehler manifold is a submanifold such that the almost complex structure J of the ambient manifold M˜ carries each tangent vector of N into the corresponding normal space of N in M˜ (cf., e.g., [57]), that is, (2) J (T p N ) ⊆ T p⊥ N , for all p ∈ N . B.-Y. Chen (B) Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, USA e-mail: [email protected] S. Uddin Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_3

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˜ the totally real submanifold N in M˜ is also known as a When dimR N = dimC M, Lagrangian submanifold. (For a detailed survey on Lagrangian submanifolds from Riemannian point of view, we refer to [30, 42].) In 1990, the first author introduced in [23, 24] another important class of sub˜ called slant submanifolds. A slant manifolds of an almost Hermitian manifold M, submanifold is defined as a submanifold of M˜ such that, for any nonzero vector X ∈ T p N , the Wintinger angle θ(X ) between J X and the tangent space T p N is a constant θ ∈ [0, π2 ] (see [106]). Clearly, holomorphic submanifolds and totally real submanifolds are special cases of slant submanifolds with θ = 0 and θ = π2 , respectively. For general results on slant surfaces and submanifolds we refer to [23, 24, 59–61, 64]. A warped product N1 × f N2 of two Riemannian manifolds (N1 , g1 ) and (N2 , g2 ) is the product manifold N1 × N2 equipped with the warped product metric g = g N1 + f 2 g N2 , where f : N1 → R+ is a positive smooth function on N1 (see, e.g., [12, 43, 46, 47, 85]). The function f is called the warping function. The well-known embedding theorem of Nash [84] states that every Riemannian manifold can be isometrically embedded in a Euclidean space with sufficiently high codimension. In particular, Nash’s theorem implies that every warped product manifold N1 × f N2 can be isometrically embedded as a Riemannian submanifold in a Euclidean space. In view of Nash’s embedding theorem, the first author proposed at the beginning of this century to study the following two fundamental questions (see [25, 26, 31, 32, 43]). Question A. What can we conclude from an arbitrary isometric immersion of a warped product manifold into a Euclidean space or, more generally, into an arbitrary Riemannian manifold ? Question B. What can we conclude from an isometric immersion of a warped product manifold into an arbitrary complex space form or, more generally, into an arbitrary Kaehler or nearly Kaehler manifold ? The main purpose of this chapter is to provide a detailed survey on these two fundamental questions from a complex slant geometric point of view. More precisely, we present in this chapter some important results on CR-products, CR-warped products, bi-slant warped products, hemi-slant warped products, semi-slant warped products, and CR-slant warped products in Kaehler and nearly Kaehler manifolds. In the last two sections, we present some related results on slant submanifolds in generalized complex space forms and in quaternionic space forms, respectively.

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2 Preliminaries Let N be an n-dimensional manifold isometrically immersed into a Riemannian ˜ We denote by ∇ and ∇˜ the Levi-Civita connections of N and M, ˜ manifold M. respectively. The Gauss and Weingarten formulas are given, respectively, by (see, for instance, [16, 43]) ∇˜ X Y = ∇ X Y + σ(X, Y ), ∇˜ X ξ = Aξ X + D X ξ

(3) (4)

for the tangent vector fields X, Y and normal vector field ξ of N , where σ denotes the second fundamental form, D the normal connection, and A the shape operator of N . The mean curvature vector H of N in M˜ is defined by H=

1 trace σ. n

(5)

The squared mean curvature is defined by H 2 = H, H  . The submanifold N is called minimal if its mean curvature vector H vanishes identically. And N is called totally geodesic if its second fundamental form σ vanishes identically. Further, N is called totally umbilical if its second fundamental form satisfies σ(X, Y ) = g(X, Y )H for X, Y tangent to N . ˜ let R and R˜ denote the Riemann curvature tensors of For a submanifold N in M, ˜ respectively. Then the equation of Gauss is given by N and M, ˜ R(X, Y ; Z , W ) = R(X, Y ; Z , W ) + σ(X, W ), σ(Y, Z ) − σ(X, Z ), σ(Y, W )

(6)

for vectors X, Y, Z , W tangent to N . For a submanifold N of a Riemannian m-manifold R m (c) of constant curvature c, the equation of Gauss becomes R(X , Y ; Z , W ) = c{X, W  Y, Z  − X, Z  Y, W } + σ(X, W ), σ(Y, Z ) − σ(X, Z ), σ(Y, W ) .

(7)

For a smooth function f defined on a Riemannian n-manifold M, the gradient of f , denoted by ∇ f , is the vector field which is dual to the differential d f . In other words, ∇ f is given by

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∇ f, X  = d f (X ) = X ( f ) for all X ∈ T M.

(8)

The Hessian of f , denoted by H ess f , is the second covariant differential ∇(∇ f ), so that H ess f (X, Y ) = X Y f − (∇ X Y ) f

(9)

for vector field X, Y tangent to M. For a given orthonormal frame fields e1 , . . . , en on M, the Laplacian  f of f is defined by n  {(∇e j e j ) f − e j e j f }. (10) f = j=1

Also, the scalar curvature τ of N at p ∈ N is defined to be τ ( p) =



K (ei ∧ e j ),

(11)

i< j

where K (ei ∧ e j ) denotes the sectional curvature of the plane section spanned by ei , e j . Similarly, one can define the scalar curvature τ (L) of N restricted to a linear subspace L ⊂ T p N with dim L ≥ 2 for some p ∈ N .

3 CR-Products in Complex Space Forms Let us denote a complex projective n-space with constant holomorphic sectional curvature c by C P n (c). Denote by (z 0 , . . . , z h ) and (w0 , . . . , w p ) the homogeneous coordinates of C P h (4) and C P p (4), respectively. Define a map: Shp : C P h (4) × C P p (4) → C P h+ p+hp (4), which carries each point ((z 0 , . . . , z h ), (w0 , . . . , w p )) in C P h (4) × C P p (4) to the point (z i w j )0≤i≤h,0≤ j≤ p in C P h+ p+hp (4). Then it is well-known that this map Shp is a Kaehlerian embedding, called the Segre embedding. The Segre embedding was introduced in 1891 by Corrado Segre (1863–1924) in [92]. A CR-submanifold of a Kaehler manifold M˜ is called a CR-product if it is locally a Riemannian product N T × N⊥ of a holomorphic submanifold N T and a totally real submanifold N⊥ . For CR-products in complex space forms, we have the following. Theorem 3.1 (a) A CR-submanifold in a complex Euclidean m-space is a CRproduct if and only if it is a direct sum of a holomorphic submanifold and a

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totally real submanifold of linear complex subspaces of the complex Euclidean m-space [18]. (b) There do not exist CR-products in complex hyperbolic spaces other than holomorphic submanifolds and totally real submanifolds [19, 68]. For a CR-product N T × N⊥ with dimC N T = h and dimR N⊥ = p in a complex projective space C P h+ p+hp (4) with constant holomorphic sectional curvature 4, we have the following. p

Theorem 3.2 ([18]) Let N Th × N⊥ be the CR-product in C P m (4). Then m ≥ h + p + hp.

(12)

The equality sign of (12) holds if and only if (a) N Th is a totally geodesic holomorphic submanifold, p (b) N⊥ is a totally real submanifold, and (c) the immersion is given by Shp

p

→ C P h (4) × C P p (4) −−−−−−−−→ C P h+ p+hp (4). N Th × N⊥ − Segreembedding

p

Theorem 3.3 Let N Th × N⊥ be the CR-product in C P m (4). Then the squared norm of the second fundamental form satisfies ||σ||2 ≥ 4hp.

(13)

The equality sign of (13) holds if and only if (a) N Th is a totally geodesic holomorphic submanifold, p (b) N⊥ is a totally geodesic totally real submanifold, and (c) the immersion is given by p totallygeodesic

N Th × N⊥ −−−−−−−→ C P h (4) × C P p (4) Shp

−−−−−−−−→ C P h+ p+hp (4) ⊂ C P m (4). Segreembedding

Remark 3.4 For further results on the Segre embedding from complex differential geometric point of view, see [35, 55, 56].

4 Generic Products in Complex Space Forms ˜ g, J ) is called a generic A real submanifold N of an almost Hermitian manifold ( M, submanifold if the maximal complex subspace, Tx N ∩ J (Tx N ), of Tx N is of constant

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dimension along N (see [21]). Generic submanifolds are “generic” in the sense that every real submanifold of an almost Hermitian manifold is the closure of the union of some generic submanifolds of the almost Hermitian manifold. ˜ g, J ). For a given Let N be a submanifold of an almost Hermitian manifold ( M, point x ∈ N , if we put Hx = Tx N ∩ J (Tx N ), then Hx is the maximal complex subspace of the tangent space Tx M˜ which is contained in Tx N . A generic submanifold is called proper if Hx  = {0} and Hx  = Tx N for any x ∈ N . ˜ J ), it is For any generic submanifold N of an almost complex manifold ( M, known that the distribution H is a differentiable distribution (see, [47, Lemma 8.1]). ˜ let H⊥ For a generic submanifold N in an almost Hermitian manifold M, x denote the ⊥ orthogonal subspace of Hx in N . The distribution H on N is called the purely real distribution of the generic submanifold N . Definition 4.1 ([21]) A submanifold N of an almost Hermitian manifold M˜ is called a generic product if locally it is the Riemannian product N0 × N T of a holomorphic ˜ submanifold N T and a purely real submanifold N0 of M. Definition 4.2 ([21]) A generic product N = N T × N0 in a C P m (4) is called a standard generic product if the following two conditions are satisfied: (a) N0 lies in a totally geodesic holomorphic submanifold C P h+ p+hp (4) of C P m (4); (b) N T is embedded in C P m (4) as a totally geodesic holomorphic submanifold, where h is the complex dimension of N T and p is the real dimension of N0 . For generic submanifolds, we put h = dimC Hx , p = dimR H⊥ x , x ∈ N. Example 4.3 (cf. [21]) Consider the Segre embedding Shp : C P h (4) × C P p (4) → C P h+ p+hp (4). If N0 is a p-dimensional purely real submanifold of C P p (4), then the product C P h (4) × N0 is a generic product in C P h+ p+hp (4) via Segre embedding Shp in which C P p (4) is embedded in C P h+ p+hp (4) as a totally geodesic, holomorphic submanifold. For generic submanifolds of a complex projective space, the first author proved the following two results in [21] which can be regarded as an extension of Theorems 3.2 and 3.3, respectively. Theorem 4.4 Let N = N T × N0 be a generic product in the complex projective m-space C P m (4). Then we have the following: (1) m ≥ h + p + hp;

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(2) Every generic product N in C P m (4c) with m = h + p + hp is a standard generic product. Theorem 4.5 Let N be a generic product in the complex projective m-space C P m (4). Then the squared norm of the second fundamental form σ satisfies ||σ||2 ≥ 4hp.

(14)

The equality sign of (14) holds identically if and only if N is locally the Riemannian product of a totally geodesic holomorphic submanifold C P h (4) and a totally geodesic totally real submanifold R P p (1) of C P m (4). By applying Theorem 4.4, we have the following result also from [21]. p

Theorem 4.6 Let N = N1h × N2 be the Riemannian product of two Kaehler manip folds with dimC N1h = h and dimC N2 = p. Then (1) N admits no Kaehler immersion into C P m (4) with m < h + p + hp; (2) If N admits a Kaehler immersion into C P h+ p+hp (4), then p

(2.1) N1h and N2 are open submanifolds of C P h (4) and C P p (4), respectively; (2.2) the Kaehler immersion ψ is given by the Segre embedding Shp .

5 Warped Products in Real Space Forms The following theorem provides a solution to Question A. Theorem 5.1 ([31]) For any isometric immersion φ : N1 × f N2 → R m (c) of a warped product N1 × f N2 into a Riemannian manifold of constant curvature c, we have n2 2 f H + n 1 c, (15) ≤ f 4n 2 where n i = dim Ni , n = n 1 + n 2 , H 2 is the squared mean curvature of φ, and  f is the Laplacian of f on N1 . The equality sign of (15) holds identically if and only if ι : N1 × f N2 → R m (c) is a mixed totally geodesic immersion satisfying trace h 1 = trace h 2 , where trace h 1 and trace h 2 denote the trace of σ restricted to N1 and N2 , respectively. By applying the method using in [41], we also have the next solution to Question A from [39, 66]. Theorem 5.2 If M˜ m (c) is a Riemannian manifold with sectional curvatures bounded from above by a constant c, then for any isometric immersion φ : N1 × f N2 → M˜ m (c) from a warped product N1 × f N2 into M˜ m (c), the warping function f satisfies

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f n2 2 H + n 1 c, ≤ f 4n 2 where n 1 = dim N1 and n 2 = dim N2 . Some easy consequences of Theorem 5.2 are the following. Corollary 5.3 There do not exist minimal immersions of a Riemannian product N1 × N2 of two Riemannian manifolds into a negatively curved Riemannian manifold M. Corollary 5.4 Let N1 × f N2 be a warped product of two Riemannian manifolds whose warping function f is a harmonic function. Then (1) N1 × f N2 admits no isometric minimal immersion into any Riemannian manifold of negative curvature. (2) Every isometric minimal immersion from N1 × f N2 into a Euclidean space is a warped product immersion. Corollary 5.5 Let f be an eigenfunction of Laplacian  on N1 with eigenvalue λ > 0. Then every Riemannian warped product N1 × f N2 does not admit any isometric minimal immersion into any Riemannian manifold of non-positive curvature. Remark 5.6 Theorems 5.1 and 5.2 were extended to multiply warped product submanifolds in [50, 51, 66]; see also [67]. Another general solution to Question A is the following result. Theorem 5.7 ([38]) For any isometric immersion φ : N1 × f N2 → R m (c), the scalar curvature τ of the warped product N1 × f N2 satisfies τ≤

n 2 (n − 2) 2 1 f + H + (n + 1)(n − 2)c. n1 f 2(n − 1) 2

(16)

If n = 2, the equality case of (16) holds automatically. If n ≥ 3, the equality sign of (16) holds identically if and only if either (i) N1 × f N2 is of constant curvature c, the warping function f is an eigenfunction with eigenvalue c, that is,  f = c f , and N1 × f N2 is immersed as a totally geodesic submanifold in R m (c), or (ii) locally, N1 × f N2 is immersed as a rotational hypersurface in a totally geodesic submanifold R n+1 (c) of R m (c) with a geodesic of R n+1 (c) as its profile curve. Remark 5.8 For further results in this direction, see [9, 41].

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6 Warped Products in Complex Space Forms For a general warped product N1 × f N2 with dim N1 = n 1 and dim N2 = n 2 in the complex projective m-space C P m (4c), we have the following general result from [37]. Theorem 6.1 Let φ : N1 × f N2 → C P m (4c) be an arbitrary isometric immersion of a warped product into the complex projective m-space C P m (4c) of constant holomorphic sectional curvature 4c. Then we have (n 1 + n 2 )2 2 f H + (3 + n 1 )c. ≤ f 4n 2

(17)

The equality sign of (17) holds identically if and only if we have the following: (1) n 1 = n 2 = 1, (2) f is an eigenfunction of the Laplacian of N1 with eigenvalue 4c, and (3) φ is totally geodesic and holomorphic. For warped products in a complex hyperbolic space C H m (4c), we have the following. Theorem 6.2 ([34]) Let φ : N1 × f N2 → C H m (4c) be an arbitrary isometric immersion of a warped product N1 × f N2 into the complex hyperbolic m-space C H m (4c) of constant holomorphic sectional curvature 4c. Then we have (n 1 + n 2 )2 2 f H + n 1 c. ≤ f 4n 2

(18)

The equality sign of (18) holds if and only if the following conditions hold. (a) φ is mixed totally geodesic, (b) trace h 1 = trace h 2 , and (c) J D1 ⊥ D2 , where J is the almost complex structure of C H m (4c). Some consequences of Theorem 6.2 are the following [34] (see also [33]). Corollary 6.3 Let N1 × f N2 be a warped product whose warping function f is harmonic. Then N1 × f N2 does not admit an isometric minimal immersion into any complex hyperbolic space. Corollary 6.4 If f is an eigenfunction of Laplacian on N1 with eigenvalue λ > 0, then N1 × f N2 does not admit an isometric minimal immersion into any complex hyperbolic space. Corollary 6.5 If N1 is compact, then every warped product N1 × f N2 does not admit an isometric minimal immersion into any complex hyperbolic space.

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7 CR-Warped Products in Kaehler Manifolds The definition of CR-submanifolds was introduced by A. Bejancu. Definition 7.1 ([11]) A submanifold N of a Kaehler manifold M˜ is called a CRsubmanifold if there exists a holomorphic distribution D on N whose orthogonal ⊥ complement D⊥ is a totally real distribution, that is, J D⊥ x ⊂ Tx N . Around the beginning of this century, the first author initiated the study of warped product CR-submanifolds of Kaehler manifolds. More precisely, he proved in [28] the following. Theorem 7.2 There do not exist warped product CR-submanifolds of the form N⊥ × f N T in any Kaehler manifold other than CR-products. Based on this theorem, the first author introduced in 2001 the notion of CR-warped products as the warped product CR-submanifolds of the form: N T × f N⊥ (see [28]). A CR-warped product N T × f N⊥ is said to be trivial if its warping function f is constant. A trivial CR-warped product N T × f N⊥ is nothing but a CR-product f f N T × N⊥ , where N⊥ denotes the manifold with metric f 2 g N⊥ which is homothetic to the original metric g N⊥ on N⊥ . A fundamental theorem on CR-warped products in Kaehler manifolds is the following. Theorem 7.3 ([28, 48]) Let N = N T × f N⊥ be a CR-warped product submanifold ˜ Then the second fundamental form σ satisfies in an arbitrary Kaehler manifold M. ||σ||2 ≥ 2 p |∇(ln f )|2 ,

(19)

where ∇(ln f ) is the gradient of ln f on N T and p = dim N⊥ . (1) If the equality sign of (19) holds identically, then N T is a totally geodesic holomorphic submanifold and N⊥ is a totally umbilical totally real submanifold of ˜ ˜ Moreover, N T × f N⊥ is minimal in M. M. ⊥ (2) When N is anti-holomorphic, that is, when J D⊥ x = Tx N , and p > 1. The equality sign of (19) holds identically if and only if N⊥ is a totally umbilical subman˜ ifold of M. (3) If N is anti-holomorphic and p = 1, then the equality sign of (19) holds identically if the characteristic vector field J ξ of N is a principal vector field with zero as its principal curvature. Conversely, if the equality sign of (19) holds, then the characteristic vector field J ξ of N is a principal vector field with zero as its principal curvature only if N = N T × f N ⊥ is a trivial CR-warped product immersed in M˜ as a totally geodesic hypersurface. (4) Also, when N is anti-holomorphic with p = 1, the equality sign of (19) holds ˜ identically if and only if N is a minimal hypersurface in M.

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Remark 7.4 CR-warped products in complex space forms satisfying the equality case of (19) have been completely classified in [28, 29]. Remark 7.5 For further results on warped product submanifolds in Kaehler manifolds, see also [45, 49, 52, 54].

8 CR-Twisted Products in Kaehler Manifolds Definition 8.1 ([17, 20]) A twisted product N1 ×λ N2 of two Riemannian manifolds (N1 , g1 ) and (N2 , g2 ) is the product manifold N1 × N2 equipped with the warped product metric g = g N 1 + λ2 g N 2 , where λ is a positive smooth function defined on N1 × N2 . The function λ is called the twisting function (see [17, p. 31] or [20, p. 66]). When the twisting function λ depends only on N1 , then the twisted product N1 ×λ N2 reduces to a warped product. The following three results on CR-twisted products were obtained by the first author in [27]. Theorem 8.2 If N = N⊥ ×λ N T is a CR-twisted product in a Kaehler manifold M˜ such that N⊥ is a totally real submanifold and N T is a holomorphic submanifold of ˜ then N is a CR-product. M, Theorem 8.3 Let N = N T ×λ N⊥ be a CR-twisted product in a Kaehler manifold M˜ such that N⊥ is a totally real submanifold and N T is a holomorphic submanifold ˜ Then we have the following: of M. (1) The second fundamental form σ of M in M˜ satisfies ||σ||2 ≥ 2 p |∇ T (ln λ)|2 , where ∇ T (ln λ) is the N T -component of the gradient of ln λ and p is the dimension of N ⊥ . (2) If ||σ||2 = 2 p |∇ T ln λ|2 holds identically, then N T is a totally geodesic sub˜ manifold and N⊥ is a totally umbilical submanifold of M. (3) If N is anti-holomorphic in M˜ and dim N⊥ > 1, then we have ||σ||2 = 2 p |∇ T ln λ|2 identically if and only if N T is a totally geodesic submanifold ˜ and N⊥ is a totally umbilical submanifold of M. Theorem 8.4 Let N = N T ×λ N⊥ be a CR-twisted product in a Kaehler manifold M˜ such that N⊥ is a totally real submanifold and N T is a holomorphic submanifold ˜ If N is mixed totally geodesic, then we have the following: of M.

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(1) The twisting function λ is a function on N⊥ . (2) N T ×λ N⊥ is a CR-product, where N⊥λ denotes the manifold N⊥ equipped with the metric g λN⊥ = λ2 g N⊥ . A doubly twisted product N1 × f N2 of two Riemannian manifolds (N1 , g1 ) and (N2 , g2 ) is the product manifold N1 × N2 equipped with the warped product metric g = λ21 g N1 + λ22 g N2 , where λ1 and λ2 are positive smooth functions defined on N1 × N2 . Sahin ¸ [87] proved the following non-existence result for doubly CR-twisted products. Theorem 8.5 There are no doubly CR-twisted products in a Kaehler manifold M˜ which are not (singly) CR-twisted product in the form λ1 N T ×λ2 N⊥ such that N T is ˜ a holomorphic submanifold and N⊥ is a totally real submanifold of M.

9 CR-Warped Products in Complex Space Forms The first author proved in [36] that CR-warped products in a complex space form also satisfy the following general optimal inequality. p

Theorem 9.1 Let N = N Th × f N⊥ be a CR-warped product in a complex space form M˜ m (4c) of constant holomorphic sectional curvature 4c. Then   ||σ||2 ≥ 2 p |∇(ln f )|2 + (ln f ) + 2hc .

(20)

If the equality sign of (20) holds identically, then N T is a totally geodesic submanifold and N⊥ is a totally umbilical submanifold. Moreover, N is a minimal submanifold in M˜ m (4c). The following three theorems completely classify CR-warped products in complex space forms which satisfy the equality case of (20) identically. p

Theorem 9.2 ([36]) Let φ : N Th × f N⊥ → Cm be a CR-warped product in Cm . Then we have   (21) ||σ||2 ≥ 2 p |∇(ln f )|2 + (ln f ) . The equality case of inequality (21) holds identically if and only if the following four statements hold. (1) N T is an open portion of C∗h := Ch − {0}; (2) N⊥ is an open portion of S p ; (3) There is α, 1 ≤α ≤ h, and complex Euclidean coordinates {z 1 , . . . , z h } on Ch α such that f = j=1 z j z¯ j ;

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(4) Up to rigid motions, the immersion φ is given by   φ = w0 z 1 , . . . , w p z 1 , . . . , w0 z α , . . . , w p z α , z α+1 , . . . , z h , 0, . . . , 0 for z = (z 1 , . . . , z h ) ∈ C∗h and w = (w0 , . . . , w p ) ∈ S p (1) ⊂ E p+1 . Theorem 9.3 ([36]) Let φ : N T × f N⊥ → C P m (4) be a CR-warped product with dimC N T = h and dimR N⊥ = p. Then we have   ||σ||2 ≥ 2 p |∇(ln f )|2 + (ln f ) + 2h .

(22)

The CR-warped product satisfies the equality case of inequality (22) identically if and only if the following three statements hold. (a) N T is an open portion of complex projective h-space C P h (4); (b) N⊥ is an open portion of unit p-sphere S p (1); (c) There exists a natural number α ≤ h such that, up to rigid motions, φ is the ˘ where composition π ◦ φ,   ˘ w) = w0 z 0 , . . . , w p z 0 , . . . , w0 z α , . . . , w p z α , z α+1 , . . . , z h , 0 . . . , 0 φ(z, for z = (z 0 , . . . , z h ) ∈ C∗h+1 and w = (w0 , . . . , w p ) ∈ S p (1) ⊂ E p+1 , where π → C Pm. is the projection π : Cm+1 ∗ Remark 9.4 For the classification of CR-warped product φ : N T × f N⊥ in C H m (−4), see [36, 44].

10 CR-Warped Products with Compact Holomorphic Factor When the holomorphic factor N T of a CR-warped product N T × f N⊥ is compact, we have the following additional sharp results from [40]. Theorem 10.1 Let N T × f N⊥ be a CR-warped product in the complex projective m-space C P m (4) of constant holomorphic sectional curvature 4. If N T is compact, then m ≥ h + p + hp. Theorem 10.2 If N T × f N⊥ is a CR-warped product in C P h+ p+hp (4) with compact N T , then N T is holomorphically isometric to C P h . Theorem 10.3 For any CR-warped product N T × f N⊥ in C P m (4) with compact N T and any q ∈ N⊥ , we have N T ×{q}

||σ||2 d VT ≥ 4hp vol(N T ),

(23)

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where ||σ|| is the norm of the second fundamental form, d VT is the volume element of N T , and vol(N T ) is the volume of N T . The equality sign of (23) holds identically if and only if we have the following: (a) The warping function f is constant. (b) (N T , gT ) is holomorphically isometric to C P h (4) and it is isometrically immersed in C P m as a totally geodesic holomorphic submanifold. (c) (N⊥ , f 2 g⊥ ) is isometric to an open portion of the real projective p-space R P p (1) of constant sectional curvature one and it is isometrically immersed in C P m as a totally geodesic totally real submanifold. (d) N T × f N⊥ is immersed linearly fully in a complex subspace C P h+ p+hp (4) of C P m (4), and moreover, the immersion is rigid. Theorem 10.4 Let N T × f N⊥ be a CR-warped product with compact N T in C P m (4). If the warping function f is a non-constant function, then for each q ∈ N⊥ we have ||σ||2 d VT ≥ 2 pλ1 (ln f )2 d VT + 4hp vol(N T ), (24) N T ×{q}

NT

where λ1 is the first positive eigenvalue of the Laplacian  of N T . Moreover, the equality sign of (24) holds identically if and only if we have (i)  ln f = λ1 ln f . (ii) The CR-warped product is N T -totally geodesic and N⊥ -totally geodesic. The following example shows that the assumption of compactness in the theorems given above cannot be removed. Example 10.5 ([40]) Let C∗ = C − {0} and Cm+1 = Cm+1 − {0}. Denote by ∗ . Consider the action of {z 0 , . . . , z h } a natural complex coordinate system on Cm+1 ∗ given by C∗ on Cm+1 ∗ λ · (z 0 , . . . , z m ) = (λz 0 , . . . , λz m ) for λ ∈ C∗ . Let π(z) be the equivalent class containing z under this action. Then the set of equivalent classes is the complex projective m-space C P m (4) with the . complex structure induced from the complex structure on Cm+1 ∗ ˘ For any two natural numbers h and p, we define a map: φ : C∗h+1 × S p (1) → h+ p+1 by C∗   ˘ 0 , . . . , z h ; w 0 , . . . , w p ) = w 0 z 0 , w1 z 0 , . . . , w p z 0 , z 1 , . . . , z h φ(z p for (z 0 , . . . , z h ) in C∗h+1 and (w0 , . . . , w p ) in S p with j=0 w 2j = 1. Since the image of φ˘ is invariant under the action of C∗ , the composition

Slant Geometry of Warped Products in Kaehler … φ˘

75 π

π ◦ φ˘ : C∗h+1 × S p − → C∗h+ p+1 − → C P h+ p (4) induces a CR-immersion of the product manifold N T × S p into C P h+ p (4), where   N T = (z 0 , . . . , z h ) ∈ C P h : z 0  = 0 is a proper open subset of C P h (4). Clearly, the induced metric on N T × S p is a warped product metric and the holomorphic factor N T is non-compact. Note that the complex dimension of the ambient space is h + p; far less than h + p + hp.

11 Bi-slant Warped Products in Kaehler Manifolds The notion of bi-slant immersions introduced by A. Carriazo provides a natural extension of the notion of slant immersions. Definition 11.1 ([5, 14]) A submanifold N of an almost Hermitian manifold ˜ g, J ) is called bi-slant if there exists a pair of orthogonal distributions D1 and ( M, D2 of N such that (a) T N = D1 ⊕ D2 ; (b) J D1 ⊥ D2 and J D2 ⊥ D1 ; (c) The distributions D1 , D2 are slant with slant angles θ1 , θ2 , respectively. The pair {θ1 , θ2 } of slant angles of a bi-slant submanifold is called the bi-slant angle. In particular, a bi-slant submanifold with bi-slant angles {θ1 , θ2 } satisfying θ1 = π2 and θ2 ∈ (0, π2 ) (respectively, θ1 = 0 and θ2 ∈ (0, π2 )) is called a hemi-slant submanifold (respectively, semi-slant submanifold). A bi-slant submanifold N is called proper if its bi-slant angle satisfies θ1 , θ2  = 0, π2 . Definition 11.2 ([14]) A warped product N1 × f N2 of two slant submanifolds N1 ˜ g, J ) is called a warped product biand N2 of an almost Hermitian manifold ( M, slant submanifold. A warped product bi-slant submanifold N1 × f N2 is called a warped product hemi-slant submanifold (respectively, warped product semi-slant ˜ submanifold) if N1 is totally real (respectively, holomorphic) in M. Remark 11.3 Hemi-slant submanifolds are also known as pseudo-slant submanifolds in some articles (see, e.g., [102]). Remark 11.4 Clearly, both CR-products N T × N⊥ and CR-warped products N T × f N⊥ are special cases of bi-slant warped product Nθ1 × f Nθ2 with bi-slant angles {θ1 = 0, θ2 = π2 }. The following result from [88] is a natural extension of Theorem 7.2.

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Theorem 11.5 There is no warped product hemi-slant submanifolds N⊥ × f Nθ in a Kaehler manifold M˜ such that N⊥ is a totally real submanifold and Nθ is a proper ˜ slant submanifold of M. Further, the following non-existence result was obtained in [3, 75]. Theorem 11.6 There do not exist non-trivial warped product submanifolds of the form N0 × f N T or N T × f N0 in a Kaehler manifold M˜ such that N T is a holomorphic ˜ submanifold and N0 is a proper purely real submanifold of M. In contrast to the existence of CR-warped products N T × f N⊥ in Kaehler manifolds, B. Sahin proved the following non-existence result for hemi-slant warped product submanifolds. Theorem 11.7 ([86]) There do not exist warped product submanifolds of the form: N T × f Nθ or Nθ × f N T in a Kaehler manifold M˜ such that N T is a holomorphic ˜ submanifold and Nθ is a proper slant submanifold of M. For warped product bi-slant submanifolds in a Kaehler manifold, both authors together with F. R. Al-Solamy proved the following in [101]. Theorem 11.8 Let Nθ1 × f Nθ2 be a warped product bi-slant submanifold with bi˜ Then one of the following two cases slant angle {θ1 , θ2 } in a Kaehler manifold M. must occur: (1) The warping function f is constant, that is, Nθ1 × f Nθ2 is a Riemannian product of two slant submanifolds; (2) θ2 = π2 , that is, N is a warped product hemi-slant submanifold such that Nθ2 is ˜ a totally real submanifold N⊥ of M.

12 Pointwise Bi-slant Warped Products in Kaehler Manifolds ˜ g, J ). For any vector Let N be a submanifold of an almost Hermitian manifold ( M, X tangent to M, we put J X = P X + F X,

(25)

where P X and F X denote the tangential and the normal components of J X , respectively. Then P is an endomorphism of the tangent bundle T N . Also, for any vector ξ normal to the submanifold N , we put J ξ = tξ + f ξ,

(26)

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where tξ and f ξ are the tangential and the normal components of J ξ, respectively. Then f is an endomorphism of the normal bundle T ⊥ N and t is a tangent-bundlevalued 1-form on T ⊥ N . Next, we mention the following definition of pointwise slant submanifolds. Definition 12.1 ([53, 70]) A submanifold N of an almost Hermitian manifold M˜ is called pointwise slant if, at each point p ∈ N , the Wirtinger angle θ(X ) is independent of the choice of the nonzero tangent vector X ∈ T p N . In this case, θ can be regarded as a function on N , which is called the slant function of the pointwise slant submanifold. We know from [53] that a submanifold N of an almost Hermitian manifold M˜ is pointwise slant if and only if the endomorphism P of the tangent bundle T N defined by (25) satisfies P 2 = −(cos2 θ)I for some real-valued function θ defined on the tangent bundle T N of N , where I denotes the identity map. Pointwise bi-slant immersions are defined as follows. ˜ g, J ) Definition 12.2 ([62]) A submanifold M of an almost Hermitian manifold ( M, is called pointwise bi-slant if there exists a pair of orthogonal distributions D1 and D2 of M, at the point p ∈ M such that (a) T N = D1 ⊕ D2 ; (b) J D1 ⊥ D2 and J D2 ⊥ D1 ; (c) The distributions D1 , D2 are pointwise slant with slant functions θ1 , θ2 , respectively. A pointwise bi-slant submanifold is bi-slant if both slant functions θ1 and θ2 are constant. A warped product Nθ1 × f Nθ2 in M˜ is a pointwise bi-slant warped product if both Nθ1 and Nθ2 are pointwise slant submanifolds of M˜ with slant functions θ1 and θ2 , respectively. A pointwise bi-slant warped product Nθ1 × f Nθ2 is called pointwise semi-slant (respectively, pointwise hemi-slant) if θ1 = 0 (respectively, θ1 = π2 ). Next, we provide an example of pointwise bi-slant warped products. Example 12.3 ([62]) Consider a submanifold N of C4 defined by φ(u, v, w, s) = ((1 + i)u + iuv, (1 − i)w + isw). If we put V1 = (1 + i + iv, 0), V2 = (iu, 0), V3 = (0, 1 − i + is), V4 = (0, iw). Let us put H1 = Span{V1 , V2 } and H2 = Span{V3 , V4 }. Then N is a pointwise bislant submanifold with slant functions given by

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θ1 =

1 1 + (1 + v)2

, θ2 =

1 1 + (1 − s)2

.

For pointwise bi-slant submanifolds in Kaehler manifolds, we have the following theorem. Theorem 12.4 ([62]) Let N = Nθ1 × f Nθ2 be a warped product pointwise bi-slant submanifold of a Kaehler manifold M˜ such that Nθ1 and Nθ2 are pointwise slant submanifolds with slant functions θ1 and θ2 , respectively. Then, if N is a mixed totally geodesic warped product submanifold, then one of the two following cases occurs: (i) either N is a Riemannian product submanifold of N1 and N2 , or (ii) θ2 = π2 , that is, N is a warped product submanifold of the from Nθ1 × f N⊥ , ˜ where N⊥ is a totally real submanifold of M.

13 Pointwise Semi-slant Warped Products in Kaehler Manifolds For pointwise semi-slant warped products in Kaehler manifolds, we have the following result from [90]. Theorem 13.1 Let N = N T × f Nθ be a non-trivial warped product pointwise semislant submanifold of a Kaehler manifold M˜ h+ p , where N T is a holomorphic submanifold with dimC N T = h and Nθ is a proper pointwise slant submanifold with dim Nθ = p. Then we have the following: (1) The second fundamental form σ of N satisfies ||σ||2 ≥ 2 p(csc2 θ + cot 2 θ)|∇ T (ln f )|2 .

(27)

(2) If the equality of (27) holds identically, then N T is a totally geodesic holomorphic submanifold and Nθ is a totally umbilical submanifold of M˜ h+ p . Moreover, N is a minimal submanifold of M˜ h+ p . Further, the following four theorems were proved in [8]. Theorem 13.2 Let φ : N = N T × f Nθ → M˜ m be an isometric immersion of an ndimensional non-trivial warped product pointwise semi-slant submanifold N into a Kaehler manifold M˜ m such that Nθ is pointwise slant submanifold and N T is invariant submanifold of M˜ m . Then (i) The squared norm of the second fundamental form of N is given by 

p∇ f ||σ||2 ≥ 2 τ˜ (T N ) − τ˜ (T N T ) − τ (T Nθ ) − , f

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where p denotes the dimension of a pointwise slant submanifold Nθ and τ˜ (L) denotes the scalar curvature of M˜ m restricted to a linear subspace L ⊂ T M˜ m . (ii) If the equality holds in the above inequality, then N T is totally geodesic and Nθ is totally umbilical in M˜ m . Theorem 13.3 Assume that φ : N = N T × f Nθ → M˜ m (c) be an isometrical immersion of an n-dimensional non-trivial warped product pointwise semi-slant submanifold M into a complex space form M˜ m (c) with constant holomorphic sectional curvature c such that Nθ is a proper pointwise slant submanifold and N T is a holomorphic submanifold of M˜ m (c). Then (i) The squared norm of the second fundamental form of N satisfies ||σ||2 ≥

1 f hpc − 2 p 2 f

(28)

where h = dimC N T and p = dimR Nθ . (ii) If the equality sign of (28) holds identically, then N T is totally geodesic and Nθ is totally umbilical in M˜ m (c). Theorem 13.4 Assume that φ : N = N T × f Nθ → M˜ m (c) be an isometrically immersion of an n-dimensional non-trivial warped product pointwise semi-slant submanifold N into a complex space form M˜ m (c) such that N is a proper pointwise slant submanifold and N T is a holomorphic submanifold of M˜ m (c). Then (i) The squared norm of the second fundamental form is given by 

 1 ||σ|| ≥ 2 p |∇(ln f )| + hc − (ln f ) , 4 2

2

(29)

where p is the dimension of pointwise slant submanifold Nθ . (ii) The equality sign of (29) holds if and only if N T is totally geodesic and Nθ is totally umbilical submanifold of M˜ m (c). Moreover, N is a minimal submanifold in M˜ m (c). Remark 13.5 It was known from [58] that every totally umbilical submanifold of a complex space form M˜ m (c) with c  = 0 is either (1) holomorphic and totally geodesic in M˜ m (c), or (2) totally real and totally umbilical in M˜ m (c). By applying this fact, we know that the factor Nθ in statement (ii) in Theorems 13.3 and 13.4 is totally real whenever c  = 0. Theorem 13.6 Let N = N T × f Nθ be a compact warped product pointwise semi˜ slant submanifold of complex space form M(c). Then N is a Riemannian product if ||σ||2 ≥ where h = dimC N T and p = dimR Nθ .

c hp, 2

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Remark 13.7 For further results on pointwise semi-slant warped products in complex space forms, see [6].

14 Pointwise Hemi-Slant Warped Products in Kaehler Manifolds Now, we provide an example of hemi-slant warped product in the complex Euclidean 3-space C3 . Example 14.1 ([88]) Consider a submanifold N in the complex Euclidean 3-space C3 defined by ψ(x, y, z) = ((x + iky) cos z, (x + iky) sin z, y + ix), where k is constant  = 0, 1. Clearly, the tangent bundle T N of N is spanned by Z 1 , Z 2 , and Z 3 , where Z 1 = (cos z, sin z, i), Z 2 = (ik cos z, ik sin z, 1), Z 3 = (−(x + iky) sin z, (x + iky) cos z, 0). And H⊥ = Span{Z 3 } is a totally real distribution and Hθ = Span{Z 1 , Z 2 } is a slant distribution with the slant angle θ = cos

−1



1−k



. 2(1 + k 2 )

It is direct to verify that Hθ is integrable. If we denote by N⊥ an integral manifold of H⊥ and by Nθ an integral manifold of Hθ , then the metric tensor g of N is given by g = g Nθ + (x 2 + k 2 y 2 )g N⊥ . Now, it is easy to see that N is a hemi-slant warped product submanifold of C3 of the form Nθ × f N⊥ with a non-constant warping function. Definition 14.2 A pointwise hemi-slant warped product N⊥ × f Nθ (or Nθ × f N⊥ ) in a Kaehler manifold M˜ is called proper if the pointwise slant factor Nθ is proper ˜ pointwise slant in M. For pointwise hemi-slant warped products of the form N⊥ × f Nθ in a Kaehler manifold, S. Uddin and M. S. Stankovi´c proved the following result in [103]. Theorem 14.3 There does not exist any proper warped product mixed totally geodesic submanifold of the form N⊥ × f Nθ of a Kaehler manifold M˜ such that

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N⊥ is a totally real submanifold and Nθ is a proper pointwise slant submanifold of ˜ M. Proposition 14.4 Let N = N⊥ × f Nθ be a warped product pointwise hemi-slant submanifold of a Kaehler manifold M˜ such that N⊥ is a totally real submanifold ˜ Then Z (ln f ) = and Nθ is a pointwise slant submanifold with slant function θ of M. tan θ Z (θ) for any Z ∈ T N⊥ . Remark 14.5 Theorems 7.2 and 11.5 can be regarded as easy consequences of Proposition 14.4. For hemi-slant warped products of the form N = Nθ × f N⊥ in a Kaehler manifold with dim Nθ = n and dim N⊥ = p, the following theorem were obtained in [88]. Theorem 14.6 Let N be an ( p + n)-dimensional mixed geodesic hemi-slant warped product submanifold Nθ × f N⊥ in a Kaehler manifold M˜ p+n such that Nθ is a proper slant submanifold and N⊥ be a totally real submanifold of M˜ p+n . Then we have the following: (1) The second fundamental form σ of N satisfies ||σ||2 ≥ p (cot 2 θ)|∇(ln f )|2 .

(30)

(2) If the equality of (30) holds identically, then Nθ is a totally geodesic submanifold ˜ Moreover, N is never a minimal and N⊥ is a totally umbilical submanifold of M. ˜ submanifold of M.

15 Pointwise CR-Slant Warped Products in Kaehler Manifolds The notion of pointwise CR-slant warped products was defined as follows. Definition 15.1 ([63]) A submanifold N of an almost Hermitian manifold M˜ is called a pointwise CR-slant warped product if it is a warped product B × f N⊥ , where the fiber N⊥ is a totally real submanifold of M˜ and the base B = N T × Nθ is the Riemannian product of a holomorphic submanifold N T and a proper pointwise ˜ slant submanifold Nθ of M. Definition 15.2 ([63]) A pointwise CR-slant warped product (N T × Nθ ) × f N⊥ is called proper if the warping function f is non-constant. Also, the pointwise CR-slant warped product (N T × Nθ ) × f N⊥ is simply called CR-slant warped product if the factor Nθ is a proper slant submanifold. Let N = (N T × Nθ ) × f N⊥ be a pointwise CR-slant warped product in an almost ˜ Then there exist three associated integrable distributions Hermitian manifold M.

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DT , Dθ , and D⊥ on N such that T N = DT ⊕ D⊥ ⊕ Dθ , where DT is a holomorphic distribution, D⊥ is a totally real distribution, and Dθ is a pointwise slant distribution whose slant function is θ. Definition 15.3 ([63]) A pointwise CR-slant warped product (N T × N⊥ ) × f Nθ in M˜ is called weakly Dθ -totally geodesic if its second fundamental form σ satisfies σ(Dθ , Dθ ), J D⊥  = {0}, that is, σ(Dθ , Dθ ) has no component in J D⊥ . Definition 15.4 ([63]) A pointwise CR-slant warped product (N T × N⊥ ) × f Nθ in an almost Hermitian manifold is called DT ⊕ Dθ -mixed totally geodesic if its second fundamental satisfies σ(DT , Dθ ) = {0}. Similarly, a pointwise CR-slant warped product (N T × N⊥ ) × f Nθ is called D⊥ ⊕ Dθ -mixed totally geodesic if its second fundamental satisfies σ(D⊥ , Dθ ) = {0}. The following results were proved in [63]. Theorem 15.5 Let N = (N T × N⊥ ) × f Nθ be a pointwise CR-slant warped prod˜ If N is DT ⊕ Dθ -mixed totally geodesic, then the uct in a Kaehler manifold M. warping function f depends only on N⊥ . The next theorem provides a sharp inequality involving the second fundamental form for CR-slant warped products in a Kaehler manifold. Theorem 15.6 ([63]) Let N = (N T × N⊥ ) × f Nθ be a pointwise CR-slant warped ˜ If N is weakly Dθ -totally geodesic, then product in a Kaehler manifold M. (i) The squared norm of the second fundamental form σ of N satisfies   σ2 ≥ 4s (csc2 θ + cot 2 θ)|∇ T (ln f )|2 + cot 2 θ |∇ ⊥ (ln f )|2 ,

(31)

where ∇ T (ln f ) and ∇ ⊥ (ln f ) denote the gradient components of ln f along N T and N⊥ , respectively, and s = 21 dim Nθ . (ii) The equality sign in (31) holds identically if and only if B is totally geodesic and ˜ Nθ is totally umbilical in M. (iii) If the warping function f in B × f Nθ is non-constant, then at least one of ˜ DT ⊕ Dθ and D⊥ ⊕ Dθ is non-mixed totally geodesic in M. Remark 15.7 Theorem 15.6 generalizes Theorems 7.3 and 13.1. If dim N T = 0, then Theorem 15.6 implies the next result from [94]. Theorem 15.8 If N⊥ × f Nθ is a D⊥ ⊕ Dθ -mixed totally geodesic, pointwise hemi˜ then second fundamental slant warped product submanifold of a Kaehler manifold M, form σ satisfies σ2 ≥ 2s cos2 θ ∇ ⊥ (ln f )|2 , dim Nθ = 2s, where ∇ ⊥ (ln f ) denotes the gradient of ln f on N⊥ .

(32)

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Remark 15.9 Theorem 15.6 improves Theorem 15.8, since if N is D⊥ ⊕ Dθ -mixed totally geodesic, then N is Riemannian product of N⊥ and Nθ , which is the original statement of Corollary 4.5 of [103]. Thus, Theorem 15.6 shows that inequality (32) obtained in [94] is not sharp. The following provides an example of pointwise CR-slant warped product in C9 which is weakly Dθ -totally geodesic. Example 15.10 ([63]) Let ψ : R5 → C9 be an isometric immersion given by ψ(u, v, w, r, s) = (u cos r, u cos s, w cos r, w cos s, u sin r, u sin s, r, w sin r, w sin s, v cos r, v cos s, w sin r, w sin s, v sin r, v sin s, s, w cos r, w cos s) with u, v, w > 0. Then the tangent bundle T N is spanned by  X 1 = cos r, cos s, 0, 0, sin r, sin s,

 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ,

  X 2 = 0, 0, 0, 0, 0, 0, 0, 0, 0, cos r, cos s, 0, 0, sin r, sin s, 0, 0, 0 ,  X 3 = 0, 0, cos r, cos s, 0, 0, 0, sin r, sin s,  0, 0, sin r, sin s, 0, 0, 0, cos r, cos s ,  X 4 = − u sin r, 0, −w sin r, 0, u cos r, 0, 1, w cos r, 0,  − v sin r, 0, w cos r, 0, v cos r, 0, 0, −w sin r, 0 ,  X 5 = 0, −u sin s, 0, −w sin s, 0, u cos s, 0, 0, w cos s,  0, −v sin s, 0, w cos s, 0, v cos s, 1, 0, −w sin s .

(33)

Then we find from (33) that   J X 1 = 0, 0, 0, 0, 0, 0, 0, 0, 0, cos r, cos s, 0, 0, sin r, sin s, 0, 0, 0 ,   J X 2 = − cos r, cos s, 0, 0, sin r, sin s, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ,  J X 3 = 0, 0, − sin r, − sin s, 0, 0, 0, − cos r, − cos s,  0, 0, cos r, cos s, 0, 0, 0, sin r, sin s ,  J X 4 = v sin r, 0, −w cos r, 0, −v cos r, 0, 0, w sin r, 0,

 − u sin r, 0, −w sin r, 0, u cos r, 0, 1, w cos r, 0 ,

 J X 5 = 0, v sin s, 0, −w cos s, 0, −v cos s, −1, 0, w sin s,

 0, −u sin s, 0, −w sin s, 0, u cos s, 0, 0, w cos s .

Since J X 3 is perpendicular to T N , D⊥ = Span{X 3 } is totally real. Moreover, the distributions

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DT = Span{X 1 , X 2 }, Dθ = Span{X 4 , X 5 } are complex and proper pointwise slant with slant function −1

θ = cos



1 1 + u 2 + v 2 + 2w2

 .

Obviously, all the three distributions are integrable. Let N T , N⊥ , and Nθ be the leaves of DT , D⊥ , and Dθ , respectively. Then the induced metric on the product manifold N T × N⊥ × Nθ is the warped product metric: g = 2(du 2 + dv 2 ) + 4dw 2 + (1 + u 2 + v 2 + 2w2 )(dr 2 + ds 2 ) = g B + f 2 g Nθ .

(34)

Thus,√N is a proper pointwise CR-slant warped product submanifold of C9 with f = 1 + u 2 + v 2 + 2w2 . We derive from (33) and (34) that σ(X i , X j ) = 0, 1 ≤ i, j ≤ 3,  −1 σ(X 1 , X 4 ) = (1 + v 2 + 2w2 ) sin r, 0, −uw sin r, 1 + u 2 + v 2 + 2w2 0, −(1 + v 2 + 2w2 ) cos r, 0, u, uw cos r, 0,  − uv sin r, 0, uw cos r, 0, uv cos r, 0, 0, −uw sin r, 0 ,  −1 0, (1 + v 2 + 2w2 ) sin s, 0, σ(X 1 , X 5 ) = 1 + u 2 + v 2 + 2w2 − uw sin s, 0, −(1 + v 2 + 2w2 ) cos s, 0, 0, uw cos s, 0,  − uv sin s, 0, uw cos s, 0, uv cos s, u, 0, −uw sin s ,  1 uv sin r, 0, vw sin r, 0, −uv cos r, σ(X 2 , X 4 ) = 1 + u 2 + v 2 + 2w2 0, −v, −vw cos r, 0, −(1 + u 2 + 2w2 ) sin r, 0, −vw cos r,  0, (1 + u 2 + 2w2 ) cos r, 0, 0, vw sin r, 0 ,  1 0, uv sin s, 0, vw sin s, 0, σ(X 2 , X 5 ) = 1 + u 2 + v 2 + 2w2 − uv cos s, 0, 0, −vw cos s, 0, −(1 + u 2 + 2w2 ) sin s, 0,  − vw cos s, 0, (1 + u 2 + 2w2 ) cos s, −v, 0, vw sin s ,

(35)

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 1 2uw sin r, 0, −(1 + u 2 + v 2 ) sin r, 1 + u 2 + v 2 + 2w2 0, −2uw cos r, 0, −2w, (1 + u 2 + v 2 ) cos r, 0, 2vw sin r, 0,  (1 + u 2 + v 2 ) cos r, 0, −2vw cos r, 0, 0, −(1 + u 2 + v 2 ) sin r, 0 ,  1 0, 2uw sin s, 0, −(1 + u 2 + v 2 ) sin s, σ(X 3 , X 5 ) = 1 + u 2 + v 2 + 2w2 0, −2uw cos s, 0, 0, (1 + u 2 + v 2 ) cos s, 0, 2vw sin s, 0,  (1 + u 2 + v 2 ) cos s, 0, −2vw cos s, −2w, 0, −(1 + u 2 + v 2 ) sin s , −1  u cos r, −u cos s, w cos r, −w cos s, u sin r, σ(X 4 , X 4 ) = 2 − u sin s, 0, w sin r, −w sin s, v cos r, −v cos s, w sin r,  − w sin s, v sin r, −v sin s, 0, w cos r, −w cos s ,

σ(X 3 , X 4 ) =

σ(X 4 , X 5 ) = 0, σ(X 5 , X 5 ) = −σ(X 4 , X 4 ). By applying (34) and (35), it is easy to verify that the pointwise CR-slant warped product defined by ψ is weakly Dθ -totally geodesic. Further, (34) and (35) imply that Nθ is not totally umbilical in C9 .

16 Basics on Nearly Kaehler Manifolds ˜ g, J ) is called nearly Kaehler if we have An almost Hermitian manifold ( M, ˜ (∇˜ X J )X = 0, for all X ∈ T M, where ∇˜ is the Levi-Civita connection of M˜ [74]. Nearly Kaehler manifolds are exactly almost Tachibana manifolds studied in [95]. Clearly, every Kaehler manifold is nearly Kaehler, but the converse is not true. A nearly Kaehler manifold is called a strict nearly Kaehler manifold if it is non-Kaehlerian. It is known that every nearly Kaehler 4-manifold is a Kaehler manifold [73]. ˜ the Nijenhuis tensor field For an almost complex structure J on the manifold M, is defined by N J (X, Y ) = [J X, J Y ] − J [J X, Y ] − J [X, J Y ] − [X, Y ] ˜ where [ , ] is the Lie bracket. It is wellfor any vector fields X, Y tangent to M, known that a necessary and sufficient condition for a nearly Kaehler manifold to be Kaehler is the vanishing of the Nijenhuis tensor N J . The best-known example of a strict nearly Kaehler manifold is the 6-sphere S 6 with the nearly Kaehlerian structure induced from the vector cross product on the

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space of purely imaginary Cayley numbers O (cf., e.g., [43, 74, 95]). The only known 6-dimensional strict nearly Kaehler manifolds are S 6 , S 3 × S 3 , Sp(2)/(SU (2) × U (1)), SU (3)/(U (1) × U (1)). In fact, these are the only homogeneous nearly Kaehler manifolds in dimension six [13]. It is also known from [13] that homogeneous and strictly nearly Kaehler manifolds are 3-symmetric. Further examples of nearly Kaehler manifolds are the homogeneous spaces G/K , where G is a compact semisimple Lie group and K is the fixed point set of an automorphism of G of order 3 (see [107]). The first complete nonhomogeneous nearly Kaehler structures on S 6 and S 3 × S 3 were recently discovered in [71]. The next properties of nearly Kaehler manifolds were proved in [74]. Theorem 16.1 We have the following: (a) A nearly Kaehler manifold M˜ is Kaehlerian (respectively, Einstein) if dimR M˜ = 4 (respectively, if dimR M˜ = 6). (b) A complete simply connected nearly Kaehler manifold with dimR M˜ = 8 is the direct product M1 × M2 of 2-dimensional Kaehler manifold M1 and a 6dimensional nearly Kaehler manifold M2 . ˜ x ∈ M} ˜ defines an integrable dis(c) {X ∈ Tx M˜ : (∇˜ X J )Y = 0, for all Y ∈ Tx M, ˜ whose leaves are Kaehler. tribution on a nearly Kaehler manifold M, (d) If a strict nearly Kaehler manifold M˜ has sufficiently large sectional curvature ˜ R) = 0, where H2 ( M; ˜ R) denotes the (or holomorphic pinching), then H2 ( M; second cohomology group over the real field. The following results are also known. Theorem 16.2 We have the following: (1) Every holomorphic submanifold of a nearly Kaehler manifold is nearly Kaehler [73]. (2) S 6 admits no 4-dimensional almost holomorphic submanifolds [73]. (3) Every Lagrangian submanifold of the nearly Kaehler S 6 is orientable and minimal [69]. (4) There does not exist a CR-product in S 6 [93]. For hypersurfaces of a nearly Kaehler manifold, we have the following result from [43, p. 396] (see also [22]). Theorem 16.3 Let N be an orientable closed hypersurface of a nearly Kaehler manifold M. If H2n−2 (N ; R) = 0 with n = 21 dim N ≥ 2, then either its holomorphic distribution D is non-integrable or its totally real distribution D⊥ is non-totally geodesic.

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17 CR-Warped Products in Nearly Kaehler Manifolds The following result from [1, 79, 91] is analogous to Theorem 7.2 for warped product CR-submanifolds of a Kaehler manifold. Theorem 17.1 If N⊥ × f N T is a warped product CR-submanifold in a nearly Kaehler manifold M˜ such that N⊥ is a totally real submanifold and N T is a holo˜ then N⊥ × f N T is non-proper. morphic in M, Analogous to Theorem 7.3, the next theorem for CR-warped products of the form N T × f N⊥ in nearly Kaehler manifolds was obtained in [1, 91] (For doubly warped product and doubly twisted product CR-submanifolds, see [97]) . Theorem 17.2 Let N = N T × f N⊥ be a CR-warped product in a nearly Kaehler ˜ Then we have the following: manifold M. (1) The second fundamental form σ of N satisfies ||σ||2 ≥ 2 p |∇(ln f )|2 ,

(36)

where p is the dimension of N⊥ . (2) If the equality sign of (36) holds identically, then N T is a totally geodesic sub˜ Moreover, N is a manifold and N⊥ is a totally umbilical submanifold of M. ˜ minimal submanifold in M. (3) If N is anti-holomorphic and p > 1, then the equality sign of (36) holds if and ˜ only if N⊥ is a totally umbilical submanifold of M. (4) If N is a real hypersurface, then the equality sign of (36) holds identically if and only if the characteristic vector field J ξ of N is a principal vector field with zero as its principal curvature, where ξ a unit normal vector field of N . Also, in this case, the equality sign of (36) holds identically if and only if N is a minimal hypersurface.

18 Hemi-Slant Warped Products in Nearly Kaehler Manifolds ˜ there is For a hemi-slant warped product N = N⊥ × f Nθ or N = Nθ × f N⊥ in M, a subbundle μ of the normal bundle T ⊥ N such that T ⊥ N = μ ⊕ FD⊥ ⊕ FDθ ,

(37)

where D⊥ and Dθ are the distributions given by the tangent bundles of N⊥ and Nθ , respectively. The next result was proved by Uddin and Chi in [102].

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Theorem 18.1 A hemi-slant warped product submanifold N = Nθ × f N⊥ of a nearly Kaehler manifold M˜ is simply a Riemannian product of Nθ and N⊥ if and only if the second fundamental form σ satisfies g(σ(X, Z ), F Z ) = g(σ(Z , Z ), F X ), for any X ∈ T Nθ and Z ∈ T N⊥ , where Nθ and N⊥ are proper slant and totally real ˜ respectively. submanifolds of M, Theorem 18.1 implies the following non-existence result. Corollary 18.2 There does not exist any hemi-slant warped product submanifold N = Nθ × f N⊥ in a nearly Kaehler manifold M˜ if the condition σ(T N , D⊥ ) ∈ μ holds. Uddin, Al-Solamy, and Khan proved in [100] the following. Theorem 18.3 Let N = Nθ × f N⊥ be a mixed geodesic, hemi-slant warped product in a nearly Kaehler manifold M˜ such that N⊥ is totally real and Nθ is proper slant ˜ Then in M. (i) The squared norm of the second fundamental form σ of N satisfies σ2 ≥ p(cot 2 θ)|∇(ln f )|2 ,

(38)

where p is the dimension of N⊥ . (ii) If the equality holds in (38), then Nθ and N⊥ are totally geodesic and totally ˜ respectively. umbilical submanifolds of M, The next result was proved by Al-Solamy and Khan in [2]. Theorem 18.4 Let N = Nθ × f N⊥ be a hemi-slant warped product in a nearly ˜ If the mean curvature vector field H of N lies in the subbundle Kaehler manifold M. μ of the normal bundle T ⊥ N (cf. (37)), then we have ||σ||2 ≥ 4 p(cos2 θ)|∇(ln f )|2 ,

p = dim N⊥ .

(39)

Remark 18.5 In [2], the authors also discussed the equality case of (39). The next non-existence result was obtained in [77]. Theorem 18.6 There does not exist a non-trivial hemi-slant warped product of the form N = N⊥ × f Nθ in a nearly Kaehler manifold M˜ with (∇˜ X J )Y orthogonal to the totally real factor for X, Y ∈ T Nθ . The following results were presented in [7].

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Theorem 18.7 Let N = N⊥ × f Nθ be a hemi-slant warped product submanifold of a nearly Kaehler manifold M˜ such that N⊥ is a totally real submanifold of dimension p and Nθ is a proper slant submanifold of dimension 2h. Then the squared norm of the second fundamental form of N satisfies ||σ||2 ≥

2h (cos2 θ)|∇(ln f )|2 . 9

(40)

Remark 18.8 The equality case of (40) is also discussed in [7].

19 Semi-slant Warped Products in Nearly Kaehler Manifolds The following inequality for semi-slant warped products in nearly Kaehler manifolds was proved by V. A. Khan and K. A. Khan in [79]. For bi-warped product submanifolds in nearly Kaehler manifolds, see [99]. Theorem 19.1 Let N = N T × f Nθ be a semi-slant warped product submanifold of a nearly Kaehler manifold M˜ such that N T is a holomorphic submanifold, Nθ is a ˜ and dim Nθ = 2q. Then the squared norm of the proper slant submanifold of M, second fundamental form σ of N satisfies   1 σ2 ≥ 4q csc2 θ + cos2 θ cot2 θ |∇(ln f )|2 . 9

(41)

On the other hand, F. R. Al-Solamy, V. A. Khan, and S. Uddin studied semi-slant warped products in nearly Kaehler manifolds and established the following result in [4] improving Theorem 19.1. Theorem 19.2 Let N = N T × f Nθ be a semi-slant warped product submanifold of a nearly Kaehler manifold M˜ such that NT is a holomorphic submanifold and Nθ is ˜ Then a proper slant submanifolds of M. (i) The squared norm of the second fundamental form σ of N satisfies   1 σ2 ≥ 4q csc2 θ + cot 2 θ |∇(ln f )|2 9

(42)

with dim Nθ = 2q. (ii) If the equality sign in (42) holds identically, then N T is totally geodesic and Nθ ˜ Further, N is a minimal is totally umbilical in the nearly Kaehler manifold M. submanifold.

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20 Generic Warped Products in Nearly Kaehler Manifolds ˜ If one of Let N1 × N2 be a warped product in an almost Hermitian manifold M. ˜ then N1 × N2 is called a generic N1 , N2 is a proper generic submanifold of M, warped product. With the extension of Theorems 7.2 and 11.6, we have the following results obtained in [78, 79] (see also [80, Theorem 3.2]). Theorem 20.1 Let N = N0 × f N T be a generic warped product submanifold of a nearly Kaehler manifold M˜ such that N0 is a generic submanifold and N T is a ˜ Then N is trivial, that is, N is a generic product in holomorphic submanifold of M. ˜ M. For doubly warped products and doubly twisted products in a nearly Kaehler manifold, we have the following results from [80]. Theorem 20.2 There are no proper doubly warped products in a nearly Kaehler ˜ manifold M˜ with one of the two factors a holomorphic submanifold of M. Theorem 20.3 There do not exist doubly twisted product generic submanifolds of a nearly Kaehler manifold which are not single twisted product generic submanifolds in the form f1 N T × f2 N0 such that N T is holomorphic and N0 is an arbitrary ˜ submanifold of M.

21 Bi-warped Products in Nearly Kaehler Manifolds Multiply warped products are defined as follows: Definition 21.1 Let N0 , N1 , · · · , N be  + 1 Riemannian manifolds and put N = N0 × N1 × · · · × N . Denote by πi : N → Ni the canonical projection of N onto Ni for i = 0, 1, . . . , . And let πi∗ denote the differential of πi . If f 1 , · · · , f  : N1 → R+ are positive-valued functions in F(N1 ), then X, Y  = π0∗ X, π0∗ Y  +

 

  ( f i ◦ π)2 πi∗ X, πi∗ Y

i=1

for X, Y ∈ T N defines a metric on N , called a multiply warped product metric. The product manifold N endowed with this multiply warped metric, denoted by N0 × f1 N1 × · · · × f N , is called a multiply warped product. A multiply warped product N0 × f1 N1 × f2 N2 with  = 2 is called a bi-warped product. A bi-warped product N = N0 × f1 N1 × f2 N2 is a Riemannian product if both warping functions f 1 and f 2 are constant. If exactly one of f 1 , f 2 is constant, then

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N is an ordinary warped product manifold. And if none of f 1 , f 2 is constant, then N is called a proper bi-warped product manifold. Now, we consider bi-warped product submanifolds of the form: N = N T × f1 ˜ where N T , N⊥ , and Nθ are holomorphic, N⊥ × f2 Nθ in a nearly Kaehler manifold M, ˜ respectively. As before, we put totally real ,and proper slant submanifolds of M, D T = T N T , D⊥ = T N ⊥ , Dθ = T N θ . Then we have the following results on bi-warped product N T × f1 N⊥ × f2 Nθ from [101]. Theorem 21.2 Let N = N T × f1 N⊥ × f2 Nθ be a bi-warped product submanifold of ˜ If N is DT ⊕ D⊥ -mixed totally geodesic, then f 1 is a nearly Kaehler manifold M. constant, and hence N is an ordinary warped product manifold. Theorem 21.3 Let N = N T × f1 N⊥ × f2 Nθ be a proper bi-warped product subman˜ If N is DT ⊕ Dθ -mixed totally geodesic, then ifold of a nearly Kaehler manifold M. f 2 is constant on M. Theorem 21.4 Let N = N T × f1 N⊥ × f2 Nθ be a bi-warped product submanifold of ˜ where N T , N⊥ , and Nθ are holomorphic, totally real, a nearly Kaehler manifold M, ˜ respectively. Then we have the following: and proper slant submanifolds of M, (i) The second fundamental form σ and the warping functions f 1 , f 2 satisfy 

 10 2 σ ≥ 2 p∇(ln f 1 ) + 4q 1 + cot θ ∇(ln f 2 )2 , 9 2

2

(43)

where p = dim N⊥ , q = 21 dim Nθ , and ∇(ln f i ) is the gradient of ln f i . ˜ and N⊥ , Nθ are (ii) If the equality sign in (43) holds, then N T is totally geodesic in M, ˜ Moreover, N is neither DT ⊕ D⊥ -mixed totally geodesic totally umbilical in M. ˜ nor DT ⊕ Dθ -mixed totally geodesic in M. Remark 21.5 For results on bi-warped products submanifolds in a Kaehler manifold, see [96].

22 CR-Slant Warped Products in Nearly Kaehler Manifolds Recall from Sect. 15 that a submanifold N of an almost Hermitian manifold M˜ is called a CR-slant warped product submanifold if it is a warped product B × f N⊥ , with B = N T × Nθ being a Riemannian product, where N T , Nθ , and N⊥ are ˜ respectively. holomorphic, proper slant, and totally real submanifolds of M,

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For a CR-slant warped product B × f N⊥ with B = N T × Nθ in a nearly Kaehler ˜ if we put DT = T (N T ), Dθ = T (Nθ ), D⊥ = T (N⊥ ) as before, then manifold M, the tangent bundle of N is decomposed as T N = DT ⊕ Dθ ⊕ D⊥ . As before, a CR-slant warped product submanifold (N T × Nθ ) × f N⊥ in a nearly Kaehler manifold is said to be DT ⊕ D⊥ -mixed totally geodesic if its second fundamental satisfies σ(DT , D⊥ ) = {0}. Similarly, the CR-slant warped product submanifold is said to be Dθ ⊕ D⊥ -mixed totally geodesic if its second fundamental satisfies σ(Dθ , D⊥ ) = {0}. Now, we present the following two results on CR-slant warped product submanifolds of nearly Kaehler manifolds obtained in [98]. Theorem 22.1 Let N = B × f N⊥ be a CR-slant warped product submanifold with B = N T × Nθ in a nearly Kaehler manifold. If N is DT ⊕ D⊥ -mixed totally geodesic, then the warping function f depends only on the factor Nθ . Theorem 22.2 Let N = B × f N⊥ be a CR-slant warped product submanifold with ˜ If N is Dθ ⊕ D⊥ -mixed totally B = N T × Nθ in a nearly Kaehler manifold M. ˜ then we have the following: geodesic in M, (1) The second fundamental form σ satisfies σ2 ≥ 2 p|∇ T (ln f )|2 + p cot 2 θ|∇ θ (ln f )|2 ,

(44)

where p = dim N⊥ and ∇ T (ln f ) and ∇ θ (ln f ) denote the gradient components of ln f along N T and Nθ , respectively. (2) If the equality sign in (44) holds identically, then ˜ (2.a) N T and Nθ are totally geodesic in M. ˜ (2.b) B is mixed totally geodesic in M. ˜ (2.c) N⊥ is totally umbilical in M. Remark 22.3 Theorem 22.1 extends the results on warped product submanifolds given in Theorem 18.3 with N T = {0} and Theorem 17.2 with Nθ = {0}. Furthermore, Theorem 2 of [89] on mixed totally geodesic skew CR-warped product submanifolds of a Kaehler manifold is a special case of Theorem 22.1 as well.

23 Generalized Complex Space Forms There exists a special class of almost Hermitian manifolds, called R K -manifolds, which contains nearly Kaehler manifolds.

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˜ g, J ) is an almost Hermitian manifold Definition 23.1 ([104]) An RK-manifold ( M, whose curvature tensor R˜ satisfies ˜ X, J Y, J Z , J W ) = R(X, ˜ R(J Y, Z , W ) ˜ for vector fields X, Y, Z , W tangent to M. ˜ we put On M, ˜ ˜ λ(X, Y ) = R(X, Y, J X, J Y ) − R(X, Y, X, Y ). Definition 23.2 ([105]) An almost Hermitian manifold M˜ is said to be of pointwise ˜ we have λ(X, Y ) = constant type if at any point x ∈ M˜ and any vector X ∈ Tx M, λ(X, Z ), where Y and Z are unit tangent vectors in Tx M˜ which are orthogonal to X and J X . And M˜ is said to be of constant type if for unit vectors fields X, Y with g(X, Y ) = g(J X, Y ) = 0, λ(X, Y ) is a constant function. A generalized complex space form is a R K -manifold of constant holomorphic sectional curvature and it is of constant type. Every complex space form is a generalized complex space form, but the converse is not true. The simplest example is the nearly Kaehler S 6 which is a generalized complex space form, but not a complex space form. Let us denote a complex m-dimensional generalized complex space form of constant holomorphic sectional curvature c and of constant type α by M˜ m (c, α). Then the curvature tensor R˜ of M˜ m (c, α) satisfies c + 3α ˜ R(X, Y )Z = {g(Y, Z )X − g(X, Z )Y } 4 c−α + {g(X, J Z )J Y − g(Y, J Z )J X + 2g(X, J Y )J Z }. 4

24 Warped Products in Generalized Complex Space Forms We need the following definition. Definition 24.1 ([41]) Let φ : N1 × f N2 → M˜ be an isometric immersion from a warped product N1 × f N2 into a Riemannian manifold M˜ with second fundamental form σ. Denote by σ1 and σ1 the restriction of σ to D1 = T N1 and D2 = T N2 , respectively. If we put dim N1 = n 1 and dim N2 = n 2 , then the partial mean curvature vectors H1 and H2 are defined, respectively, by H1 =

n1 1  σ(ea , ea ), n 1 a=1

H2 =

n2 1  σ(eb , eb ), n 2 b=1

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where {ea }, a = 1, · · · , n 1 and {eb }, b = n 1 + 1, · · · , n 2 are the frame fields on N1 and N2 , respectively. For warped products in generalized complex space forms, we have the following result from [83]. Theorem 24.2 Let φ : N1 × f N2 → M˜ m (c, α) be an isometric immersion of a warped product N1 × f N2 into M˜ m (c, α). Then we have the following: (i) If c < α, then f c + 3α n2 2 H + n1 ≤ , f 4n 2 4

n = n1 + n2,

(45)

where H 2 is the squared mean curvature of φ. The equality case of (45) holds if and only if φ is a mixed totally geodesic immersion satisfying n 1 H1 = n 2 H2 , where H1 , H2 are the partial mean curvature vectors of N1 , N2 and also J (T N1 ) ⊥ T N2 . (ii) If c = α, then c + 3α n2 2 f H + n1 ≤ . f 4n 2 4

(46)

Moreover, the equality case of (46) holds if and only if φ is a mixed totally geodesic immersion satisfying n 1 H1 = n 2 H2 . (iii) If c > α, then c + 3α c−α n2 2 f H + n1 ≤ + n1 ||P||2 . f 4n 2 4 8

(47)

Moreover, the equality case of (47) holds if and only if φ is a mixed totally geodesic immersion satisfying n 1 H1 = n 2 H2 . The next result from [83] is a consequence of Theorem 24.2. Corollary 24.3 Let M˜ m (c, α) be a generalized complex space form with c ≤ α. If N1 is a Riemannian n 1 -manifold and f is a positive function on N1 such that ( f )(x) > c+3α n 1 f (x) holds at some point x ∈ N1 , then the warped product N1 × f N2 does 4 not admit a minimal isometric immersion into M˜ m (c, α). For CR-warped products in generalized complex space forms, we have the next result which extends the inequality (20) in Theorem 9.1 (cf. [36]).

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Theorem 24.4 ([1]) Let φ : N = N T × f N⊥ → M˜ m (c, α) be a CR-warped product submanifold of a generalized complex space form M˜ m (c, α). Then the second fundamental form σ of N satisfies 

 1 h(c − α) , ||σ|| ≥ 2 p |∇(ln f )| + (ln f ) + 2 4 2

2

(48)

where p = dim N⊥ and h = dimC N T .

25 Casorati Curvature of Bi-slant Submanifolds in Generalized Complex Space Forms The Casorati curvature of a surface in E3 was introduced by Felice Casorati (1835– 1890) in [15] which extends the concept of the principal direction. The Casorati curvature of an n-dimensional submanifold N of a Riemannian mmanifold M˜ is defined to be the normalized square length of the second fundamental form σ of N in M˜ as C=

||σ||2 . n

(49)

If L is a k-subspace of Tx M, k ≥ 2 and {e1 , . . . , ek } is an orthonormal basis of L, then the Casorati curvature C(L) of the subspace L is defined as C(L) =

k 1  |σ(ei , e j )|2 . k i, j=1

δc (n − 1) of the submaniThe normalized δ-Casorati curvatures δc (n − 1) and  fold N are defined by [δc (n − 1)] (x) =

1 n+1 C(x) + inf{C(L) : L a hyperplane of Tx N } and 2 2n

  2n − 1  sup{C(L) : L a hyperplane of Tx N }. δc (n − 1) (x) = 2C(x) − 2n For a positive real number t  = n(n − 1), if we put a(t) =

1 (n − 1)(n + t)(n 2 − n − t), nt

δc (t; n − 1) are then the generalized normalized δ-Casorati curvatures δc (t; n − 1),  defined as

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[δc (t; n − 1)] (x) = tC(x) + a(t) inf{C(L) : L a hyperplane of Tx M}, for 0 < t < n 2 − n; and for t > n 2 − n as    δc (t; n − 1) (x) = tC(x) − a(t) sup{C(L) : L a hyperplane of Tx M}. Recall that if N is a bi-slant submanifold N , then there exists a pair of orthogonal distributions D1 and D2 of N such that (a) T N = D1 ⊕ D2 , (b) J D1 ⊥ D2 and J D2 ⊥ D1 , and (c) the distributions D1 , D2 are slant distributions with slant angles θ1 , θ2 , respectively. The pair {θ1 , θ2 } is called the bi-slant angle of N . Definition 25.1 An n-dimensional submanifold N of a Riemannian manifold of dimension m is called invariantly quasi-umbilical if there exist m − n mutually orthogonal unit normal vectors ξn+1 , . . . , ξm such that the shape operators with respect to all directions ξα have an eigenvalue of multiplicity n − 1 and that for each ξα , the distinguished eigen-direction is the same. For a bi-slant submanifold N , we put dim D1 = n 1 and dim D2 = n 2 . The following result on bi-slant proper submanifold was obtained in [10]. Theorem 25.2 Let N be an n-dimensional proper bi-slant submanifold of a generalized complex space form M˜ m (c, α). Then we have the following: (i) For any real number t with 0 < t < n(n − 1), the generalized normalized δCasorati curvature δC (r ; n − 1) satisfies τ≤

 3(c − α)  δC (t; n − 1) c + 3α + + n 1 cos2 θ1 + n 2 cos2 θ2 , n(n − 1) 4 4n(n − 1)

(50)

where τ is the scalar curvature of N . (ii) For any real number t > n(n − 1), the generalized normalized δ-Casorati curvature  δC (t; n − 1) satisfies τ≤

  3(c − α)  δC (t; n − 1) c + 3α + + n 1 cos2 θ1 + n 2 cos2 θ2 . n(n − 1) 4 4n(n − 1)

(51)

Remark 25.3 Theorem 25.2 improves the result in [82]. Remark 25.4 The equality cases of (50) and (51) were discussed in [10]. For further results on slant submanifolds in generalized complex space forms, see [10, 72, 76, 81].

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Slant Geometry of Riemannian Submersions from Almost Hermitian Manifolds Bang-Yen Chen, Bogdan D. Suceavˇa, and Mohammad Hasan Shahid

2020 AMS Mathematics Subject Classification 53C15 · 53C40 · 53C55 · 53D15

1 Introduction The notion of a submersion is dual to the notion of an immersion. More precisely, a submersion π : M → B is a smooth map between differentiable manifolds whose differential π∗ is everywhere surjective. The notion of submersion is a fundamental concept in differential topology. A Riemannian submersion π : M → B is a submersion which equips compatible Riemannian metrics on the total space M and on the base manifold B. The concept of Riemannian submersions arose from [34, 35, 40, 42, 43], and it was further studied from various perspectives in the decades following the aforementioned first papers, see, e.g., [29, 30, 36, 46, 51], or the references cited in [44] or those referred to in the specialized monographs [32, 58]. One important property of a Riemannian submersion π : M → B is that the lower bound for the sectional curvature of the

B.-Y. Chen (B) Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, USA e-mail: [email protected] B. D. Suceavˇa Department of Mathematics, California State University, Fullerton, CA 92854, USA e-mail: [email protected] M. H. Shahid Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_4

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base manifold B is at least as big as the lower bound for the sectional curvature of total space M. It is well known that Riemannian submersions are natural generalizations of warped products, which occur widely in geometry (see, e.g., [9]). For instance, when a Lie group acts isometrically, freely, and properly on a Riemannian manifold M, then the projection π from M onto the quotient space B = M/G with the quotient metric g B is a Riemannian submersion. Let π : M → B be a Riemannian submersion with dim M = m, dim B = n, and m > n. Then, for each point x ∈ B, π −1 (x) is an (m − n)-dimensional submanifold of M. Each π −1 (x) is called a fiber of π. A vector field on the total space M is called vertical if it is always tangent to fibers, and a vector field on M is called horizontal if it is always orthogonal to fibers. A vector field X on the total space M is called basic if it is horizontal and π-related to a vector field X ∗ on B, i.e., π∗ X x = X ∗π(x) for each x ∈ M. If π : M → B is a Riemannian submersion, then at a point x ∈ M, the vertical space Vx and the horizontal space Hx at x are given, respectively, by Vx = ker (π∗ )x , Hx = (Vx )⊥ .

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Let V and H denote the vertical and the horizontal distributions as well as for the orthogonal projections of T M on its horizontal and vertical subspaces, respectively. Clearly, at each point x ∈ M, we have Tx M = Vx ⊕ Hx . Analogue of holomorphic submanifolds, Watson defined and studied in [66] almost Hermitian submersions between almost Hermitian manifolds. He proved that if π : M → B is an almost Hermitian submersion such that the total space M is either Hermitian, nearly Kaehler or Kaehler, then the base manifold B has the same property. In [53], Sahin ¸ defined and studied the notion of anti-invariant Riemannian submersions from an almost Hermitian manifold onto Riemannian manifold. On the other hand, Chen introduced in [16] the notion of slant submanifolds in almost Hermitian manifolds which generalizes the notions of holomorphic and totally real (or anti-invariant) submanifolds. As a natural generalization of almost Hermitian and anti-invariant submersions, Sahin ¸ introduced in [54] the notion of slant submersions. The main purpose of this chapter is to provide a survey of recent results in complex slant geometry of Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds.

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2 Hermitian and Almost Hermitian Manifolds According to a well-known result of Newlander and Nirenberg [41], an almost complex manifold (M, J ) is a complex manifold if and only if the Nijenhuis tensor N J of J on M vanishes identically, where the Nijenhuis tensor N J is defined by N J (X, Y ) = [J X, J Y ] − [X, Y ] − J ([X, J Y ] + [J X, Y ]) for vector fields X, Y tangent to M. An almost complex manifold (M, J ) is called an almost Hermitian manifold if it admits a compatible Riemannian metric g, i.e., it satisfies g(X, Y ) = g(J X, J Y ).

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In particular, when the almost complex structure on an almost Hermitian manifold (M, g, J ) is a complex manifold, then (M, g, J ) is called a Hermitian manifold. The fundamental 2-form  of an almost Hermitian manifold (M, g, J ) is defined by (X, Y ) = g(X, J Y ) (3) ˜ for vector fields X, Y tangent to M. Definition 2.1 A Hermitian manifold (M, J, g) is called a Kaehler manifold if its fundamental 2-form  is a closed form, i.e., d = 0. This condition is equivalent to (∇ X J )Y = 0

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for any vector fields X, Y tangent to M, where ∇ denotes the Levi-Civita connection on M. Definition 2.2 An almost Hermitian manifold (M, J, g) is called a almost semiKaehler manifold its fundamental 2-form  is co-closed, i.e., δ = 0, where δ denotes the co-differential operator. In particular, a Hermitian manifold (M, J, g) is called a semi-Kaehler manifold if its fundamental 2-form  is co-closed. Definition 2.3 An almost Hermitian manifold (M, g, J ) is called a nearly Kaehler manifold if it satisfies (5) (∇ X J )Y + (∇Y J )X = 0 for any vector fields X, Y tangent to M. Remark 2.4 Every Kaehler manifold is nearly Kaehler, but the converse is not always true. For instance, the unit 6-sphere S 6 (1) admits a nearly Kaehler structure and it does not admit any Kaehler structure.

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Definition 2.5 An almost Hermitian manifold (M, g, J ) is called quasi-Kaehler if it satisfies (6) (∇ X J )Y + (∇ J X J )J Y = 0 for any vector fields X, Y tangent to M. For a Kaehler manifold (M, g, J ) and for a unit vector X ∈ T M, the holomorphic sectional curvature H (X ) is defined by H (X ) = R(X, J X, J X, X ),

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where R denotes the Riemann curvature tensor of M.

3 Submanifolds of Almost Hermitian Manifolds Let N be a submanifold of an almost Hermitian manifold (M, g, J ). For any vector ˜ let us put X tangent to M, J X = P X + F X,

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where P X and F X denote the tangential and the normal components of J X , respectively. Then P is an endomorphism of the tangent bundle T N . For any non-zero vector X ∈ T p N at p ∈ N , the angle θ (X ) between J X and the tangent space T p N is called the Wirtinger angle of X . We recall the following definition from [16, 21–23, 31]. Definition 3.1 A submanifold N of an almost Hermitian manifold (M, g, J ) is called pointwise slant if, for each given point p ∈ M˜ and a non-zero vector X ∈ T p N , the Wirtinger angle θ (X ) is independent of the choice of the vector X ∈ T p N . The Wirtinger angle function θ of a pointwise slant submanifold is called the slant function. In particular, if the slant function θ of a pointwise slant submanifold N is a constant function on N , then N is called a slant submanifold. A slant submanifold with slant angle θ is simply called a θ-slant submanifold. Pointwise slant submanifolds (in particular, slant submanifolds) are simply characterized by the following (cf. [16, 17, 23]). Proposition 3.2 An immersion φ : N → M of a manifold M into an almost Hermitian manifold M˜ is a pointwise slant (respectively, slant) immersion if and only if P 2 = −(cos2 θ )I for some function θ (respectively, some real number) defined on the tangent bundle T N of N .

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Definition 3.3 A θ -slant submanifold in a almost Hermitian manifold (M, g, J ) is called a holomorphic submanifold (respectively, totally real submanifold) of M if θ = 0 (respectively, θ = π2 ). A slant submanifold is called proper slant if it is neither holomorphic nor totally real. Definition 3.4 Let N be a submanifold of a almost Hermitian manifold (M, g, J ). A distribution D defined on N is called a holomorphic (or invariant) distribution if it satisfies J D = D. And the distribution D is called an anti-invariant (or totally real) distribution if J D ⊥ D. More generally, the distribution D is called a pointwise slant distribution if, at each given point p ∈ N and each unit vector X ∈ Dx , the angle θ (X ) between J X and Dx is independent of the choice of the vector X ∈ Dx . In particular, if the angle θ (X ) between J X and Dx is independent of the choice of the vector X ∈ Dx and also independent of the point x ∈ N , then D is called a slant distribution. Definition 3.5 A submanifold N of an almost Hermitian manifold (M, g, J ) is called bi-slant if there exist two orthogonal distributions D1 and D2 of N such that (a) T N = D1 ⊕ D2 ; (b) J D1 ⊥ D2 and J D2 ⊥ D1 ; (c) The distributions D1 , D2 are slant with slant angle θ1 , θ2 , respectively. The pair {θ1 , θ2 } of slant angles of a bi-slant submanifold is called the bi-slant angle. In particular, a bi-slant submanifold with bi-slant angles {θ1 , θ2 } satisfying θ1 = π2 and θ2 ∈ (0, π2 ) (respectively, θ1 = 0 and θ2 ∈ (0, π2 )) is called a hemi-slant submanifold (respectively, semi-slant submanifold). ˜ g, J ) is called A bi-slant submanifold N of an almost Hermitian manifold ( M, proper if its bi-slant angle satisfies θ1 , θ2  = 0, π2 . Remark 3.6 Bi-slant immersions were defined by Carriazo in [11]. Such notion provides a natural extension of the notion of slant submanifolds. For a detailed survey on recent results on warped product bi-slant submanifolds, see [25].

4 Basics on Riemannian Submersions For general references on Riemannian submersions, we refer to the books [20, 32, 44, 58]. Definition 4.1 A Riemannian submersion π : M → B is a differentiable map π between Riemannian manifold such that (a) the differential π∗ is everywhere surjective and (b) π∗ preserves scalar products of vectors normal to fibers.

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For a given Riemannian submersion π : M → B, we denote by g M and g B the metric tensors of M and B, respectively. Also we denote by g F be the induced metric on fiber π −1 (π( p)), p ∈ M. Further, we denote the curvature tensors of g M , g B and g F by R, R B and R F be, respectively. For a Riemannian submersion, the O’Neill fundamental tensors A and T are defined, respectively, by A E F = H∇HE V F + V∇HE HF, TE F = H∇V E V F + V∇V E HF

(9) (10)

for vector fields E, F tangent to M, where ∇ is the Levi-Civita connection of M, and HZ and HZ denote the horizontal and vertical components of a vector field Z tangent to M. The following theorem characterizes a Riemannian submersion in terms of the two O’Neill fundamental tensors. Theorem 4.2 [42] Assume that π1 , π2 : (M, g) → (B, g B ) are two Riemannian submersions. If π1 and π2 have the same fundamental tensors A and T and also π1∗ ( p) = π2∗ ( p) at a point p ∈ M, then π1 and π2 coincide. For a given Riemannian submersion π : M → B with dim B ≥ 2, leaves do not necessarily exist, even locally. This failure can be measured by using the vector field V[X, Y ] = A X Y − AY X. The tensor A is called the O’Neill integrability tensor. O’Neill’s integrability tensor satisfies the following properties. Theorem 4.3 [42] Let X, Y be any two horizontal vector fields and E, F be vector fields on M. Then we have the following. (a) A X Y = −AY X , AHE F = A E F and g(A X E, F) = −g(E, A X F). (b) A E maps each horizontal subspace into a vertical one, and each vertical subspace into a horizontal one. (c) If X is a basic field and V a vertical vector field, then A X V = H∇V X . (d) g((∇Y A) X E, F) = g(E, (∇Y A) X F). The most interesting and important family of Riemannian submersions consists of those with totally geodesic fibers and with minimal fibers. The Hopf fibration is typical example of Riemannian submersions with totally geodesic fibers. It is well known that Riemannian submersions with totally geodesic fibers from a unit spheres S m (1) onto a Riemannian manifold were classified in [27, 28, 49]. Further, it is also known from [18] that if a Riemannian manifold admits a non-trivial Riemannian submersion with totally geodesic fibers, then it cannot be isometrically immersed in any Riemannian manifold with non-positive sectional curvature as a minimal submanifold. For Riemannian submersions with minimal fibers, we have

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Theorem 4.4 [19] Suppose that π : M → B is a Riemannian submersion with minimal fibers. If M is a positively curved Riemannian manifold, then the horizontal distribution of the Riemannian submersion π : M → B is a non-totally geodesic distribution. Theorem 4.5 [19] Assume that π : M → B is a Riemannian submersion with minimal fibers. If M is a non-positively curved (respectively, negatively curved) Riemannian manifold, then fibers of π : M → B have non-positive (respectively, negative) scalar curvature. Further, if M is a non-positively curved Riemannian and the fibers of π : M → B has zero scalar curvature, then the horizontal distribution of π : M → B is totally geodesic; and M, B, F are flat. Moreover, the Riemannian submersion π : M → B is trivial, i.e., M is locally the direct product of B and F.

5 Almost Hermitian Submersions The following notion was defined by Watson in [66]. Definition 5.1 Let (M m , g M , JM ) and (B n , g B , J B ) be two almost Hermitian manifolds of complex dimensions m and n, respectively. Then a Riemannian submersion π : (M m , g M , JM ) → (B n , g B , J B ) is called an almost Hermitian submersion if π is an almost complex map, i.e., it satisfies π∗ JM = J B π∗ . The following results on holomorphic submersions were proved in [66]. Theorem 5.2 Let π : M m → B n be an almost Hermitian submersion. Then the fibers of π are almost complexly embedded closed submanifolds of M m of dimension 2(m − n). Theorem 5.3 Let π : M → B be an almost Hermitian submersion. If the total space M is Hermitian, quasi-Kaehler, nearly Kaehler, or Kaehler, then the base manifold B has the same property. Theorem 5.4 Let π : M → B be an almost Hermitian submersion. If the total space M is an almost semi-Kaehler manifold, then the base manifold B is almost semiKaehler if and only if the submersion has minimal fibers. As a consequence of Theorem 5.4, Watson obtained the following. Corollary 5.5 Let π : M → B be an almost Hermitian submersion. If the total space M is an semi-Kaehler manifold, then the base manifold B is semi-Kaehler if and only if the submersion has minimal fibers. Definition 5.6 An almost Hermitian submersion π : M → B is called a Kaehler submersion (respectively, almost semi-Kaehler submersion or quasi-Kaehler submersion) if the total space and the base manifold are both Kaehler (respectively, both almost semi-Kaehler or both quasi-Kaehler).

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Remark 5.7 Since holomorphic submanifolds of quasi-Kaehler (in particular, Kaehler) manifolds are minimal, it is obvious that a quasi-Kaehler submersion or a Kaehler submersion have minimal fibers. Watson also proved the following results in [66]. Theorem 5.8 The horizontal distribution of a Kaehler submersion is completely integrable. Theorem 5.9 Let π : M → B be an almost Hermitian submersion between compact semi-Kaehler manifolds. Then the first Betti numbers of the total space and of base manifolds satisfy b1 (B) ≤ b1 (M). Theorem 5.10 If M is a Kaehler manifold with strictly negative holomorphic bisectional curvature, then there do not exist any Kaehler submersions π : M → B. Theorem 5.11 If M is a Kaehler manifold with non-positive holomorphic bisectional curvature and if π : M → B is a Kaehler submersion, the π is a totally geodesic mapping and M is a locally product. Let π : (M, g M , J ) → (B, g B ) be a Riemannian submanifold from an almost Hermitian manifold into a Riemannian manifold. For a vertical vector field V , we put J V = ϕV + ωV,

(11)

where ϕV and ωV denote the vertical and the horizontal components of J V , respectively. Similarly, for horizontal vector field Z , we put J Z = B Z + C Z,

(12)

where B Z and C Z denote the vertical and the horizontal components of J Z , respectively.

6 Invariant Submersions The notion of invariant submersions was defined by Sahin ¸ in [55]. Definition 6.1 Let π : M → B be an almost Hermitian submersion. Then π : M → B is called an invariant submersion if the vertical distribution V of π is an invariant distribution, i.e., JM (V p ) = V p , p ∈ M m . Sahin ¸ proved the next three results in [55] for invariant submersions. Theorem 6.2 Let π : M → B be an almost Hermitian submersion. Then the horizontal distribution H of π is invariant under the action of JM .

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Theorem 6.3 Let π : M → B be an almost Hermitian submersion. Then π : M → B is an invariant submersion. However, an invariant submersion may not be an almost Hermitian submersion. Theorem 6.4 If π : M → B is an invariant submersion from an almost Hermitian manifold onto a Riemannian manifold, then the submersion has minimal fibers. Let En be the Euclidean n-space equipped with the Euclidean metric. On E2m with Euclidean coordinates (x1 , . . . , x2m ), if J is the almost complex structure defined by J (x1 , x2 , . . . , x2m−1 , x2m ) = (−x2 , x1 , . . . , x2m , −x2m−1 ), then (E2m , J ) is the complex Euclidean m-space Cm . Sahin ¸ provided in [55] the following example of invariant submersion which is not almost Hermitian. Example 6.5 Consider the map π : R8 → R4 defined by  π(x1 , . . . , x8 ) =

x1 − x5 x2 − x6 x3 − x7 x4 − x8 √ , √ , √ , √ 2 2 2 2

 .

By direct computation we see that the horizontal and vertical distributions are given by H = Span{H1 , H2 , H3 , H4 }, V = Span{V1 , V2 , V3 , V4 }, where ∂ ∂ − , ∂ x1 ∂ x5 ∂ ∂ − , H3 = ∂ x3 ∂ x7 ∂ ∂ + , V1 = ∂ x1 ∂ x5 ∂ ∂ + , V3 = ∂ x3 ∂ x7 H1 =

∂ ∂ − , ∂ x2 ∂ x6 ∂ ∂ H4 = − , ∂ x4 ∂ x8 ∂ ∂ V2 = + , ∂ x2 ∂ x6 ∂ ∂ V4 = + . ∂ x4 ∂ x8 H2 =

Now, let us consider the complex structures J8 and J4 on R8 and R4 by J8 (a1 , . . . , a8 ) = (−a2 , a1 , −a4 , a3 , −a6 , a5 , −a8 , a7 ), J4 (a1 , a2 , a3 , a4 ) = (−a3 , a4 , a1 , a2 ). It is easy to verify that both horizontal and vertical distribution of π are invariant under the action of J8 . Although π is an invariant submersion, it is not an almost Hermitian submersion.

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7 Anti-invariant Submersions The study of anti-invariant Riemannian submersions from almost Hermitian manifolds was initiated by Sahin ¸ in [53]. In this case, the fibers are totally real with respect to the almost complex structure of the total manifold. Definition 7.1 Assume that π : M → B is a Riemannian submersion from an almost Hermitian manifold (M, g M , JM ) onto a Riemannian manifold (B, g). Then the Riemannian submersion π : M → B is called an anti-invariant submersion if its vertical distribution V of π : M → B is anti-invariant under the action of JM , i.e., JM (V) ⊂ H. H. M. Ta¸stan proved the following characterization theorem in [63] for Riemannian submersions to be either invariant or anti-invariant. Theorem 7.2 Assume that π : M m (c) → B is a Riemannian submersion from a non-flat complex space form M m (c) onto a Riemannian manifold B. Then the fibers of π : M m (c) → B are either holomorphic or totally real with respect to the almost complex structure JM of M m (c) if and only if g M ((∇U T )V W, X ) = g M ((∇V T )U W, X )

(13)

for any vertical vector fields U, V, W and any horizontal vector field X on M m (c). Corollary 7.3 If π : M n (c) → B is a Riemannian submersion from a non-flat complex space form onto a Riemannian manifold. Then we have (13). Sahin ¸ provided the following example of anti-invariant submersion. Example 7.4 Consider the map π : R4 → R2 defined by  π(x1 , x2 , x3 , x4 ) =

x1 + x4 x2 + x3 √ , √ 2 2

 .

By direct computation we see that the horizontal distribution H and vertical distribution V are given by H = Span{H1 , H2 }, V = Span{V1 , V2 }, where ∂ ∂ ∂ ∂ + , H2 = + , ∂ x1 ∂ x4 ∂ x2 ∂ x3 ∂ ∂ ∂ ∂ − , V2 = − . V1 = ∂ x1 ∂ x4 ∂ x2 ∂ x3 H1 =

One may verify directly that π is a Riemannian submersion. Moreover, from J V1 = H2 and J V2 = H1 we see that J (V) = H. Hence, π : R4 → R2 is an anti-invariant submersion.

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8 Lagrangian Submersions We recall the following definitions. Definition 8.1 A smooth map π : M → N between two Riemannian manifolds is called a totally geodesic map if π carries each geodesic in M to a geodesic in N . Equivalently, we have ∇π∗ = 0. Definition 8.2 A twisted product N1 × f N2 of two Riemannian manifolds (N1 , g1 ) and (N2 , g2 ) is the product manifold N1 × N2 equipped with the twisted product metric g = g N1 + λ2 g N2 , where λ is a positive smooth function defined on N1 × N2 . The function λ is called the twisting function. When the twisting function λ depends on the first factor N1 , then N1 ×λ N2 reduces to a warped product (see, e.g., [12, p. 31] or [14, p. 66]). A totally real (or anti-invariant) submanifold N of an almost Hermitian manifold (M, g, J ) is called a Lagrangian submanifold if J (T N ) = T ⊥ N . Similarly, an antiinvariant submersion π : M → B is called a Lagrangian submersion if it satisfies J (V) = H (cf. [63]), where V and H be the vertical and horizontal distributions of πs. The following three lemmas can be found in [63]. Lemma 8.3 Assume that π : (M, J, g) → B is a submersion from an almost Hermitian manifold onto a manifold. Then the fibers of the submersion π : (M, J, g) → B are Lagrangian submanifolds if and only if π satisfies J (ker π∗ ) = (ker π∗ )⊥ . Furthermore, the horizontal distribution (ker π∗ )⊥ of the submersion π : (M, J, g) → B is Lagrangian. Lemma 8.4 Assume that π : M → B is a Lagrangian submersion from a Kaehler manifold M onto a Riemannian manifold B. Then we have the following properties: TV J W = J TV W and A X J W = J A X W,

(14)

for any vertical vector field V and tangent vector field W of M and any horizontal vector field X . Lemma 8.5 Under the hypothesis of Lemma 8.4, the holomorphic sectional curvature of M satisfies the following: H (E) = g M ((∇ E T ) J E J E, E) − ||TJ E E||2 , H (V ) = g M ((∇ J V T )V V, J V ) − ||TV V ||2 , for any unit horizontal E and unit vertical vector field V on M. Ta¸stan also proved the following results in [63].

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Proposition 8.6 Under the hypothesis of Lemma 8.4, we have A E J F = −A F J E for horizontal vector fields E, F on M. Theorem 8.7 Under the hypothesis of Lemma 8.4, the Lagrangian submersion π : M → B is a totally geodesic map if and only if it has totally geodesic fibers. Theorem 8.8 Under the hypothesis of Lemma 8.4, the horizontal distribution H of the Lagrangian submersion π : M → B is an integrable distribution and it defines a totally geodesic foliation. Theorem 8.9 Under the hypothesis of Lemma 8.4, we have the following two properties: (a) The Kaehler manifold M is locally a twisted product manifold of the form M1H × f M2V if and only if the Lagrangian submersion π has totally umbilical fibers, where M1H and M2V are integral manifolds of the horizontal distribution H and the vertical distribution V of the Lagrangian submersion π : M → B, respectively. (b) The Kaehler manifold M is locally a product manifold if and only if the Lagrangian submersion π : M → B has totally geodesic fibers. Theorem 8.10 Under the hypothesis of Lemma 8.4, if the O’Neill tensor field T of the Lagrangian submersion π : M → B is parallel, then M has flat holomorphic sectional curvature. Theorem 8.11 Suppose that M is a non-flat Kaehler manifold. Then there does exist any Lagrangian submersion π : M → B from M onto a Riemannian manifold with totally geodesic fibers. Fibers of a Riemannian submersion π : M → N from a Riemannian manifold onto another is said to be totally umbilical if it satisfies TU V = g(U, V )η

(15)

for any U,V ∈ ker π∗ , where η denotes the mean curvature vector field of the fiber in M. The fiber is called minimal if its mean curvature vector field η vanishes identically. Proposition 8.12 Under the hypothesis of Lemma 8.4, if the fibers of π are totally umbilical, then either dim Hx = 0 or 1 for x ∈ M, or π has minimal fibers. The next theorem was obtained in [38, 63]. Theorem 8.13 Let π : M → B be a a Lagrangian submersion from a Kaehler manifold (M, g M , J ) onto a Riemannian manifold (B, g B ) with totally umbilical fibers and with dim V > 1. Then the fibers are totally geodesic.

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The following result was proved in [38]. Theorem 8.14 Let π : M → B be a Lagrangian submersion with totally geodesic fibers from a Kaehler manifold (M, g M , J ) onto a Riemannian manifold (B, g B ). Then (M, g M , J ) is an Einstein space if and only if the fibers and the base space (B, g B ) are Einstein spaces.

9 Semi-invariant Submersions First, we mention the following definition from [56]. Definition 9.1 Let π : (M, g M , J ) → (B, g B ) be a Riemannian submersion from an almost Hermitian manifold (M, g M , J ) onto a Riemannian manifold (B, g). Then π is called a semi-invariant submersion if there is a distribution DT ⊂ V such that V = DT ⊕ D⊥ , J (DT ) = DT and J (D⊥ ) ⊂ H, where D⊥ is the orthogonal complementary distribution to DT in V. A semi-invariant submersion is called proper if it is neither an invariant submersion nor an anti-invariant submersion. For a semi-invariant submersion π : (M, g M , J ) → (B, g B ) given above, the horizontal distribution H is decomposed as H = J (D⊥ ) ⊕ μ,

(16)

where μ is the orthogonal complementary distribution of J (D⊥ ) in H and μ is invariant under J. The following results on semi-invariant submersions were proved in [56]. Proposition 9.2 Let π : M → B be a semi-invariant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then we have (1) The distribution D⊥ is always integrable. (2) The distribution DT is integrable if and only if g M (TX J Y − TY J X, J Z ) = 0 holds for vectors X, Y ∈ DT and Z ∈ D⊥ . Theorem 9.3 Let π : (M m (c), g M , J ) → (B, g) be a semi-invariant submersion with totally umbilical fibers from a complex space form M m (c) onto a Riemannian manifold B. Then c = 0. Theorem 9.4 Let π : (M, g M , J ) → (B, g B ) be a semi-invariant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then either D⊥ is onedimensional or the fibers are totally geodesic.

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The following example of semi-slant submersion was given in [56]. Example 9.5 Consider the map π : R6 → R3 defined by  π(x1 , x2 , x3 , x4 , x5 , x6 ) =

x1 + x2 x3 + x5 x4 + x6 √ , √ , √ 2 2 2

 .

Then it follows that the horizontal distribution H and vertical distribution V are given by H = Span{H1 , H2 , H3 }, V = Span{V1 , V2 , V3 }, where ∂ ∂ ∂ ∂ ∂ ∂ + , H2 = + , H3 = + , ∂ x1 ∂ x2 ∂ x3 ∂ x5 ∂ x4 ∂ x6 ∂ ∂ ∂ ∂ ∂ ∂ + , V2 = − + , V3 = − + . V1 = − ∂ x1 ∂ x2 ∂ x3 ∂ x5 ∂ x4 ∂ x6 H1 =

Thus J V2 = V3 and J V1 = −H1 . Hence we find D1 = Span{V2 , V3 } and D2 = Span{V1 }. By a direct verification that π is a semi-invariant submersion. Remark 9.6 S. Kobayashi defined CR-submersions in [37], which are different from semi-invariant submersions discussed in this section. Remark 9.7 A Riemannian submersion π : M → B from an almost Hermitian manifold onto a Riemannian manifold is called a generic submersion if there exists an invariant distribution DT in V and if its orthogonal complementary distribution D⊥ in V is purely real in the sense of [15], i.e., V = DT ⊕ D⊥ , J (DT ) = DT , and J (D⊥ ) ∩ D⊥ = {0}. For such generic submersions, see [33]. See [61] for a different notion of “generic submersions” via the generic submanifolds in the sense of [50].

10 Pointwise Slant Submersions The following definition of slant submersions was given in [54]. Definition 10.1 Assume that π : (M, g M , J ) → (B, g) is a Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold. If, at a given point x ∈ M and for a given unit vector X ∈ Vx , the Wirtinger angle θ (X ) between J X and the vertical space Vx is independent of the choice of vector X ∈ Vx at x, then π is called a pointwise slant submersion. In this case, the function θ is called the slant function. In particular, for the pointwise slant submersion π , if the slant function θ of π is a global constant on M, then π is called a slant submersion and the constant θ is called the slant angle of the slant submersion. Definition 10.2 For a given pointwise slant submersion π : M → B, a point x ∈ M is called a totally real point if its slant function satisfies θ = π2 at x. Similarly, a point x ∈ M is called a complex point if we have θ = 0 at x. A pointwise slant submersion is called proper if it is free from complex and totally real points.

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Suppose that π : M → B is a Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold. For any given vertical vector V of π , let us put J V = ϕV + ωV,

(17)

where ϕV and ωV denote the vertical and horizontal component of J V . We have the following result from [39]. Proposition 10.3 Let π : M → B be a Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold. Then the submersion π is called a pointwise slant submersion if and only if there exists a real-valued function θ defined on V which satisfies the condition: ϕ 2 = −(cos2 θ )I. For pointwise slant submersions, we have the following results from [39]. Theorem 10.4 Under the hypothesis of Proposition 10.3, the Riemannian submersion π is pointwise slant if and only if ϕ : V → V preserves orthogonality. Hence, ϕ carries each pair of orthogonal vectors in V into orthogonal vectors in V. Theorem 10.5 Under the hypothesis of Proposition 10.3, if the Riemannian submersion π has totally geodesic fibers, then it is slant. Theorem 10.6 Suppose that a Riemannian submersion π : M → B is a pointwise slant submersion with totally umbilical fibers from a Kaehler manifold onto a Riemannian manifold. If π has no totally geodesic fibers, then it is non-slant. The second fundamental form σφ of a map φ : (M, g M ) → (N , g N ) between two Riemannian manifolds is defined by σφ (X, Y ) = ∇¯ X φ∗ (Y ) − φ∗ (∇ X Y ), for vector fields X, Y tangent to M, where ∇ is the Levi-Civita connection of M and ∇¯ is the pullback of the connection ∇  of N to the induced vector bundle φ −1 (T N ). The tension field τφ of a map φ : (M, g M ) → (N , g N ) is the trace of σφ . A map φ : (M, g M ) → (N , g N ) is called a harmonic map if its tension field τφ vanishes identically (see, e.g., [45]). For slant submersions, we have the next result form [54]. Theorem 10.7 Assume that π : M → B is a slant submersion from a Kaehler manifold onto a Riemannian manifold. If the ω defined by (17) is parallel, then the slant submersion π is a harmonic map.

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11 Bi-Slant Submersions The notion of bi-slant submersions was given in [60] as follows. Definition 11.1 Let π : (M, g M , J ) → (B, g B ) be a Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold. Then π is called a bislant submersion if the vertical distribution V of π admits two orthogonal slant distributions Dθ1 and Dθ2 such that V = Dθ1 ⊕ Dθ2 , where Dθ1 and Dθ2 have slant angles θ1 and θ2 , respectively. The bi-slant submersion π : M → B is called proper if we have θ1 , θ2 ∈ (0, π2 ). For the bi-slant submersion π : (M, g M , J ) → (B, g B ), the fibers of π are said to be (Dθ1 , Dθ2 )-mixed geodesic if the O’Neill tensor T satisfies TX U = 0 for X ∈ Dθ1 and U ∈ Dθ2 . Remark 11.2 For a bi-slant submersion π : (M, g M , J ) → (B, g B ), let m 1 and m 2 denote the dimension of the two slant distributions Dθ1 and Dθ2 , respectively. Bi-slant submersions include anti-invariant, invariant, semi-slant, and slant submersions as special cases. In fact, we have (a) If m 1 = 0 and θ2 = 0, then π is an invariant submersion (see [53]). (b) If m 1 = 0 and θ = π2 , then π is an anti-invariant submersion (see [53]). (c) If m 1 , m 2  = 0, θ1 = 0 and θ2 = π2 , then π is a semi-invariant submersion (see [57]). (d) If m 1 = 0 and θ2 ∈ (0, π2 ), then π is a proper slant submersion (see [54]). (e) If m 1 , m 2  = 0, θ1 = 0 and θ2 ∈ (0, π2 ), then π is a semi-slant submersion (see [47]). (f) If m 1 , m 2  = 0, θ1 = π2 and θ2 ∈ (0, π2 ), then π is a hemi-slant submersion (see [65]). Sayar, Akyol and Prasad [60] investigated the integrability of Dθ1 for a bi-slant submersion. They obtained the following. Theorem 11.3 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the slant distribution Dθ1 is integrable if and only if g(TPU FY − TU F PY + A FU FY, X ) = g(TPU F X − TU F P X − A FU F X, Y ) for vector fields X, Y in Dθ1 and U in Dθ2 . Sayar, Akyol and Prasad also studied the conditions for a slant distribution to be totally geodesic and obtained the following.

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Theorem 11.4 [60] Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the slant distribution Dθ1 defines a totally geodesic foliation on V if and only if g(TPU FY − TU F PY + A FU FY, X ) = 0,

(18)

for vector fields X, Y in Dθ1 and U in Dθ2 . Remark 11.5 Clearly, statements analogous to Theorems 11.3 and 11.4 work for the other slant distribution Dθ2 as well. In [60], Sayar, Akyol and Prasad also proved the following. Theorem 11.6 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the vertical distribution V defines a totally geodesic foliation if and only if ω(TW P V + A F V W ) + F(∇ˆ W P V + TW F V ) = 0

(19)

for vertical vector fields V, W , where ∇ˆ X Y denotes the vertical component of ∇ XM Y and ∇ M is the Levi-Civita connection of M. Theorem 11.7 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the horizontal distribution H defines a totally geodesic foliation if and only if φ(A X ϕY + H∇ X ωY ) + P(A X ωY + V∇ X ϕY ) = 0

(20)

for horizontal vector fields X, Y . Theorems 11.6 and 11.7 imply the following. Theorem 11.8 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the total space M is locally the Riemannian product MV × MH of a leaf MV of V and MH of H if and only if conditions (19) and (20) are satisfied. The following two results were also obtained in [60]. Theorem 11.9 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then F is parallel if and only if ϕ is parallel. Theorem 11.10 Let π : (M, g, J ) → (B, g  ) be a bi-slant submersion with slant angle (θ1 , θ2 ) from a Kaehler manifold onto a Riemannian manifold such that the canonical structure F given by (8) is parallel. Then the fibers of π are (Dθ1 , Dθ2 )mixed geodesic.

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12 Hemi and Semi-slant Submersions 12.1 Hemi-slant Submersions A Riemannian submersion π : M → B from a Kaehler manifold M onto a Riemannian manifold B is called hemi-slant if its vertical distribution V admits a slant distribution Dθ whose orthogonal complementary distribution D⊥ in V is anti-invariant (see, e.g., [47]). For semi-slant submersions, we have the following three results obtained by Ta¸stan, Sahin ¸ and Yanan in [65]. Theorem 12.1 Let π : M → B be a hemi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then its anti-invariant distribution D⊥ is integrable. Theorem 12.2 Let π : M → B be a hemi-slant submersion from a Kaehler manifold onto a Riemannian manifold. If ω is parallel with respect to the connection ∇ on V, then π is a harmonic map. Theorem 12.3 Let π : M → B be a hemi-slant submersion from a Kaehler manifold onto a Riemannian manifold. If π has totally umbilical fibers, then either the anti-invariant distribution D⊥ is 1-dimensional or the mean curvature vector H of a fiber π −1 (y) is perpendicular to J (D⊥ ) for y ∈ B. Moreover, if ϕ is parallel, then H ∈ μ. Further, if ω is parallel, then T = 0.

12.2 Semi-slant Submersions Recall that a Riemannian submersion π : M → B from an almost Hermitian manifold onto a Riemannian manifold is called semi-slant if its vertical distribution V admits an invariant distribution DT , i.e., J DT ⊂ DT , such that its orthogonal complementary distribution Dθ in V is a θ -slant distribution. The angle θ associated with the semi-slant submersion π is called the semi-slant angle (cf. [47]). For a semi-slant submersion π : M → B, the semi-slant angle θ satisfies ϕ 2 X = − cos2 θ X

(21)

for any unit vector X in the slant distribution Dθ . Therefore, if θ ∈ [0, π2 ), the base manifold B must be even-dimensional (see [47]). Park and Prasad studied in [47] the integrability conditions of the distributions DT and Dθ and proved the following.

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Theorem 12.4 Let π : (M, g M , J ) → (N , g N ) be a semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. Then the invariant distribution DT is integrable if and only if ω(∇ˆ X Y − ∇ˆ Y X ) = C(TY X − TX Y ) for vector field X, Y in the invariant distribution DT , where ∇ˆ X Y denotes the vertical component of ∇ XM Y and ∇ M is the Levi-Civita connection of M. Theorem 12.5 Let π : (M, g M , J ) → (N , g N ) be a semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. Then the slant distribution Dθ is integrable if and only if P(ϕ(∇ˆ X Y − ∇ˆ Y X ) + B(TX Y − TY X )) = 0 for vector fields X, Y in the slant distribution Dθ . Park and Prasad also studied in [47] the conditions for DT and Dθ to be totally geodesic distribution. They obtained the following. Theorem 12.6 Let π : (M, g M , J ) → (N , g N ) be a semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. Then the vertical distribution V defines a totally geodesic foliation if and only if ω(∇ˆ V ϕW + TV ωW ) + C(TV ϕW + H∇ X ωY ) = 0 for vertical vector fields V, W of π. Theorem 12.7 Let π : (M, g M , J ) → (N , g N ) be a semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. Then the horizontal distribution H defines a totally geodesic foliation if and only if φ(∇ˆ X BY + TX CY ) + B(TX BY + H∇ X CY ) = 0 for horizontal vector fields V, W of π. Park and Prasad [47] also obtained the condition for a semi-slant submersion to be a totally geodesic map. Theorem 12.8 π : (M, g M , J ) → (N , g N ) be a semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. Then π is totally geodesic map if and only if ω(∇ˆ V ϕW + TV ωW ) + C(TV ϕW + H∇V ωW ) = 0, ω(∇ˆ V B X + TV C X ) + C(TV B X + H∇V C X ) = 0, for vertical vector fields V, W and horizontal vector field X .

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Park and Prasad also proved in [47] the following. Theorem 12.9 Let π : (M, g M , J ) → (B, g B ) be a semi-slant submersion from a Kaehler manifold onto a Riemannian manifold. If DT is integrable, then π is a harmonic map if and only if Trace (∇π∗ ) = 0 on Dθ . Several examples of semi-slant submersions were also constructed by Park and Prasad in [47].

13 Quasi Bi-Slant Submersions Quasi bi-slant submersions are defined in [48] as follows. Definition 13.1 Let π : (M, g, J ) → (B, g  ) be a Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold. Then π is called a quasi bi-slant submersion if the vertical distribution V admits three mutually orthogonal distributions DT , Dθ1 and Dθ2 such that V = DT ⊕ Dθ1 ⊕ Dθ2 , J DT = DT , J Dθ1 ⊥ Dθ2 , where Dθ1 , Dθ2 are slant distributions with slant angles θ1 , θ2 , respectively. The pair (θ1 , θ2 ) is simply called the bi-slant angle of π . A quasi bi-slant submersion is called proper if DT , Dθ1 and Dθ2 are of positive dimension and θ1 , θ2 ∈ (0, π2 ). Let π : (M, g, J ) → (B, g  ) be a quasi bi-slant submersion. For a vertical vector field V , we put V = P V + QV + RV,

(22)

where P, Q, and R are the projection of the vertical distribution V onto DT , Dθ1 and Dθ2 , respectively. As before, we put J V = ϕV + ωV,

(23)

where ϕV and ωV are the vertical and the horizontal components of J V . In [48], Prasad, Shukla, and Kumar studied the integrability conditions of DT and Dθ1 and proved the following. Theorem 13.2 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then DT is integrable if and only if g(TW φV − TV φW, ωQ Z + ω R Z ) = g(V∇V ϕW − V∇W ϕV, ϕ Q Z + ϕ R Z ) for V, W in DT and Z in Dθ1 ⊕ Dθ2 .

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Theorem 13.3 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then Dθ1 is integrable if and only if g(TZ ωϕW − TW ωϕ Z , U ) = g(TZ ωW − TW ωZ , J PU + ϕ RU ) + g(H∇ Z ωW − H∇W ωZ , ω RU ) for Z , W in DT and U in DT ⊕ Dθ2 . Similar condition works for the other slant distribution Dθ2 . Prasad, Shukla, and Kumar [48] also studied the conditions for H and V to be totally geodesic. In this respect, they obtained the following. Theorem 13.4 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold onto a Riemannian manifold. Then the horizontal distribution H defines a totally geodesic foliation if and only if g(A X Y, P V + cos2 θ1 QV + cos2 θ2 RV ) = g(H∇ X Y, ωϕ P V + ωϕ QV + ωϕ RV ) + g(A X BY + H∇ X CY, ωV )

(24)

for horizontal vector fields X, Y and vertical vector field V . Theorem 13.5 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then the vertical distribution V defines a totally geodesic foliation if and only if g(TU V, X ) + cos2 θ1 g(TU QV, X ) + cos2 θ2 g(TU RV, X ) = g(H∇U ωϕ P V + H∇U ωϕ QV + H∇U ωϕ RV, X ) + g(TU ωV, B X ) + g M (H∇U ωV, C X )

(25)

for horizontal vector field X and vertical vector fields U, V . Theorems 13.4 and 13.5 imply the following. Theorem 13.6 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then the total space M is locally the Riemannian product MV × MH of a leaf MV of V and a leaf MH of H if and only if (24) and (25) are satisfied. For DT and Dθ1 , Prasad, Shukla, and Kumar [48] proved the following. Theorem 13.7 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then DT defines a totally geodesic foliation if and only if g(TX ϕ PY, ωQ Z + ω R Z ) = −g(V∇ X ϕ PY, ϕ Q Z + ϕ R Z ),

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and g(TX ϕ PY, C H ) = −g(V∇ X ϕ PY, B H ), for vector fields X, Y in DT , Z in Dθ1 ⊕ Dθ2 , and horizontal vector field H . Theorem 13.8 Let π : (M, g, J ) → (B, g  ) be a proper quasi bi-slant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then the slant distribution Dθ1 defines a totally geodesic foliation if and only if g(TZ ωϕW, X ) = g(TZ ωQW, J P X + ϕ R X ) + g(H∇ Z ωQW, ω R X ) and g(H∇ Z ωϕW, H ) = g(H∇ Z ωW, C H ) + g(TZ ωW, B H ) for vector fields Z , W in Dθ1 , X in DT ⊕ Dθ2 , and horizontal vector field H . The following examples are given in [48]. Example 13.9 Define a map π : C7 → E6 by π(x1 , . . . , x14 ) = (x3 sin θ1 − x5 cos θ1 , x6 , x7 sin θ2 − x9 cos θ2 , x10 , x13 , x14 ). The π is a quasi bi-slant submersion with bi-slant angle (θ1 , θ2 ) such that V = DT ⊕ D1 ⊕ D2 with DT = Span{X 1 , X 2 , X 7 , X 8 }, D1 = Span{X 3 , X 4 }, D2 = Span{X 5 , X 6 }, where ∂ ∂ ∂ ∂ ∂ , X2 = , X 3 = cos θ1 + sin θ1 , X4 = , ∂ x1 ∂ x2 ∂ x3 ∂ x5 ∂ x4 ∂ ∂ ∂ ∂ ∂ + sin θ2 , X6 = , X7 = , X8 = . X 5 = cos θ2 ∂ x7 ∂ x9 ∂ x8 ∂ x11 ∂ x12 X1 =

It is direct to verify that the horizontal distribution H is spanned by ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , sin θ1 − cos θ1 , sin θ2 − cos θ2 , , , . ∂ x6 ∂ x3 ∂ x5 ∂ x7 ∂ x9 ∂ x10 ∂ x13 ∂ x14 Example 13.10 Define a map π : C6 → E6 by  π(x1 , . . . , x12 ) =

x1 − x3 √ , x2 , 2

 √ 3x7 − x9 , x10 , x11 , x12 2

The π is a quasi bi-slant submersion with bi-slant angle (θ1 , θ2 ) = V = DT ⊕ D1 ⊕ D2 with

π 4

 , π3 such that

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DT = Span{X 3 , X 4 }, D1 = Span{X 1 , X 2 }, D2 = Span{X 5 , X 6 }, where   1 ∂ ∂ ∂ ∂ X1 = √ + , X3 = , , X2 = ∂ x3 ∂ x4 ∂ x5 2 ∂ x1   √ ∂ ∂ 1 ∂ ∂ , X5 = + 3 . , X6 = X4 = ∂ x6 2 ∂ x7 ∂ x9 ∂ x8 The horizontal distribution H is spanned by 1 ∂ , √ ∂ x2 2



∂ ∂ + ∂ x1 ∂ x3



  ∂ ∂ ∂ 1 √ ∂ ∂ 3 − , , . , , 2 ∂ x7 ∂ x9 ∂ x10 ∂ x11 ∂ x12

14 Conformal Submersions We need the following definition from [8]. Definition 14.1 Let ψ : (M, g M ) → (N , g N ) be smooth map from a Riemannian manifold (M, g M ) onto another Riemannian manifold (N , g N ). Then ψ is called horizontally conformal if there exists a positive function λ such that ψ∗ maps each horizontal space Hx , x ∈ M, conformally onto Tψ(x) N , i.e., for any horizontal vectors X, Y in Hx , x ∈ M, we have λ2 g M (X, Y ) = g N (ψ∗ (X ), ψ∗ (Y )). In this case, the function λ is called the dilation of ψ.

14.1 Conformal Slant Submersions Definition 14.2 Let φ : M → N be a smooth map between Riemannian manifolds. Then φ is called a harmonic morphism if, for each harmonic function f : D → R defined on an subset D of N with φ −1 (C) non-empty, the composition f ◦ φ is harmonic on φ −1 (D) (cf. [8]). Definition 14.3 Let π : (M, g M , J ) → (N , g N ) be a horizontally conformal submersion from an almost Hermitian manifold M onto a Riemannian manifold N . If, for any non-zero vertical vector X ∈ Vx , x ∈ M, the angle θ (X ) between J X and the vertical space Vx is a constant, i.e., it is independent of the choice of the point x ∈ M and the choice of the vertical vector X ∈ Vx , then π is called a conformal slant submersion. In this case, the angle θ is called the slant angle of the confor-

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mal slant submersion (cf. [4]). A conformal slant submersion π : M → N is called horizontally homothetic if the gradient of its dilation λ is vertical, i.e., H(grad λ) = 0. The following result for a conformal slant submersion to be harmonic morphism was obtained in [4] by M. A. Akyol and B. Sahin. ¸ Theorem 14.4 Let π : (M 2m+r , g M , J ) → (B m+2r , g B ) be a conformal slant submersion from a Kaehler manifold M 2m+r onto a Riemannian manifold B m+2r such 2 . Then any two conditions below imply the third: that λ2  = m+2r (a) π a harmonic morphism. (b) ω is parallel with respect to the connection ∇ on V. (c) π is horizontally homothetic. Akyol and Sahin ¸ also studied the geometry of foliations arisen from a conformal slant submersion and derived a decomposition theorem on the total space. Further, they obtained the necessary and sufficient conditions for a conformal slant submersion to be totally geodesic.

14.2 Conformal Anti-invariant Submersions Definition 14.5 [2] A conformal slant submersion π : M → B from an almost Hermitian manifold M onto a Riemannian manifold B is called a conformal antiinvariant submersion it its vertical distribution V is anti-invariant, i.e., J (V) ⊂ V ⊥ . The following conditions for a conformal anti-invariant submersion to be a harmonic map were obtained in [4]. Theorem 14.6 Let π : (M, g M , J ) → (B, g B ) be a conformal anti-invariant submersion from a Kaehler manifold M onto a Riemannian manifold B. Then any two conditions below imply the third: (a) π a harmonic map. (b) The fibers are minimal. (c) π is a horizontally homothetic map. Akyol and Sahin ¸ also studied in [2] the geometry of the foliations arisen from conformal anti-invariant submersions. They obtained the necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesic. Further, they derived curvature relations between the base space and the total space, and found geometric implications of these relations.

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14.3 Conformal Semi-slant Submersions The next definition was given in [3]. Definition 14.7 A horizontal conformal submersion π : (M, g M , J ) → (N , g N ) from an almost Hermitian manifold M onto a Riemannian manifold N is called conformal semi-slant if there exists a distribution DT ⊂ V such that V = DT ⊕ D⊥ , J (DT ) = DT , J (D⊥ ) ⊂ (V)⊥ , where D⊥ is the orthogonal complementary to DT in V. In [3], Akyol and Sahin ¸ investigated geometry of foliations arisen from conformal semi-slant submersions. They proved that there are certain product structures on the total space and derived the necessary and sufficient conditions of a conformal semiinvariant submersion to be totally geodesic. Moreover, they proved the following result in [3]. Theorem 14.8 Let π : (M 2(m+n+r) , g M , J ) → (B n+2r , g B ) be a conformal semislant submersion from a Kaehler manifold M onto a Riemannian manifold B. If n + 2r  = 2, then any two conditions below imply the third: (a) π a harmonic map. (b) The fibers are minimal. (c) π is a horizontally homothetic map. Remark 14.9 For further results in this respect, see also [1, 5].

References 1. Akyol, M.A.: Conformal semi-slant submersions. Int. J. Geom. Methods Mod. Phys. 14, Article 1750114, 25 pp (2017) 2. Akyol, M.A., Sahin, ¸ B.: Conformal anti-invariant submersions from almost Hermitian manifolds. Turk. J. Math. 40, 43–70 (2016) 3. Akyol, M.A., Sahin, ¸ B.: Conformal semi-invariant submersions. Commun. Contemp. Math. 19, Article 1650011, 22 pp (2017) 4. Akyol, M.A., Sahin, ¸ B.: Conformal slant submersions. Hacet. J. Math. Stat. 48, 28–44 (2019) 5. Akyol, M.A., Sahin, ¸ B.: Conformal semi-invariant Riemannian maps to Kähler manifolds. Revista Un. Mat. Argentina 60, 439–468 (2019) 6. Alegre, P., Chen, B.-Y., Munteanu, M.I.: Riemannian submersions, δ-invariants, and optimal inequality. Ann. Global Anal. Geom. 42, 317–331 (2012) 7. Ali, S., Fatima, T.: Anti-invariant Riemannian submersions from nearly Kaehler manifolds. Filomat 27(7), 1219–1235 (2013) 8. Baird, P., Wood, J.C.: Harmonic Morphisms between Riemannian Manifolds. The Clarendon Press, Oxford University Press, Oxford (2003) 9. Besse, A.: Einstein Manifolds. Springer, Berlin (1987)

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10. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Riemannian submersions and slant submanifolds. Publ. Math. Debrecen 61, 523–532 (2002) 11. Carriazo, A.: Bi-slant immersions. in: Proc. ICRAMS 2000, Kharagpur, India, 88–97 (2000) 12. Chen, B.Y.: Totally umbilical submanifolds. Soochow J. Math. 5, 9–37 (1979) 13. Chen, B.Y.: Geometry of Submanifolds. Marcel Dekker, New York, NY (1973) 14. Chen, B.Y.: Geometry of Submanifolds and Its Applications. Science University of Tokyo, Japan (1981) 15. Chen, B.Y.: Differential geometry of real submanifolds in a Kähler manifold. Monatsh. Math. 91, 257–274 (1981) 16. Chen, B.Y.: Slant immersions. Bull. Austral. Math. Soc. 41, 135–147 (1990) 17. Chen, B.Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven, Belgium (1990) 18. Chen, B.Y.: Riemannian submersions, minimal immersions and cohomology class. Proc. Japan Acad. Ser. A Math. Sci. 81(10), 162–167 (2005) 19. Chen, B.Y.: Riemannian submersions with minimal fibers and affine hypersurfaces. Monatsh. Math. 151(2), 143–152 (2007) 20. Chen, B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific Publ, Hackensack, NJ (2011) 21. Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. World Scientific Publ, Hackensack, NJ (2015) 22. Chen, B.Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific Publ, Hackensack, NJ (2017) 23. Chen, B.Y., Garay, O.J.: Pointwise slant submanifolds in almost Hermitian manifolds. Turkish J. Math. 36, 63–640 (2012) 24. Chen, B.Y., Tazawa, Y.: Slant submanifolds of complex projective and complex hyperbolic spaces. Glasgow Math. J. 42, 439–454 (2000) 25. Chen, B.Y., Uddin, S.: Slant geometry of warped products in Kaehler and nearly Kaehler manifolds. In: Complex Geometry of Slant Submanifolds, pp. 61–100, Springer (2022) 26. Deshmukh, S., Ali, S., Husain, S.I.: Submersions of CR-submanifolds of a Kaehler manifold. Indian J. Pure Appl. Math. 19(12), 1185–1205 (1988) 27. Escobales, R.: Submersions from spheres. Bull. Am. Math. Soc. 79, 71–74 (1973) 28. Escobales, R.: Riemannian submersions with totally geodesic fibers. J. Differ. Geom. 10, 253– 276 (1975) 29. Escobales, R.: Riemannian submersions from complex projective space. J. Differ. Geom. 13, 93–107 (1978) 30. Escobales, R.: Sufficient conditions for a bundle-like foliation to admit a Riemannian submersion onto its leaf space. Proc. Am. Math. Soc. 84, 280–284 (1982) 31. Etayo, F.: On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. (Debrecen) 53, 217–223 (1998) 32. Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific Publ, River Edge, NJ (2004) 33. Fatima, T., Ali, S.: Submersions of generic submanifolds of a Kaehler manifold. Arab J. Math. Sci. 20, 119–131 (2014) 34. Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967) 35. Hermann, R.: A sufficient condition that a mapping of Riemannian manifolds be fibre bundle. Proc. Am. Math. Soc. 11, 236–242 (1960) 36. Johnson, D.: Kähler submersions and holomorphic connections. J. Diff. Geom. 15, 71–79 (1980) 37. Kobayashi, S.: Submersions of CR submanifolds. Tohoku Math. J. 39, 95–100 (1987) 38. Lee, J.C., Park, J.H., Sahin, ¸ B., Song, D,-Y.: Einstein conditions for the base space of antiinvariant Riemannian submersions and Clairaut submersions. Taiwanese J. Math. 19, 1145– 1160 (2015)

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39. Lee, J.W., Sahin, ¸ B.: Pointwise slant submersions. Bull. Korean Math. Soc. 51, 1115–1126 (2014) 40. Nagano, T.: On fibred Riemann manifolds. Sci. Papers Coll. Gen. Ed. Univ. Tokyo 10, 17–27 (1960) 41. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 65, 391–404 (1957) 42. O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966) 43. O’Neill, B.: Submersions and geodesics. Duke Math. J. 34, 363–373 (1967) 44. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic, New York (1983) 45. Ou, Y.-L., Chen, B.-Y.: Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry. World Scientific Publ, Hackensack, NJ (2020) 46. Papaghiuc, N.: Submersions of semi-invariant submanifolds of a Sasakian manifold, An. Stiint. Univ "Al. I. Cuza" Ia¸si, Sect. I, Mat. 35, 281–288 (1989) 47. Park, K.-S., Prasad, R.: Semi-slant submersions. Bull. Korean Math. Soc. 50, 951–962 (2013) 48. Prasad, R., Shukla, S. S., Kumar, S.: On quasi-bi-slant submersions. Mediterr. J. Math. 16, Article 155, 18 pp (2019) 49. Ranjan, A.: Riemannian submersions of spheres with totally geodesics fibers. Osaka J. Math. 22, 243–260 (1985) 50. Ronsse, G.B.: Generic and skew CR-submanifolds of a Kaehler manifold. Bull. Inst. Math. Acad. Sinica 18, 127–141 (1990) 51. Rovenskii, V.Y.: Classes of submersions of Riemannian manifolds with compact fibers. Siberian Math. J. 35 1027–1035 (1994); translated from Sibirsk Mat. Zh., 35, 1154–1164 (1994) 52. Sahin, ¸ B.: Horizontally conformal submersions of CR-submanifolds. Kodai Math. J. 31, 46–53 (2008) 53. Sahin, ¸ B.: Anti-invariant Riemannian submersions from almost Hermitian manifolds. Cent. Eur. J. Math. 8, 437–447 (2010) 54. Sahin, ¸ B.: Slant submersions from almost Hermitian manifolds. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102) (2011), no. 1, 93–105 55. Sahin, ¸ B.: Biharmonic Riemannian maps. Ann. Polon. Mat. 102, 39–49 (2011) 56. Sahin, ¸ B.: Riemannian submersions from almost Hermitian manifolds. Taiwanese J. Math. 17, 629–659 (2013) 57. Sahin, ¸ B.: Semi-invariant submersions from almost Hermitian manifolds. Canad. Math. Bull. 56, 173–183 (2013) 58. Sahin, ¸ B.: Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and Their Applications. Elsevier/Academic, London (2017) 59. Sayar, C.: Generalized Skew Semi-invariant Submersions. Mediterr. J. Math. 17(3), Article 81 (2020) 60. Sayar, C., Akyol, M. A., Prasad, R.: Bi-slant submersions in complex geometry. Int. J. Geom. Methods Mod. Phys. 17(4), Article 2050055, 17 pp (2020) 61. Sayar, C., Ta¸stan, H.M., Özdemir, F., Tripathi, M.M.: Generic submersions from Kaehler manifolds. Bull. Malays. Math. Sci. Soc. 43, 809–831 (2020) 62. Shahid, M.H., Al-Solamy, F.R.: Submersion of quaternion CR-submanifolds of a quaternion Kaehler manifold. Math. J. Toyama Univ. 23, 93–113 (2000) 63. Ta¸stan, H.M.: On Lagrangian submersions. Hacet. J. Math. Stat. 43(6), 993–1000 (2014) 64. Ta¸stan, H.M.: Biwarped product submanifolds of a Kähler manifold. Filomat 32(7), 2349–2365 (2018) 65. Ta¸stan, H.M., Sahin, ¸ B., Yanan, S: ¸ Hemi-slant submersions. Mediterr. J. Math. 13, 2171–2184 (2016) 66. Watson, B.: Almost Hermitian submersions. J. Differ. Geom. 11, 147–165 (1976)

Slant Submanifolds of the Nearly Kaehler 6-Sphere Luc Vrancken

2000 Mathematics Subject Classification Primary 53B20; Secondary 53B21 · 53B25

1 Introduction By a result of Butruille, [4], the nearly Kaehler 6-sphere is one of 4 homogeneous 6-dimensional strict nearly Kaehler manifolds. Its almost complex structure is introduced by considering R7 as the imaginary Cayley numbers. This almost complex structure is compatible with the usual metric on S 6 (1). Details of the construction will be given in Sect. 2. It was shown by Calabi and Gluck, see [5], that this structure, from a geometric viewpoint, is the best possible almost complex structure on S 6 (1). Note that recently Foscolo and Haskins constructed a non-homogeneous nearly Kaehler structure on S 6 . In order to study submanifolds in this space, we have to look at how the almost complex structure J behaves with respect to the submanifold. For example, it is natural to study submanifolds for which J maps the tangent space into the tangent space (and hence also the normal space into the normal space) and those for which J maps the tangent into the normal space. The first class is called almost complex submanifolds and the second class of submanifolds mentioned is called totally real submanifolds. Both of these classes can be considered as special cases of so-called slant submanifolds. These were introduced by Chen in [6]. A slant submanifold M is defined as a submanifold such that, for any nonzero vector X ∈ T p M, the angle θ (X ) between L. Vrancken (B) Université Polytechnique Hauts de France, LMI, 59313 Valenciennes, France Departement Wiskunde, KU Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_5

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J X and the tangent space T p M is a constant, which is independent of the choice of the point p of M and the choice of the tangent vector X in the tangent plane. It is obvious that complex submanifolds and totally real submanifolds are special classes of slant submanifolds. A slant submanifold is called proper if it is neither a complex submanifold nor a totally real submanifold.

2 Preliminaries We give a brief exposition of how the standard nearly Kaehler structure on S 6 (1) arises in a natural manner from the Cayley multiplication. For further details about the Cayley numbers and their automorphism group G 2 , we refer the reader to [9, 11]. The multiplication on the Cayley numbers O may be used to define a vector cross product × on the purely imaginary Cayley numbers R7 using the formula u×v =

1 (uv − vu), 2

(1)

while the standard inner product on R7 is given by 1 u, v = − (uv + vu). 2

(2)

It is now elementary [9] to show that u × (v × w) + (u × v) × w = 2u, wv − u, vw − w, vu,

(3)

and that the triple scalar product u × v, w is skew-symmetric in u, v, w. From this, it also follows that u × v, u × w = u, uv, w − u, vu, w.

(4)

The Cayley multiplication on O is given in terms of the vector cross product and the inner product by (r + u)(s + v) = r s − u, v + r v + su + u × v, r, s ∈ Re(O), u, v ∈ I m(O). (5) In view of (1), (2) and (5), it is clear that the group G 2 of automorphisms of O is precisely the group of isometries of R7 preserving the vector cross product. The associative 3-form φ on R7 is defined by φ(x, y, z) = x, y.z,

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whereas the associator is given by [x, y, z] = (x y)z − x(yz). An ordered basis e0 , ..., e6 is said to be a G 2 -frame if e2 = e0 × e1 , e4 = e0 × e3 , e5 = e1 × e3 , e6 = e2 × e3 .

(6)

For example, the standard basis e0 , . . . , e6 of R7 is a G 2 -frame. Two G 2 -frames are related by a unique element of G 2 . Moreover, if e0 , e1 , e3 are mutually orthogonal unit vectors with e3 orthogonal to e0 × e1 , then e0 , e1 , e3 determine a unique G 2 -frame e0 , . . . , e6 and (R7 , ×) is generated by e0 , e1 , e3 subject to the relations : ei × (e j × ek ) + (ei × e j ) × ek = 2δik e j − δi j ek − δ jk ei .

(7)

Therefore, for any G 2 -frame, we have the following very useful multiplication table [11] : × e0 e1 e2 e3 e4 e5 e6

e0 0 −e2 e1 −e4 e3 e6 −e5

e1 e2 0 −e0 −e5 −e6 e3 e4

e2 −e1 e0 0 −e6 e5 −e4 e3

e3 e4 e5 e6 0 −e0 −e1 −e2

e4 −e3 e6 −e5 e0 0 e2 −e1

e5 −e6 −e3 e4 e1 −e2 0 e0

e6 e5 −e4 −e3 e2 e1 −e0 0

The standard nearly Kaehler structure on S 6 (1) is then obtained as follows: J u = x × u, u ∈ Tx S 6 (1), x ∈ S 6 (1). It is clear that J is an orthogonal almost complex structure on S 6 (1). In fact, J is a nearly Kähler structure in the sense that the (2, 1)-tensor field G on S 6 (1) defined by G(X, Y ) = (∇˜ X J )Y, where ∇˜ is the Levi-Civita connection on S 6 (1), is skew-symmetric. A straightforward computation also shows that G(X, Y ) = X × Y − x × X, Y x, This tensor field has the following properties:

X, Y ∈ Tx S 6 (1).

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G(X, J Y ) + J G(X, Y ) = 0, ˜ (∇G)(X, Y, Z ) = Y, J Z X + X, Z J Y − X, Y J Z , G(X, Y ), Z  + G(X, Z ), Y  = 0, G(X, Y ), G(Z , W ) = X, Z Y, W  − X, W Z , Y  + J X, Z Y, J W  − J X, W Y, J Z . Then the Gauss, Codazzi and Ricci equations state that R(X, Y, Z , W ) = X, W Y, Z  − X, Z Y, W + + h(X, W ), h(Y, Z ) − h(X, Z ), h(Y, W ), (∇h)(X, Y, Z ) = (∇h)(Y, X, Z ), R ⊥ (X, Y )ξ, μ = [Aξ , Aμ ]X, Y .

(8) (9) (10)

Also the following lemma holds. Lemma 2.1 D X (Y × Z ) = D X Y × Z + Y × D X Z .

3 Slant Submanifolds For any vector X tangent to M, we put J X = P X + F X, where P X and F X are the tangential and the normal components of J X , respectively. Thus, P is an endomorphism of the tangent bundle T M and F a normal-bundlevalued 1-form on T M. It is a consequence of, for example, Proposition 7.4 of [7] that a submanifold of the nearly Kaehler 6-sphere is slant if and only if P 2 X = − cos2 θ X, where θ is the Wirtinger angle. As a consequence, following the construction of a slant frame, we get the following: Theorem 3.1 ([10]) There does not exist a 4-dimensional slant submanifold of the nearly Kaehler 6-sphere. Proof Note first that it is well known that 4-dimensional almost complex submanifolds do not exist. We start with e1 a unit tangent vector and construct a frame as follows. We take

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e2 = sec θ Pe1 , ξ1 = csc θ Fe1 , ξ2 = csc θ Fe2 . As ξ1 and ξ2 span the normal space, it then follows that G(e1 , e2 ) and J G(e1 , e2 ) are tangent vectors. This contradicts the fact that, because of the slant condition, J G(e1 , e2 ) must have non-trivial tangent and normal components. Note that the above argument of course remains valid in any 6-dimensional strict nearly Kaehler manifold. As a consequence of the previously mentioned Proposition 7.4, together with the previous theorem, we, therefore, must have the following: Theorem 3.2 Let M n be a slant submanifold of the nearly Kaehler S 6 (1), then either 1. n = 3 and M is a Lagrangian submanifold or 2. n = 2 and M is a slant surface in S 6 (1). Therefore, for the remainder of the paper, we will assume that M is a slant surface in S 6 (1). We will list all known examples of proper slant surfaces in S 6 (1). First we recall the following classification of [10]. Theorem 3.3 Let S 2 (r ) = π  ∩ S 6 be a 2-dimensional sphere of radius r. Denote by π the 3-dimensional plane parallel to the affine plane π  containing S 2 (r ). 1. If S 2 (1) is a great sphere, that is π = π  , then it is slant with the slant angle arccos φ(π). 2. If the plane π is associative, then the small sphere S 2 (r ) is slant with the slant angle arccos r . slant 3. If the plane π is not associative, then the small sphere S 2 (r ) is slant √ with the [π] . angle arccos(r φ(π)) if and only if its center is the point C = ± 1 − r 2 |[π ]| In the same paper [10], the authors also consider the orbits of flat tori. Due to the homogeniety, these are automatically slant and can be obtained starting from a point p = (x1 , x2 , x3 , y0 , y1 , y2 , y3 ), and its immersion is given by f (s, t) = (x1 , x2 cos t − x3 sin t, x3 cos t + x2 sin t, y0 cos s + y1 sin s, y1 cos s − y0 sin s, y2 cos(s − t) + y3 sin(s − t), y3 cos(s − t) − y2 sin(s − t)).

Note that for the orbit to be two-dimensional, we must have that at most one of the couples (x1 , x2 ), (y0 , y1 ), (y2 , y3 ) may vanish.This follows from the fact that W =  f s , f s  f t , f t  −  f s , f t 2 = (y02 + y12 )(y22 + y32 ) + (x22 + x32 )(y02 + y12 + y22 + y32 ). It is now straightforward to check that J fsW, ft  is independent of the choice of basis. Therefore, it follows that the slant angle is given by 2

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cos2 φ =

J f s , f t 2 3 = (3(x2 y0 y2 − x3 y1 y2 + x3 y0 y3 + x2 y1 y3 ))2 . W W

Note that the factor 3 in the numerator is missing in [10] which leads in [10] to the false claims that there are no almost complex orbits and that not all slant angles can be obtained. By choosing the initial point of the action properly, we may also assume that x2 = y0 = 0, and therefore, we get a 4 parameter family of examples x3 = y1 = y2 = y3 =

   

1 − x12 sin θ cos ϕ, 1 − x12 sin ϕ, 1 − x12 cos θ cos ϕ cos ψ, 1 − x12 cos θ cos ϕ sin ψ,

where the parameters x1 ∈] − 1, 1[, θ, ϕ ∈ [− π2 , π2 ] and ψ ∈ [0, 2π ]. Note that the immersion is minimal if and only if p=

√1 (0, 0, 1, 0, 1, cos ψ, sin ψ). 3

We get in this case that the slant angle satisfies cos2 θ = cos2 ψ. Note that in that case the tori are all contained in a totally geodesic S 5 (1) and have ellipse of curvature a circle centered at the origin. Recall that for a surface, the set {h(v, v)| v = 1} always describes an ellipse in the normal space with center the mean curvature vector of the immersion. Such slant surfaces were studied in [1]. It was shown that Theorem 3.4 Let f : M 2 → S 6 (1) be a proper slant minimal immersion with ellipse of curvature a circle. Then M 2 is contained in a totally geodesic S 5 (1) in S 6 (1). Of course, we may assume that the normal of S 5 (1) in S 6 (1) is given by e4 . In this case, we have the following characterization using the Hopf map. Theorem 3.5 Let f : M 2 → S 5 (1) ⊂ S 6 (1) be a slant minimal immersion with ellipse of curvature a circle. Suppose that e4 is the normal of S 5 (1) in S 6 (1). Denote by the Hopf map from S 5 (1) to C P 2 (4). Then ◦ f is a minimal Lagrangian immersion. Conversely, if M 2 is simply connected and g : M 2 → C P 2 (4) is a minimal Lagrangian immersion, then any horizontal lift is a slant minimal immersion with ellipse of curvature a circle. Moreover, by choosing the lift appropriately, all slant angles occur. Note that if we start with the flat Lagrangian torus in CP 2 (4), we obtain in this way the minimal tori of [10].

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We would like to conclude this survey with the following conjecture. Conjecture 3.6 Let f : M 2 → S 6 (1) be a proper slant minimal immersion. Then M 2 has ellipse of curvature a circle and must be obtained as described in the previous theorem. Note that the above result is not true for totally real immersions. A complete classification of totally real immersions with ellipse of curvature not a circle was obtained in [2]. Trying to apply the same technique in the proper slant case leads to an overdetermined system of differential equations for which the compatibility conditions become highly complicated. We will assume now that the surface is a proper slant surface. Then, starting with e1 a unit tangent vector, we can construct a frame as follows. We take e2 = sec θ Pe1 , ξ1 = csc θ Fe1 , ξ2 = csc θ Fe2 . From the properties of the nearly structure G, it follows that the remaining normals are given by η1 = csc G(e1 , e2 ), η2 = csc J G(e1 , e2 ). As G is skew-symmetric, it immediately follows: Lemma 3.7 Let M 2 be a proper slant surface in S 6 (1). Then η1 and η2 are globally defined normal vector fields on M. It, therefore, makes sense to look at the shape operators Aη1 and Aη2 with respect to these vector fields η1 and η2 . We then have the following theorems: Theorem 3.8 Let f : M 2 → S 6 (1) be a proper slant surface in S 6 (1). If Aη1 = Aη2 = 0, then M 2 is minimal with ellipse of curvature a circle. Moreover, f is obtained as constructed in Theorem 3.5. Theorem 3.9 There does not exist a proper slant surface in S 6 (1) such that Aη1 = 0 and Aη2 = μI , where μ  = 0. Theorem 3.10 Let f : M 2 → S 6 (1) be a proper slant surface in S 6 (1). If Aη1 = λI ,λ  = 0 and Aη2 = 0, then M 2 is non-minimal with ellipse of curvature a circle. Moreover, the slant angle θ and λ are related by λ = − tan θ. Moreover, f is contained in a small sphere and f can be written as

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f = cos θ g − sin θ ξ, where g is an almost complex surface in S 6 (1) contained in a totally geodesic S 5 (1) and ξ is the constant normal of S 5 (1) into S 6 (1). Remark 3.11 Note that such almost complex surfaces are obtained as described in Theorem 3.5. The above theorems can be proved using the previously mentioned orthonormal frame. As M has constant Kaehler angle, it follows that < h(X, Y ), J Z > is totally symmetric. As such, if necessary, by taking a suitable rotation, we may assume that < h(e1 , e1 ), J e2 >= 0. Therefore, taking into account the assumptions of the previous theorems, we may assume that the second fundamental form is given by h(e1 , e1 ) = aξ1 + λη1 + μη2 , h(e1 , e2 ) = bξ2 , h(e2 , e2 ) = bξ1 + cξ2 + λη1 + μη2 . If we denote the connection coefficients by ∇e1 e1 = αe2 and ∇e2 e2 = βe1 , the fact that M is a slant surface in the nearly Kaehler 6-sphere then implies that the derivatives of the normal vectors are given by ∇˜ e1 ξ1 = ξ2 (−a cot(θ ) + α − b cot(θ )) − ae1 + η2 λ csc(θ ) − η1 μ csc(θ ), ∇˜ e2 ξ1 = −be2 + ξ2 (−β − c cot(θ )) + η1 (−λ cot(θ ) − 1) − η2 μ cot(θ ), ∇˜ e1 ξ2 = ξ1 (a cot(θ ) − α + b cot(θ )) − be2 + η1 (λ cot(θ ) + 1) + η2 μ cot(θ ), ∇˜ e2 ξ2 = −be1 + ξ1 (β + c cot(θ )) − ce2 + η2 λ csc(θ ) − η1 μ csc(θ ), ∇˜ e1 η1 = η2 (−a − b) csc(θ ) + e1 (−λ) + ξ2 (−λ cot(θ ) − 1) + μξ1 csc(θ ), ∇˜ e2 η1 = −cη2 csc(θ ) + e2 (−λ) + ξ1 (λ cot(θ ) + 1) + μξ2 csc(θ ), ∇˜ e1 η2 = − csc(θ ) (−η1 (a + b) + e1 μ sin(θ ) + μξ2 cos(θ ) + λξ1 ) , ∇˜ e2 η2 = csc(θ ) (cη1 + μξ1 cos(θ ) − λξ2 ) − e2 μ. Applying now the Codazzi equation, it follows that λ(a + b) cot(θ ) + a + b − cμ csc(θ ) − e2 (λ) = 0, μ(a + b) cot(θ ) + cλ csc(θ ) − e2 (μ) = 0, aμ csc(θ ) + bμ csc(θ ) + cλ cot(θ ) + c + e1 (λ) = 0, − λ(a + b) csc(θ ) + cμ cot(θ ) + e1 (μ) = 0. Recall that depending on the theorem, we either have that 1. λ = μ = 0, 2. λ = 0 and μ  = 0, or

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3. λ  = 0 and μ = 0. In all three cases, it follows from the Codazzi equation that c = 0 and b = −a. From this, Theorem 3.10 is immediate as the immersion is now minimal with ellipse of curvature a circle. To conclude the proof of Theorem 3.9, we look at the Ricci equation corresponding to the normal vectors η1 and η2 . In this case ,this becomes 2μ2 cot(θ ) csc(θ ), from which a contradiction follows. For the final theorem, we look again at the Ricci equation corresponding to the normal vectors η1 and η2 . This yields   csc(θ ) 2λ2 cot(θ ) + 2λ . Hence, we deduce that λ = − tan θ . The Codazzi equation now reduces to e1 (a) = 3aβ, e2 (a) = 3aα. 1

1

From the integrability conditions of a, it then follows that the vector fields a 3 e1 a 3 e2 are coordinate vector fields on the surface. In we denote the original immersion by f , we also get that the map g = cos θ f + sin θ η1 is an almost complex immersion in S 6 (1) with ellipse of curvature a circle. Moreover, this almost complex immersion is contained in a totally geodesic S 5 (1). The normal vector of S 5 (1) is given by − sin θ f + cos θ η1 . As such the immersion g is obtained as described in Theorem 3.5. From this, we can now immediately deduce the expression of f .

References 1. Bolton, J., Vrancken, L., Woodward, L.M.: On almost complex curves in the nearly Kaehler 6-sphere. Quart. J. Math. Oxf. Ser. 2(45), 407–427 (1994) 2. Bolton, J., Vrancken, L., Woodward, L.M.: Totally real minimal surfaces with non-circular ellipse of curvature in the nearly Kaehler S 6 . J. Lond. Math. Soc. 56, 625–644 (1997) 3. Bryant, R.L.: Submanifolds and special structures on the octonians. J. Diff. Geom. 17, 185–232 (1982) 4. Butruille, J.: Homogeneous nearly Kaehler manifolds, Handbook of pseudo-Riemannian geometry and supersymmetry, pp. 399–423, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich (2010)

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5. Calabi, E., Gluck, H.: What are the best almost complex structures on the 6-sphere in Differential Geometry: geometry in mathematical physics and related topics. Am. Math. Soc. 99–106 (1993) 6. Chen, B.-Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Louvain, Belgium (1990) 7. Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific Publishing, Hackensack, NJ (2017) 8. Frölicher, A.: Zur Differentialgeometrie der komplexen Strukturen. Math. Ann. 129, 151–156 (1955) 9. Harvey, R., Lawson, H.B.: Calibrated Geometries. Acta Math. 148, 47–157 (1982) 10. Obrenovic, K., Vukmirovic, S.: Two classes of slant surfaces in the nearly Kaehler six sphere. Rev. Un. Mat. Argentina 54, 111–121 (2013) 11. Wood, R.M.W.: Framing the exceptional Lie group G 2 . Topology, 15, 303–320 (1976)

Slant Submanifolds of Para Hermitian Manifolds Pablo Alegre

2000 Mathematics Subject Classification. 53C15 · 53C25 · 53C40 · 53C50

1 Introduction In [11], Chen introduced slant submanifolds of an almost Hermitian manifold, as those submanifolds for which the angle between J X and the tangent space is constant, for any tangent vector field X . These submanifolds play an intermediate role between complex submanifolds and totally real ones. The study of slant submanifolds has produced an incredible amount of results and examples. There have been two remarkable lines of studies: studying what happens when the environment is changed and studying other types of submanifolds involving the slant case. For example, Cabrerizo, Carriazo, Fernández, and Fernández analyzed slant submanifolds of a Sasakian manifold in [7], and Sahin did the same for slant submanifolds of almost product manifolds in [22]. On the other hand, some generalizations of both slant and CR-submanifolds have also been defined in different ambient spaces, such as semi-slant [6, 15, 19], hemi-slant [23, 24], bi-slant [8], or generic submanifolds [21]. Moreover, the study of slant submanifolds of a semi-Riemannian manifold has also been initiated: Chen, Garay, and Mihai classified slant surfaces in Lorentzian complex space forms in [12, 13]. Arslan, Carriazo, Chen, and Murathan defined slant submanifolds of a neutral Kaehler manifold in [3], while Carriazo and Pérez-García did in neutral almost contact pseudo-metric manifolds in [9].

P. Alegre (B) Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_6

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The present survey compiles the studies made by A. Carriazo and the author in [1, 2]. In the first one, we introduced slant submanifolds of para Hermitian manifolds. These ambient spaces have a rich structure, similar to that of almost Hermitian ones, but also with very interesting differences. In the second, we introduced semi-slant, hemi-slant, and bi-slant submanifolds of para Hermitian manifolds. Finally, we point out that recently Chanyal has recently followed these steps in order to define slant submanifolds of para contact manifolds [10].

2 Basic Formulas and Definitions  A para Hermitian manifold is a 2n-dimensional semi-Riemannian manifold, M, endowed with a structure (J, g), where J is a (1, 1) tensor, and g is a semi-defined metric, satisfying J 2 X = X, g(J X, Y ) + g(X, J Y ) = 0, (1)  It is called para Kaehlerian if, in addition, ∇  J = 0. for any vector fields X, Y on M.  J, g), the Gauss and Weingarten formulas are given For a submanifold M of ( M, by X Y = ∇ X Y + h(X, Y ), (2) ∇ X V = −A V X + ∇ X⊥ N , ∇

(3)

for any tangent vector fields X, Y and any normal vector field V , where h is the second fundamental form of M, A V is the Weingarten endomorphism associated with V and ∇ ⊥ is the normal connection. For every tangent vector field X , we decompose J X = P X + F X,

(4)

where P X is the tangential component of J X and F X is the normal one. And for every normal vector field V , J V = t V + f V, where t V and f V are the tangential and normal components of J V , respectively. For such a submanifold of a para Kaehler manifold, taking the tangent and normal part and using the Gauss and Weingarten formulas (2) and (3) (∇ X P)Y = ∇ X PY − P∇ X Y = A FY X + th(X, Y ), (∇ X F)Y = ∇ X⊥ FY − F∇ X Y = −h(X, PY ) + f h(X, Y ), for all tangent vector fields X, Y .

(5) (6)

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Slant submanifolds of an almost Hermitian manifold were introduced by Chen at [11] as those submanifolds with Wirtinger angle θ (X ) independent of the choice of X ∈ T p M and of p ∈ M. The Wirtinger angle of a slant submanifold is called the slant angle. Complex submanifolds and totally real submanifolds are θ -slant π submanifolds with θ = 0 and θ = , respectively. A slant submanifold is called a 2 proper slant if it is neither complex nor totally real. Considering the Cauchy–Schwarz inequality, on a Riemannian manifold, |g(J X, P X )| ≤ |J X ||P X |, for any tangent vector field X , therefore cos θ =

g(J X, P X ) |P X | = , |J X ||P X | |J X |

for any non-null tangent vector field X . Actually, |J X | = |X | > 0. Proper slant submanifolds are characterized by the following simple condition: P 2 = λI , where λ is a real number λ ∈ (−1, 0) and I is the identity map. However, in the semi-Riemannian setting, the above definition has no meaning for light-like vector fields. So we will just measure the Wirtinger angle for those tangent vector fields for which it can be considered:  J, g) is called Definition 2.1 A submanifold M of a para Hermitian manifold ( M, slant if for every space-like or time-like tangent vector field X , the quotient g(P X, P X )/g(J X, J X ) is constant. Again, it will be soon clarified, complex and totally real submanifolds are particular cases of slant submanifolds. And we call proper slant to a neither complex nor totally real slant submanifold. Remark 2.2 There was a first attempt to define slant submanifolds of slant para Hermitian manifolds made by Gunduzalp, [14]. However, he considered the slant angle given by cos θ (X ) = |P X |/|J X | for any tangent vector field X and follows [11]. However, this quotient is not defined for light-like vector fields, and it is not always compressed in [0, 1]. It is necessary to be very careful and study each case separately, depending on the causal character of the involved vector fields. Let us begin by examining a low dimensional case: consider a two-dimensional submanifold M of a para Hermitian  and let {e1 , e2 } be a local orthonormal frame of tangent vector fields manifold M, with g(e1 , e1 ) = 1, i.e., such that e1 is space-like (if both e1 and e2 are time-like, the situation would be similar). From (1) to (4), we have −1 = g(J e1 , J e1 ) = g(Pe1 , Pe1 ) + g(Fe1 , Fe1 ). On the other hand, Pe1 = μe2 . Let us suppose μ  = 0, ±1; these cases would correspond to complex and totally real submanifolds, as we will explain later. Obviously, Pe1 and e2 have the same causal character. Depending on it and the value of μ, we can distinguish the following three cases:

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(1) If Pe1 is time-like and |μ| > 1, then g(Fe1 , Fe1 ) = −1 + μ2 and so Fe1 is space-like. (2) If Pe1 is time-like and |μ| < 1, then g(Fe1 , Fe1 ) = −1 + μ2 and so Fe1 is time-like. (3) Finally, if Pe1 is a space-like, g(Fe1 , Fe1 ) = −1 − μ2 , and Fe1 is a time-like vector field. Therefore, we distinguish three different types of proper slant submanifolds: Definition 2.3 Let M be a proper slant submanifold of a para Hermitian manifold  J, g). We say that it is: ( M, (1) slant of type 1 if, for any space-like (time-like) vector field X , P X is time-like |P X | (space-like), and > 1, |J X | (2) slant of type 2 if, for any space-like (time-like) vector field X , P X is time-like |P X | (space-like), and < 1, |J X | (3) slant of type 3 if, for any space-like (time-like) vector field X , P X is space-like (time-like). Remark 2.4 Let us point out that the case g(P X, P X ) = 0 (g(P X, P X ) = g(J X, J X ), respectively) for any space-like or time-like X , corresponds to the totally real case (complex case), that is P ≡ 0 (F ≡ 0 or equivalently P ≡ J ). This implies that the case P X being light-like is excluded. Indeed, let us suppose that g(P X, P X ) = 0, for any space-like or time-like tangent vector field X . Since any light-like vector field can be approximated by a sequence of space-like or time-like vector fields, it is clear that g(P X, P X ) = 0, for any vector field X , independently of its causal character. Then, 0 = g(P(X + Y ), P(X + Y )) = g(P X + PY, P X + PY ) = 2g(P X, PY ), for any X, Y . Thus, it follows from the previous equation that g(P X, PY ) = 0 for any tangent vector field Y . But also g(P X, Z ) = 0 for any Z ∈ D , where we have written T M = P(T M) ⊕ D. Hence, P X ∈ rad(T M, g) and then, since g is non degenerate, P X = 0. Therefore, M is totally real. The complex case can be similarly proved.

3 Characterization Results As it happens in the complex case, slant submanifolds can be characterized by means of P 2 . Theorem 3.1 Let M be a semi-Riemannian submanifold of a para Hermitian man J, g). Then, ifold ( M,

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(1) M is slant of type 1 if and only if for any space-like (time-like) vector field X , P X is time-like (space-like), and there exists a constant λ ∈ (1, +∞) such that P 2 = λI d.

(7)

We write λ = cosh2 θ , with θ > 0. (2) M is a slant of type 2 if and only if for any space-like (time-like) vector field X , P X is time-like (space-like), and there exists a constant λ ∈ (0, 1) such that P 2 = λI d.

(8)

We write λ = cos2 θ , with 0 < θ < 2π. (3) M is a slant of type 3 if and only if for any space-like (time-like) vector field X , P X is space-like (time-like), and there exists a constant λ ∈ (−∞, 0) such that P 2 = λI d.

(9)

We write λ = − sinh2 θ , with θ > 0. In every case, we call θ the slant angle. Proof In the first case, let M be slant of type 1, for any space-like tangent vector field X , P X is time-like, and, by virtue of (1), J X also is. Moreover, as they satisfy |P X |/|J X | > 1, there exists θ > 0 such that cosh θ =

√ |P X | −g(P X, P X ) = √ . |J X | −g(J X, J X )

(10)

If we now consider P X , we obtain cosh θ =

|P 2 X | |P 2 X | = . |J P X | |P X |

(11)

On the one hand g(P 2 X, X ) = g(J P X, X ) = −g(P X, J X ) = −g(P X, P X ) = |P X |2 .

(12)

Therefore, using (10)–(12) g(P 2 X, X ) = |P X |2 = |P 2 X ||J X | = |P 2 X ||X |. On the other hand since both X and P 2 X are space-like, it follows that they are collinear, that is P 2 X = λX . Finally, from (10), we deduce that λ = cosh2 θ . For a time-like tangent vector field Y everything works in a similar way, but now PY and J Y are space-like and so, instead of (10), we should write

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√ g(PY, PY ) |PY | cosh θ = . = √ |J Y | g(J Y, J Y ) Since P 2 X = λX , for any space-like or time-like X , it also holds for light-like vector fields and so we have that P 2 = λI d. The converse is a simple computation. In the second case, let M be slant of type 2, for any space-like or time-like vector field X , |P X |/|J X | < 1, and so there exists θ > 0 such that √ −g(P X, P X ) |P X | . = √ cos θ = |J X | −g(J X, J X ) Proceeding as before, we can prove that g(P 2 X, X ) = |P 2 X ||X | and, as both X and P 2 X are space-like vector fields, it follows that they are collinear, that is P 2 X = λX . Again, the converse is a direct computation. Finally, let M be slant of type 3, for any space-like vector field X , P X is also space-like, and there exists θ > 0 such that √ g(P X, P X ) |P X | . =√ sinh θ = |J X | −g(J X, J X ) Once more, we can prove that g(P 2 X, X ) = |P 2 X ||X | and P 2 X = λX . And again, the converse is just a direct computation. Note that, for slant submanifolds of type 2, the slant angle really coincides with the Wirtinger angle, i.e., the angle between J X and P X . Moreover, as every light-like vector field can be decomposed as the sum of one space-like and one time-like vector field, we directly obtain that conditions (7)–(9) also hold for every light-like vector field. We can deduce another characterization result:  J, g). Corollary 3.2 Let M be a submanifold of a para Hermitian manifold ( M, Then: (1) M is a slant submanifold of type 1 if and only if t F X = − sinh2 θ X for every space-like (time-like) vector field, (2) M is a slant submanifold of type 2 if and only if t F X = sin2 θ X for every spacelike (time-like) vector field, (3) M is a slant submanifold of type 3 if and only if t F X = cosh2 θ X for every space-like (time-like) vector field. Moreover, we notice that every slant submanifold of type 1 or 2 must be a neutral semi-Riemannian manifold, because if X is space-like, then P X is time-like. Therefore, for types 1 and 2, it is only necessary to ask space-like vector fields to satisfy conditions of Theorem 3.1 and Corollary 3.2. Finally, we can characterize the special case of neutral slant submanifolds of type 1 and 2 with half the dimension of the ambient space, by studying the normal

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145

4s fields. We write M2s , where the superindex indicates the dimension and the subindex indicates the tensor metric index. 4s 2s Corollary 3.3 Let Ms2s be a submanifold of a para Hermitian manifold ( M , J, g). Then:

(1) M is a slant submanifold of type 1 if and only if f 2 V = cosh2 θ V for every space-like (time-like) normal vector field V , (2) M is a slant submanifold of type 2 if and only if f 2 V = cos2 θ V for every space-like (time-like) normal vector field V . However, type 3 slant submanifolds are not always neutral, so the equivalent result only holds for those that are neutral. Corollary 3.4 Let M2i2s (0 < i < s) be a submanifold of a para Hermitian manifold 4s 2s , J, g). Then, M is a slant submanifold of type 3 if and only if f 2 V = − sinh2 θ V (M for every normal vector field V .

4 Examples of Slant Submanifolds We can present easy examples of all three types of slant submanifolds. Let us consider two different para Hermitian structures over R4 : ⎛

0 ⎜1 J =⎜ ⎝0 0 and:

⎛

0 ⎜0 ⎜ J1 = ⎝ 1 0

1 0 0 0

0 0 0 1

⎞ 0 0⎟ ⎟, 1⎠ 0

0 0 0 1

1 0 0 0

⎞ 0 1⎟ ⎟, 0⎠ 0

⎛

1 ⎜0 g=⎜ ⎝0 0 ⎛

0 0 −1 0 0 1 0 0

1 ⎜0 ⎜ g1 = ⎝ 0 0

0 1 0 0

⎞ 0 0 ⎟ ⎟, 0 ⎠ −1

0 0 −1 0

⎞ 0 0 ⎟ ⎟. 0 ⎠ −1

Then, considering the first para Hermitian structure: Example 4.1 For any a, b ∈ R with a 2 + b2  = 1, x(u, v) = (au, v, bu, u) defines a slant submanifold in (R4 , J, g), with P 2 = (1) of type 1 if a 2 + b2 > 1 and b2 < 1, (2) of type 2 if a 2 + b2 > 1 and b2 > 1, (3) time-like of type 3 if a 2 + b2 < 1.

a2 I d, and it is: −1 + a 2 + b2

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Therefore, with a proper election of a and b we obtain the following three nice examples: Example 4.2 For any θ > 0, x(u, v) = (u cosh2 θ, v, u 1 − sinh2 θ , u) defines a slant submanifold of type 1 in (R4 , J, g), with slant angle θ . Example 4.3 For any 0 < θ < π/2, x(u, v) = (u cos2 θ, v, u sin2 θ + 1, u) defines a slant submanifold of type 2 in (R4 , J, g), with slant angle θ . Example 4.4 For any θ > 0, x(u, v) = (u sinh2 θ, v, u 1 − cosh2 θ , u) defines a slant submanifold of type 3 in (R4 , J, g), with slant angle θ . We can present more examples using the second para Hermitian structure: Example 4.5 For any a, b with a 2 − b2  = 1, x(u, v) = (u, av, bv, v), defines a slant submanifold in (R4 , J1 , g1 ), with P 2 =

b2 I d, and it is: 1 − a 2 + b2

(1) of type 1 if a 2 − b2 < 1 and a 2 > 1, (2) of type 2 if a 2 − b2 < 1 and a 2 < 1, (3) space-like of type 3 if a 2 − b2 > 1. Moreover, it also defines a slant submanifold in (R4 , J, g), with P 2 = a2 I d, and it is: 1 + a 2 − b2 1) of type 1 if b2 − a 2 < 1 and a 2 > 1, 2) of type 2 if b2 − a 2 < 1 and a 2 < 1, 3) space-like of type 3 if b2 − a 2 > 1. All the given examples of type 3 slant submanifolds are space-like or time-like. Now we present a neutral one in a higher dimension. Combining Examples 4.1 and 4.5, we obtain Example 4.6 For any a, b with a 2 + b2 < 1, x(u, v, z, t) = (u, av, bv, v, z, at,



2 − b2 t, t),

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147

defines a neutral slant submanifold of type 3 in (R8 , J, g), with P 2 = a2 I d. −1 + a 2 + b2 And with a proper election of the coefficients, like in Example 4.4, we have Example 4.7 For any θ > 0, x(u, v, z, t) = (u, sinh2 θ v, 1 − cosh2 θ v, v, z, sinh2 θ t, 1 + cosh2 θt, t), defines a neutral slant submanifold of type 3 in (R8 , J, g), with slant angle θ.

5 Bi-Slant, Semi-slant, Hemi-slant, and CR-Submanifolds Slant distributions of a Kaehler manifold were introduced by N. Papaghiuc, [19].  J, g), and a differentiable distribution D Given an almost Hermitian manifold, ( M, , the angle is called a slant distribution if, for any nonzero vector X ∈ Dx , x ∈ N between J X and the vector space Dx is constant, that is, it is independent of the choice of X ∈ Dx and of the point x. They are characterized by PD2 = λI , where PD X denotes the projection of J X over D. Following these ideas, we introduced the equivalent concept for distributions of para Hermitian manifolds, [2]. Definition 5.1 A differentiable distribution D on a para Hermitian manifold  J, g) is called a slant distribution if for every non-light-like X ∈ D, the quotient ( M, g(PD X, PD X )/g(J X, J X ) is constant. Obviously, a submanifold M is a slant submanifold if and only if T M is a slant distribution. Moreover, a distribution is called invariant if it is a slant with slant angle zero, that is if g(PD X, PD X )/g(J X, J X ) = 1 for all non-light-like X ∈ D. It is called anti-invariant if PD X = 0 for all X ∈ D. In other cases, it is called a proper slant distribution. With this definition, every one-dimensional distribution defines  so we are just going to study distributions with an anti-invariant distribution in M, dimensions greater than one. Remember that an holomorphic distribution satisfies J D = D, so every holomorphic distribution is a slant distribution with angle 0, but the converse is not true. And the distribution D is called totally real distribution if J D ⊆ T ⊥ M, therefore, every totally real distribution is anti-invariant but the converse does not always hold. Let us clarify the relations between these concepts. For holomorphic and totally real distributions, the following necessary conditions are easy to prove: Theorem 5.2 Let D be a distribution of a submanifold of a para Hermitian metric  manifold M. (1) If D is a holomorphic distribution, then |PD X | = |J X |, for all X ∈ D. (2) If D is a totally real distribution, then |PD X | = 0, for all X ∈ D.

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However, the converse results do not hold if D is not T M; in such a case, T M = D ⊕ ν, and for a unit vector field X J X = PD X + Pν X + F X. Therefore, from g(J X, J X ) = g(PD X, PD X ) + g(Pν X, Pν X ) + g(F X, F X ), and |PD X | = |J X |, in the case that PD X is also space-like, we can only deduce that g(Pν X, Pν X ) + g(F X, F X ) = −2, or, in the case that PD X is time-like, g(Pν X, Pν X ) + g(F X, F X ) = 0. Therefore, in general, F X  = 0, and D is not invariant. Similarly, it can be shown that the converse of the second statement does not always hold. It is easy to prove the following result: Theorem 5.3 Let M be a semi-Riemannian submanifold of a para Hermitian metric  The maximal holomorphic (respectively, totally real) distribution is manifold M. characterized as D = {X/F X = 0} (resp. D ⊥ = {X/P X = 0}). Once the relation between holomorphic, invariant, totally real, and anti-invariant distributions has been established, we study proper slant distributions. The same way we distinguished three types of slant submanifolds, we now consider three types of slant distributions. Definition 5.4 Let D be a proper slant distribution of a para Hermitian manifold  J, g). We say that it is: ( M, (1) slant of type 1 if, for every space-like (time-like) vector field X , PD X is time-like |PD X | (space-like), and > 1, |J X | (2) slant of type 2 if, for every space-like (time-like) vector field X , if PD X is |PD X | time-like (space-like), and < 1, |J X | (3) slant of type 3 if, for every space-like (time-like) vector field X , PD X is spacelike (time-like). Slant distributions can be characterized by means of PD2 , as slant submanifolds were in Theorem 3.1, the proof is quite similar and hence it is omitted.  Then: Theorem 5.5 Let D be a distribution of a para Hermitian metric manifold M.

Slant Submanifolds of Para Hermitian Manifolds

149

(1) D is a slant distribution of type 1 if and only if for any space-like (time-like) vector field X , PD X is time-like (space-like), and there exists a constant λ ∈ (1, +∞) such that (13) PD2 = λI d We write λ = cosh2 θ . (2) D is a slant distribution of type 2 if and only if for any space-like (time-like) vector field X , PD X is time-like (space-like), and there exists a constant λ ∈ (0, 1) such that (14) PD2 = λI d We write λ = cos2 θ . (3) D is a slant distribution of type 3 if and only if for any space-like (time-like) vector field X , PD X is space-like (time-like), and there exists a constant λ ∈ (0, +∞) such that (15) PD2 = λI d We write λ = sinh2 θ . In every case, we call θ the slant angle. In [19], semi-slant submanifolds of an almost Hermitian manifold were introduced as those submanifolds whose tangent space could be decomposed as a direct sum of two distributions, one holomorphic and the other a slant distribution. In [8], antislant submanifolds were introduced as those whose tangent space is decomposed as a direct sum of a totally real and a slant distribution; they were called hemi-slant submanifolds in [23]. Finally, in [6], the authors defined bi-slant submanifolds when both distributions are slant ones. Definition 5.6 Let M be a semi-Riemannian submanifold of a para Hermitian man J, g), with the tangent space admitting a decomposition T M = D1 ⊕ D2 , ifold ( M, M is called: (i) a bi-slant if both D1 and D2 are slant distributions. (ii) a semi-slant submanifold if D1 is a holomorphic distribution and D2 a proper slant distribution. In such a case, we will write D1 = DT . (iii) a hemi-slant submanifold if D1 is a totally real distribution and D2 a proper slant distribution. In such a case, we will write D1 = D⊥ . (iv) CR-submanifold if D1 a holomorphic distribution and D2 a totally real distribution. In such a case, we will write D1 = DT and D2 = D⊥ . As we have said before, being holomorphic (totally real) is a stronger condition than being slant with slant angle zero (π/2). CR-submanifolds have been intensively studied in many environments [5]. Moreover, there are also some works about CRsubmanifolds of para Kaehler manifolds [17].

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6 Examples of Bi-Slant, Semi-slant, Hemi-slant, and CR-Submanifolds We can obtain examples of bi-slant submanifolds in R8 using the examples of slant submanifolds of R4 given in Sect. 4 and taking products. We consider the following para Kaehler structures over R8 in order to present different examples with all the combinations of slant distributions:



 J  g  , g2 = , J2 =  J  g

J3 =

J4 =

 J1  ,  J J1   J1

g3 =

 ,

g4 =

 g1  ,  g g1   g1

 ,

where  is the corresponding null matrix. Example 6.1 For any a, b, c, d ∈ R with a 2 + b2  = 1, and c2 + d 2  = 1, x(u 1 , v1 , u 2 , v2 ) = (au 1 , v1 , bu 1 , u 1 , cu 2 , v2 , du 2 , u 2 ) 8 defines a bi-slant

submanifold in (R , J2 , g 2 ), with slant distributions D1 = ∂ ∂ ∂ ∂ Span , , and D2 = Span . We can see the different types in ∂u 1 ∂v1 ∂u 2 ∂v2 the table 1. The type 3 case is a time-like slant distribution.We write Pi and I di for PDi and I d Di

As it also happens working with bi-slant submanifolds at [6], the decomposition of T M into two slant distributions is not if, in the

unique. For example

previous example, ∂ ∂ ∂ ∂ , , and D˜ 2 = Span , both distributions we choose D˜ 1 = Span ∂u 1 ∂v2 ∂u 2 ∂v1 are anti-invariant, that is P( D˜ 1 ) = D˜ 2 and P( D˜ 2 ) = D˜ 1 ; therefore, P1 = 0 and P2 = 0. However, they are not totally real distributions.

Table 1 Types for Example 1 D1 + b2

>

D2

type 1

a2

1, c2 < 1

(R8 , J2 , g2 )

type 2

a 2 + b2 > 1, b2 > 1

c2 + d 2 > 1, c2 > 1

P12 =

type 3

a 2 + b2 < 1

c2 + d 2 < 1

1, b2

a2 I d1 −1 + a 2 + b2 2 c P22 = I d2 −1 + c2 + d 2

Slant Submanifolds of Para Hermitian Manifolds Table 2 Types for Example 3 D1 − a2


1

d 2 − c2 < 1, d 2 > 1

type 2

b2 − a 2 < 1, b2 < 1

d 2 − c2 < 1, d 2 < 1

type 3

b2 − a 2 > 1

d 2 − c2 > 1

type 1

b2 − a 2 > 1, a 2 > 1

d 2 − c2 < 1, d 2 > 1

type 2

b2 − a 2 > 1, a 2 < 1

d 2 − c2 < 1, d 2 < 1

type 3

b2 − a 2 < 1

d 2 − c2 > 1

type 1

b2 − a 2 > 1, a 2 > 1

d 2 − c2 > 1, c2 > 1

type 2

b2 − a 2 > 1, a 2 < 1

d 2 − c2 > 1, c2 < 1

type 3

b2 − a 2 < 1

d 2 − c2 < 1

1, b2

151

(R8 , J2 , g2 ) a2 P12 = I d1 1 + a 2 − b2 2 c P22 = I d2 1 + c2 − d 2 8 (R , J3 , g3 ) a2 P12 = I d1 1 + a 2 − b2 2 c P22 = I d2 1 + c2 − d 2 8 (R , J4 , g4 ) a2 P12 = I d1 1 + a 2 − b2 2 c P22 = I d2 1 + c2 − d 2

Example 6.2 Taking a = 0 in the previous example, we obtain a semi-slant submanifold, and taking b = 1, we obtain a hemi-slant submanifold. Example 6.3 For any a, b, c, d with a 2 − b2  = 1, c2 − d 2  = 1: x(u 1 , v1 , u 2 , v2 ) = (u 1 , av1 , bv1 , v1 , u 2 , cv2 , dv2 , v2 ),

∂ ∂ defines a bi-slant submanifold, with slant distributions D1 = Span , and ∂u 1 ∂v1

∂ ∂ , . We can see the different types in the table 2. In the first D2 = Span ∂u 2 ∂v2 two cases, the type 3 distribution is space-like, and in the third, it is time-like. Among the semi-Riemannian manifolds, the Lorentzian manifolds are specifically interesting. It is possible to present examples of bi-slant submanifolds of a Lorentzian almost para Hermitian manifold. Taking into account that the only odddimensional slant distributions are the totally real ones and that type 1 and 2 are neutral distributions, there are only two possible cases: (i) M12s+1 with T M = D1 ⊕ D2 , where D1 is a one-dimensional, time-like, antiinvariant distribution and D2 is a space-like, type 3 slant distribution. (ii) M12s+2 with T M = D1 ⊕ D2 , where D1 is a two-dimensional, neutral, slant distribution of Type 1 or 2 and D2 is a space-like, type 3 slant distribution. From Examples 6.1 and 6.3, we can obtain examples of Case (ii). Now we will construct an example of the Case (i). Let us consider R6 with the Lorentzian almost para Hermitian structure given by

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⎛

⎞ J  J5 = ⎝ 0 1 ⎠ ,  10

⎛

⎞  g5 = ⎝ 1 0 ⎠ ,  0 −1 g

with  the corresponding null matrix. Example 6.4 For any k > 1, x(u, v, w) = (u, k cosh v, v, k sinh v, w, 0)

∂ defines a bi-slant submanifold in (R , J5 , g5 ) with D1 = Span a totally ∂w

∂ ∂ , a type 3 slant distribution with P22 = real distribution and D2 = Span ∂u ∂v 1 I d|D2 . 2 k −1 6

We can even present a bi-slant submanifold, with the same angle for both slant distributions, which is not a slant submanifold. Example 6.5 The submanifold of (R8 , J2 , g2 ) defined by x(u 1 , v1 , u 2 , v2 ) = (u 1 , v1 + u 2 , u 1 , u 1 , u 2 , v2 ,

√

3u 2 , u 2 − v1 ),



∂ ∂ is a bi-slant submanifold. The slant distributions are D1 = Span , and ∂u 1 ∂v1

∂ ∂ 1 1 , , with P12 = I d1 and P22 = I d2 . It is not a slant submanD2 = Span ∂u 2 ∂v2 2 2 ifold. Finally, examples of CR-submanifolds

can be obtained from Example 6.1. Taking ∂ ∂ , is a totally real distribution, and D2 = a = 1andd = 0, D1 = Span ∂u 1 ∂v1

∂ ∂ , is a holomorphic distribution. Moreover, D1 is of type 1 if b2 < 1, Span ∂u 2 ∂v2 and of type 2 if b2 > 1, and D2 is of type 2 if c2 > 1 and of type 3 if c2 < 1. Therefore, we have CR-submanifolds of types 1–2, 1–3, 2-2, and 2–3. Taking a = 0andd = 1, we can obtain the types 2-1, 2-2, 3-1, and again 3-2 examples.

7 Slant Submanifolds of a Para Kaehler Manifold In the almost Hermitian case, a proper slant submanifold can be endowed with an almost complex structure, compatible with the induced metric. Now we obtain the equivalent result for slant submanifolds of an almost para Hermitian environment.

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153

Theorem 7.1 Let M be a slant submanifold of an almost para Hermitian manifold  J, g). Then: ( M, 1 P is an almost para Hermitian structure over M, cosh θ 1 (2) if M is of type 2, J˙ = P is an almost para Hermitian structure over M, cos θ 1 (3) and if M is of type 3, J˙ = P is an almost Hermitian structure over M. sinh θ (1) if M is of type 1, J˙ =

Proof If M is a type 1 slant submanifold, considering such a structure J˙2 X =

1 P 2 X = X, cosh2 θ

where we have used Theorem 3.1. On the other hand, 

1 1 ˙ P X, Y = − g(X, PY ) = −g(X, J˙Y ). g( J X, Y ) = g cosh θ cosh θ The same holds for cases 2 and 3 with the corresponding structures. From both statements, it follows that the given structure is an almost para Hermitian (almost para Hermitian and almost Hermitian, respectively) structure. Again, following Chen’s steps for the study of slant submanifolds at [11], we study the case ∇ P = 0. So we find, that proper slant surfaces of para Hermitian manifolds admit different characterizations: 24 , that is neither Theorem 7.2 Let M be a surface of a para Hermitian manifold M totally real nor complex. The following conditions are equivalent: (i) M is a slant submanifold, (ii) ∇ P = 0, (iii) ∇ P 2 = 0. Proof Firstly, let us suppose that M is a slant submanifold, and let us prove that ∇ P = 0. We consider a local orthonormal basis {e1 , e2 } in T M, with e1 space-like 1 and e2 = Pe1 (μ = cosh θ, cos θ, sinh θ for type 1, 2 and 3 respectively). Since e1 μ and Pe1 are orthogonal, g(∇ X Pe1 , e1 ) = −g(Pe1 , ∇ X e1 ), and, given that g(Pe1 , Pe1 ) is constant, g(∇ X Pe1 , e2 ) =

1 g(∇ X Pe1 , Pe1 ) = 0, μ

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and g(∇ X e1 , Pe2 ) =

1 g(∇ X e1 , P 2 e1 ) = μg(∇ X e1 , e1 ) = 0. μ

From these equations, it is deduced that ∇ X Pe1 = P∇ X e1 , for any tangent vector field X , and the same holds for e2 . Then, for a general tangent vector field Y = ae1 + be2 , a, b being functions, we have P∇ X Y =P∇ X (ae1 + be2 ) = P(X (a)e1 + a∇ X e1 + X (b)e2 + b∇ X e2 ) = X (a)Pe1 + a P∇ X e1 + X (b)Pe2 + b P∇ X e2 = X (a)Pe1 + a∇ X Pe1 + X (b)Pe2 + b∇ X Pe2 = ∇ X PY. Therefore, ∇ P = 0. Obviously, ∇ P = 0 implies ∇ P 2 = 0. Finally, if ∇ P 2 = 0, let us prove that M is a slant surface. Considering a local orthonormal basis {e1 , e2 }, since Pe1 is orthogonal to e1 , then, Pe1 = λe2 and orthogonal so P 2 e1 = λPe2 for a certain function λ. Hence, P 2 e1 is

 to e2 and μ 1 P 2 e1 = μe1 for a certain function μ. Moreover, P 2 e2 = P 2 Pe1 = Pe1 = λ λ μe2 . Therefore, P 2 X = μX for any tangent vector field X . If we consider another unit tangent vector field Y , then X (μ) =X g(P 2 Y, Y ) = g(∇ X P 2 Y, Y ) + g(P 2 Y, ∇ X Y ) = g(P 2 ∇ X Y, Y ) + g(P 2 Y, ∇ X Y ) = 2μg(∇ X Y, Y ) = 0, which proves that μ is constant, and therefore, M is a slant submanifold. Let us point out that if the ambient space is para Kaehler and ∇ P = 0, then the induced structure on the submanifold is para Kaehler or Kaehler, accordingly to the different cases of Theorem 7.1. As we have mentioned before, all slant submanifolds of type 1 and 2 are neutral semi-Riemannian manifolds, but this does not hold for type 3 slant submanifolds.This makes the type 3 case specially interesting to study. For such a submanifold, M2i2s , we can choose an adapted orthonormal slant frame  e1 , ..., es , e1∗ , ..., es∗ ,



1 Pe j . Recall that, for type 3, both e j and e∗j have the same causal where e∗j = sinh θ character. Moreover, if the codimension coincides with the dimension of the submanifold,

1 1 1 1 Fe1 , ..., Fes , Fe1∗ , ..., Fes∗ cosh θ cosh θ cosh θ cosh θ is an adapted orthonormal frame in T ⊥ M.

Slant Submanifolds of Para Hermitian Manifolds

155

We recall that Chen proved in [11] that a surface in C 2 , which is neither holomorphic nor totally real, is a minimal slant surface if and only if ∇ F = 0. We can give a similar result Theorem 7.3 Let M2i2s be a slant submanifold of type 3 of a para Kaehler manifold 4s 2s . If ∇ F = 0, then the submanifold is minimal. M  J = 0. From (6), it follows that ∇ F = 0 if Proof For a para Kaehler manifold ∇ and only if h(X, PY ) = f h(X, Y ), for any tangent vector fields X, Y . Therefore, by taking an adapted orthonormal frame, and using Corollary 3.4, we have h(e∗j , e∗j )

=h

1 1 Pe j , Pe j sinh θ sinh θ

Consequently, H =

s 

 =

1 f 2 h(e j , e j ) = −h(e j , e j ). sinh2 θ

ε j (h(e j , e j ) + h(e∗j , e∗j )) = 0, where ε j = g(e j , e j ), and M

j=1

is minimal. Furthermore, for type 3 slant surfaces, the converse result also holds: 24 is minimal Theorem 7.4 A type 3 slant surface M of a para Kaehler manifold M if and only if ∇ F = 0. Proof We assume that M is space-like. In the case that it is time-like, our proof would be similar. We can consider orthonormal bases



1 1 1 Pe1 Fe1 , e4 = Fe2 and e3 = e1 , e2 = sinh θ cosh θ cosh θ of T M and T ⊥ M. We already know that ∇ F = 0 if and only if h(X, PY ) = f h(X, Y ), for any tangent vector fields X, Y , that is, if and only if A f V X = −A V P X for any X tangent and any V normal. We will prove that condition for the given basis. For such a basis te3 = cosh θ e1 , f e3 = − sinh θ e4 , te4 = cosh θ e2 , f e4 = sinh θe3 . On the one hand A f e3 X =

1 −1 1 sinh θ A f Fe1 X = A F Pe1 X = th(X, Pe1 ) = th(X, e2 ), cosh θ cosh θ cosh θ cosh θ

(16) where we have used (∇ X P)Y = A FY X + th(X, Y ) = 0, by virtue of Theorem 7.2. On the other hand Ae3 P X =

1 −1 A Fe1 P X = th(P X, e1 ). cosh θ cosh θ

(17)

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Then for X = e1 , (16) and (17) gives Ae3 Pe1 =

−1 − sinh θ th(Pe1 , e1 ) = th(e2 , e1 ) = −A f e3 e1 , cosh θ cosh θ

and for X = e2 , Ae3 Pe2 =

−1 −1 th(Pe2 , e1 ) = th cosh θ cosh θ

and A f e3 e2 =



1 P 2 e1 , e1 sinh θ

 =

sinh θ th(e1 , e1 ) cosh θ

sinh θ th(e2 , e2 ). cosh θ

Since M is minimal, these last equations imply A f e3 e2 = −Ae3 Pe2 . Then, A f e3 X = −Ae3 P X for any tangent vector field X . Proceeding in the same way, it is proved for e4 , so A f V X = −A V P X for any X tangent and V normal to M, what finishes the proof.

8 Semi-slant and Hemi-slant Submanifolds of a Para Kaehler Manifold In this last section, we study the integrability of the distributions and conditions for being totally geodesic. Most of the proofs are direct computations, and they can be directly obtained reasoning as [6] or [19] so we omit some proofs. First, we deal with semi-slant submanifolds. Proposition 8.1 Let M be a semi-slant submanifold of a para Hermitian manifold. Then, both the holomorphic and the slant distributions are P invariant. Proof Let be T M = DT ⊕ D2 the decomposition with DT holomorphic and D2 the slant distribution. Of course DT is invariant as J DT = DT implies P DT = DT . Now, consider X ∈ D2 , we have this decomposition J X = P1 X + P2 X + F X. For any Y ∈ DT , g(P1 X, Y ) = g(J X, Y ) = −g(X, J Y ) = 0, as DT is invariant. But also, for all Z ∈ D2 , g(P1 X, Z ) = 0. So we conclude that P1 X = 0 and P X = P2 X , therefore, P D2 ⊆ D2 and consequently D2 is P invariant. From (5) to (6), it is easy to prove the following result: Theorem 8.2 Let M be a semi-slant submanifold of a para Kaehler manifold. (i) The holomorphic distribution is integrable if and only if h(X, J Y ) = h(J X, Y ) for all X, Y ∈ DT .

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(ii) The slant distribution is integrable if and only if: π1 (∇ X PY − ∇Y P X ) = π1 (A FY X − A F X Y ),

(18)

for all X, Y ∈ D2 , where π1 is the projection over the invariant distribution DT . Now, we study the conditions for the distributions to be totally geodesic.  Theorem 8.3 Let M be a semi-slant submanifold of a para Kaehler manifold M. (i) If the holomorphic distribution DT is totally geodesic, then (∇ X P)Y = 0, and ∇ X Y ∈ DT for any X, Y ∈ DT . (ii) The slant distribution D2 is totally geodesic if and only if (∇ X F)Y = 0, and (∇ X P)Y = A FY X for any X, Y ∈ D2 . Proof (i) On the one hand, for a para Kaehler manifold, taking X, Y ∈ DT in equations (5)–(6) leads to ∇ X PY − P∇ X Y − th(X, Y ) = 0,

(19)

− F∇ X Y + h(X, PY ) − f h(X, Y ) = 0.

(20)

So, if DT is totally geodesic, (∇ X P)Y = 0 and F∇ X Y = 0, which imply the result. (ii) On the other hand, if D2 is a totally geodesic distribution, again taking X, Y ∈ D2 in equations (5) and (6) ∇ X PY − A FY X − P∇ X Y = 0,

(21)

∇ X⊥ FY − F∇ X Y = 0.

(22)

which imply the given conditions. Conversely, if (∇ X P)Y = A FY X , then th(X, Y ) = 0, so J h(X, Y ) = f h(X, Y ). From (6) and ∇ F = 0, it holds that h(X, PY ) = nh(X, Y ). Then, for PY ∈ D2 λh(X, Y ) = h(X, P 2 Y )= f 2 h(X, Y ) = J 2 h(X, Y ) = h(X, Y ), and as D2 is a proper slant distribution, λ  = 1, so h(X, Y ) = 0 for all X, Y ∈ D2 . Now we present the equivalent results for a hemi-slant submanifold. First is the study of the integrability of the distributions. Proposition 8.4 Let M be a hemi-slant submanifold of a para Hermitian manifold. Then, the slant distribution is P invariant.

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Theorem 8.5 Let M be a hemi-slant submanifold of a para Kaehler manifold. Then: (i) The totally real distribution is always integrable. (ii) The slant distribution is integrable if and only if π1 (∇ X PY − ∇Y P X ) = π1 (A FY X − A F X Y ),

(23)

for all X, Y ∈ D2 , where π1 is the projection over the totally real distribution D⊥ .  Theorem 8.6 Let M be a hemi-slant submanifold of a para Kaehler manifold M. (i) The totally real distribution D⊥ is totally geodesic if and only if (∇ X F)Y = 0, and P∇ X Y = −A FY X for any X, Y ∈ D⊥ . (ii) The slant distribution D2 is totally geodesic if and only if (∇ X F)Y = 0, and P∇ X Y = −A FY X for any X, Y ∈ D2 . The classical De Rham–Wu Theorem is well known [20, 25]. It says that two orthogonal, complementary, and geodesic foliations (called a direct product structure) in a complete and simply connected semi-Riemannian manifold give rise to a global decomposition as a direct product of two leaves. Therefore, the following theorem follows from the previous results, it is directly deduced: Theorem 8.7 Let M be a complete and simply connected hemi-slant submanifold of  Then, M is locally the product of the integral submana para Kaehler manifold M. ifolds of the slant distributions if and only if (∇ X F)Y = 0, and P∇ X Y = −A FY X for both any X, Y ∈ D⊥ or X, Y ∈ D2 . The existence of warped product bi-slant, semi-slant, and hemi-slant submanifolds of a para Hermitian manifold is an open problem. Acknowledgements The author wants to express his gratitude to the referees of this paper for some very pertinent suggestions which have improved it.

References 1. Alegre, P., Carriazo, A.: Slant submanifolds of para Hermitian manifolds. Mediterr. J. Math. 14, 214 (2017). https://doi.org/10.1007/s00009-017-1018-3 2. Alegre, P., Carriazo, A.: Bi-Slant submanifolds of para Hermitian manifolds. Mathematics 7(7), 618 (2019). https://doi.org/10.3390/math7070618 3. Arslan, K., Carriazo, A., Chen, B.-Y., Murathan, C.: On slant submanifolds of neutral Kaehler manifolds. Taiwanesse J. Math. 14(2), 561–584 (2010) 4. Barros, M., Urbano, F.: CR-submanifolds of generalized complex space forms. An. Stiint. Al. I. Cuza. Univ. Iasi. 25, 855–863 (1979) 5. Bejancu, A.: CR submanifolds of a Kaehler manifold. Trans. Am. Math. Soc. 250, 333–345 (1979) 6. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Semi-slant submanifolds of a Sasakian manifolds. Geom. Dedicata. 78, 183–199 (1999)

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7. Cabrerizo, J.L., Carriazo, A., Fernández, L.M., Fernández, M.: Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42, 125–138 (2000) 8. Carriazo, A.: Bi-slant immersions. In Proceedings of the ICRAMS, Kharagpur, India 20–22, 88–97 (2000) 9. Carriazo, A., Pérez-García, M.J.: Slant submanifolds in neutral almost contact pseudo-metric manifolds. Differ. Geom. Appl.54, 71–80 Part A (2017) 10. Chanyal, S.K.: Slant submanifolds in an almost paracontact metric manifold. An. Stiint. Univ. I. Cuza Iasi Math. (N.S.) 67(2), 213–229 (2021) 11. Chen, B.-Y.: Slant inmersions. Bull. Aust. Math. Soc. 41, 135–147 (1990) 12. Chen, B.-Y., Garay, O.: Classification of quasi-minimal surfaces with parallel mean curvature vector in pseudo-Euclidean 4-space E42 . Results Math. 55, 23–38 (2009) 13. Chen, B.-Y., Mihai, I.: Classification of quasi-minimal slant surfaces in Lorentzian complex space forms. Acta Math. Hungar. 122, 307–328 (2009) 14. Gunduzalp, Y.: Neutral slant submanifolds of a para-Kaehler manifold. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 752650, 8 p. https://doi. org/10.1155/2013/752650 15. Li, H., Liu, X.: Semi-slant submanifolds of a locally product manifold. Ga. Math. J. 12, 273–282 (2005) 16. Matsumoto, K.: On Lorentzian paracontact manifolds. Bull. Yamagata Univ. Nat. Sci. 12, 151–156 (1989) 17. Mihai, A., Rosca, R.: Skew-symmetric vector fields on a CR-submanifold of a para-Kaehlerian manifold. IJMMS 10, 535–540 (2004) 18. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics, vol. 103. Academic, New York (1983) 19. Papaghiuc, N.: Semi-slant submanifolds of a Kaehlerian manifold. Ann. St. Univ. Iasi. Tom. 40, 55–61 (1994) 20. Ponge, R., Reckziegel, H.: Twisted products in pseudo-Riemannian geometry. Geom. Dedicata. 48, 15–25 (1993) 21. Ronsse, G.S.: Generic and skew CR-submanifolds of a Kaehler manifold. Bull. Inst. Math. Acad. Sin. 18, 127–141 (1990) 22. Sahin, B.: Slant submanifolds of an almost product Riemannian manifold. J. Korean Math. Soc. 43(4), 717–732 (2006) 23. Sahin, B.: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Pol. Math. 95, 207–226 (2009) 24. Ta¸stan, H.M., Gerdan, S.: Hemi-slant submanifolds of a locally conformal Kähler manifold. Int. Electron. J. Geo. 8, 46–56 (2015) 25. Wu, H.: On the de Rham decomposition theorem. Ill. J. Math. 8, 291–311 (1964)

Hemi-slant and Semi-slant Submanifolds in Locally Conformal Kaehler Manifolds Hakan Mete Ta¸stan and Sibel Gerdan Aydın

2010 Mathematics Subject classification: 53C05 · 53C25 · 53C50.

1 Hemi-slant Submanifolds in Locally Conformal Kaehler Manifolds One of the popular research areas in differential geometry is the theory of submanifolds. Indeed, there are well-known classes of submanifolds such as holomorphic (invariant), totally real (anti-invariant) [32], slant [8] and CR-submanifold [2]. Carriazo [6], introduced the notion of bi-slant submanifold as a generalization of previous classes. One of the subclasses of bi-slant submanifolds is that of anti-slant submanifolds which are studied by Carriazo [6], but Sahin ¸ [26] named these submanifolds as hemi-slant submanifolds because of that the name anti-slant seems to refer that they have no slant factor. Also, some authors study such submanifolds under the name of pseudo-slant submanifolds, see [12, 30]. The notion of hemi-slant submanifold is a a special case of the notion of generic submanifold which was introduced by Ronsse [23]. Hemi-slant submanifolds have been studied actively in different kinds of structures by many geometers. For example, see [1, 30].

H. Mete Ta¸stan (B) · S. Gerdan Aydın Department of Mathematics, ˙Istanbul University, 34134 Vezneciler ˙Istanbul, Turkey e-mail: [email protected] S. Gerdan Aydın e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_7

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1.1 Locally and Globally Conformal Kaehler Manifolds ¯ J, g) be a Hermitian manifold of dimension 2m. Then it is called a locally Let ( M, conformal Kaehler manifold (briefly l.c.K. manifold) [13], if each point of p ∈ M¯ has an open neighborhood U with smooth function σ : U → R such that g˜ = e−σ g|U ¯ then ( M, ¯ J, g) is called a globally is a Kaehler metric on U. If one choose U = M, conformal Kaehler manifold (briefly g.c.K. manifold). ¯ J, g) be a Hermitian manifold and let  be a 2− form Theorem 1.1 ([13]) Let ( M, ¯ ¯ ¯ ¯ Then ( M, ¯ J, g) is defined by ( X , Y ) = g( X , J Y¯ ) for any vector fields X, Y in M. a l.c.K. manifold if and only if there exists a globally defined 1− form ω such that d = ω ∧ 

and

dω = 0 .

(1)

¯ J, g). In The closed 1− form ω called the Lee form of the l.c.K. manifold ( M, ¯ J, g) is g.c.K., if its Lee form ω is also exact. In this case, addition, the manifold ( M, we have ω = dσ [31]. The Lee vector field B is defined by ω( X¯ ) = g(B, X¯ ) ,

(2)

¯ We denote by ∇˜ (resp. ∇) ¯ the Levi-Civita connection for any vector fields X¯ on M. −σ ¯ on M with respect to g˜ = e g (resp. g). Then we have [13] ∇˜ X¯ Y¯ = ∇¯ X¯ Y¯ −

 1 2

 ¯ ¯ ¯ ¯ ¯ ¯ ω( X )Y + ω(Y ) X − g( X , Y )B ,

(3)

¯ The connection ∇˜ is a torsionless linear confor any vector fields X¯ and Y¯ on M. ¯ nection on M which is called the Weyl connection of g. It is easy to see that the Weyl connection ∇˜ satisfies the condition ∇˜ J = 0 .

(4)

For examples and more details on l.c.K. manifolds we refer to [13]. ¯ J, ω, g) the l.c.K. manifold Remark 1.2 Throughout this chapter, we denote by ( M, with the Lee form ω.

1.2 Submanifolds of Riemannian Manifolds ¯ g). Let M be an isometrically immersed submanifold in a Riemannian manifold ( M, Let ∇¯ is the Levi-Civita connection of M¯ with respect to the metric g and let ∇ and ∇ ⊥ be the induced, and induced normal connection on M, respectively. Then,

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for all X, Y ∈ T M and ξ ∈ T ⊥ M, the Gauss and Weingarten formulas are given respectively by (5) ∇¯ X Y = ∇ X Y + h(X, Y ) , ∇¯ X ξ = −Aξ X + ∇ X⊥ ξ ,

(6)

¯ Additionwhere T M is the tangent bundle and T ⊥ M is the normal bundle of M in M. ally, h is the second fundamental form of M and Aξ is the Weingarten endomorphism associated with ξ. The second fundamental form h and the shape operator A related by g(h(X, Y ), ξ) = g(Aξ X, Y ) . (7) The mean curvature vector field H of M is given by H = m1 (trace h), where dim(M) = m. We say that the submanifold M is totally geodesic in M¯ if h = 0, and minimal if H = 0. The submanifold M is called totally umbilical if h(X, Y ) = g(X, Y )H for all X, Y ∈ T M. ¯ J, ω, g). Then the Gauss and Let M be any submanifold of a l.c.K. manifold ( M, Weingarten formulas with respect to ∇˜ are given by ˜ Y), ∇˜ X Y = ∇ˆ X Y + h(X,

(8)

∇˜ X ξ = − A˜ ξ X + ∇˜ X⊥ ξ ,

(9)

for X, Y ∈ T M and ξ ∈ T ⊥ M. A distribution D on a manifold M¯ is called autoparallel if ∇ X Y ∈ D for any ¯ If a distriX, Y ∈ D and called parallel if ∇¯ U X ∈ D for any X ∈ D and U ∈ T M. bution D on M¯ is autoparallel, then it is clearly integrable, and by Gauss formula D ¯ If D is parallel then the orthogonal complementary distribuis totally geodesic in M. tion D⊥ is also parallel, which implies that D is parallel if and only if D⊥ is parallel. In this case M¯ is locally product of the leaves of D and D⊥ . For two distributions D1 ¯ we say that M is (D1 , D2 )- mixed geodesic, if for and D2 on a submanifold M of M, all X ∈ D1 and Y ∈ D2 we have h(X, Y ) = 0, where h is the second fundamental form of M [3].

1.3 Hemi-slant Submanifolds of an Almost Hermitian Manifold ¯ J, g) be an almost Hermitian manifold and let M be a Riemannian manifold Let ( M, ¯ A distribution D on M is called a slant distribution if isometrically immersed in M. for V ∈ D p , the angle θ between J V and D p is constant, i.e., independent of p ∈ M

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and V ∈ D p . The constant angle θ is called the slant angle of the slant distribution D. We know that holomorphic and totally real distributions on M are slant distributions with θ = 0 and θ = π2 , respectively. A slant distribution is called proper if it is neither holomorphic nor totally real. A submanifold M of M¯ is said to be a slant submanifold [8] if the tangent bundle T M of M is slant. For examples and more details, see [8]. ¯ J, g) is A hemi-slant submanifold M [6, 26] of an almost Hermitian manifold ( M, a submanifold its tangent bundle T M admits two orthogonal complementary totally real distribution D⊥ and slant distribution Dθ , i.e., we have T M = D⊥ ⊕ Dθ .

(10)

We say that the hemi-slant submanifold M is proper if dim(D⊥ )  = {0} and θ  = 0, π2 . For any X ∈ T M we write JX = PX + FX , (11) where P X is the tangential part of J X, and F X is the normal part of J X. Similarly, for any ξ ∈ T ⊥ M, we put J ξ = tξ + nξ , (12) where tξ is the tangential part of J ξ, and nξ is the normal part of J ξ. Then the normal bundle T ⊥ M of M is decomposed as T ⊥ M = J D⊥ ⊕ FDθ ⊕ D ,

(13)

where D is the orthogonal complementary distribution of J D⊥ ⊕ FDθ in T ⊥ M and it is invariant subbundle of T ⊥ M with respect to J. In this case, we have [26] P 2 V = − cos2 θV , g(PU, P V ) = cos2 θg(U, V )

and

g(FU, F V ) = sin2 θg(U, V ) .

(14) (15)

for any U, V ∈ (Dθ ). The following results on hemi-slant submanifolds of an almost Hermitian manifold were proved in [27]. Lemma 1.3 Let M be a hemi-slant submanifold of an almost Hermitian manifold ¯ J, g). Then, we have ( M, (a) P 2 + t F = −I, (c) F P + n F = 0,

(b) n 2 + Ft = −I, (d) tn + Pt = 0.

(16)

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Lemma 1.4 Let M be a hemi-slant submanifold of an almost Hermitian manifold ¯ J, g). Then we have, ( M, J D⊥ ⊥ FDθ . (17) Lemma 1.5 Let M be a hemi-slant submanifold of an almost Hermitian manifold ¯ J, g). Then we have, ( M, (a) PD⊥ = {0},

(b) PDθ = Dθ .

(18)

1.4 Hemi-slant Submanifolds of a Locally Conformal Kaehler Manifold ¯ J, ω, g) with anti-invariant Let M be any submanifold of a l.c.K. manifold ( M, ⊥ θ ¯ we put distribution D and slant distribution D . For the Lee vector field B of M, B = B M + B N (along M) ,

(19)

where B M and B N are respectively tangential and normal part of B. Using (3), (5), (6), (8) and (9), we have following lemma. ¯ J, ω, g). Then we Lemma 1.6 Let M be any submanifold of a l.c.K. manifold ( M, have (20) ∇ˆ U V = ∇U V − 21 {ω(U )V + ω(V )U − g(U, V )B M } , ˜ V ) = h(U, V ) + 21 g(U, V )B N , h(U,

(21)

A˜ ξ U = Aξ U + 21 ω(ξ)U ,

(22)

∇˜ U⊥ ξ = ∇U⊥ ξ − 21 ω(U )ξ ,

(23)

for any U, V ∈ T M and ξ ∈ T ⊥ M. We remark that Lemma 1.6 also appeared as Lemma 2.1 in ([24]) for generic submanifolds (in the sense of Chen [7]) of a l.c.K. manifold. First, we give an example. Example 1.7 Let (y1 , ..., y6 ) be natural coordinates of the six-dimensional Euclidean space R6 and let (J, g0 ) be a usual Kaehler structure on R6 . Then (R6 , J, g0 ) is a Kaehler manifold. Now, we consider the Riemannian metric g = eσ g0 conformal to Kaehler metric g0 on R6 , where σ is a smooth function on R6 . Thus, (R6 , J, g) is a g.c.K. manifold.

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Next, let M be a submanifold of (R6 , J, g) defined by u u u y1 = √ cos v , y2 = √ sin v , y3 = √ , y4 = 0 , y5 = x , y6 = 0 , 2 2 2 where u, x  = 0. Then, a local frame of T M is given by   1 U = √ cos v∂1 + sin v∂2 + ∂3 , 2 V = − sin v∂1 + cos v∂2 , X = ∂5 , where ∂i = ∂∂yi for i = 1, 2, ..., 6. Then D⊥ = span{X } is a totally real and Dθ = span{U, V } is a (proper) slant distribution with the slant angle θ = π/4 . Thus, M is a proper hemi-slant submanifold of a g.c.K. manifold (R6 , J, g). Now, we study the integrability of totally real distribution D⊥ and slant distribution Dθ . ¯ J, ω, g). Lemma 1.8 ([27]) Let M be any submanifold of a l.c.K. manifold ( M, Then we have ∇U P V − A F V U − 21 ω(J V )U + 21 g(U, P V )B M = P∇U V + th(U, V ) − 21 ω(V )PU + 21 g(U, V )(P B M + t B N )

(24)

∇U⊥ F V + h(U, P V ) + 21 g(U, P V )B N = F∇U V + nh(U, V ) − 21 ω(V )FU + 21 g(U, V )(F B M + n B N )

(25)

and

for any U, V ∈ T M. Proof Substituting V by J V in (4) and using (11) and (5) we obtain ∇U P V + h(U, P V ) + ∇U⊥ F V − A F V U − 21 ω(U )J V − 21 ω(J V )U + 21 g(U, J V )B = P∇U V + F∇U V + th(U, V ) + nh(U, V )

(26)

− 21 ω(U )J V − 21 ω(V )J U + 21 g(U, V )J B Thus, (24) and (25) follow from (26) by using (11), (12) and identifying the compo nents from T M and T ⊥ M. By Lemma 1.8, we get the following result.

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Lemma 1.9 ([27]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the totally real D⊥ is integrable if and only if ( M, A F X Y + 21 ω(F X )Y = A FY X + 21 ω(FY )X

(27)

for any X, Y ∈ (D⊥ ). From Lemma 1.9 and (22), we have the following result. Corollary 1.10 ([27]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the totally real distribution D⊥ is integrable if and only if ( M, A˜ F X Y = A˜ FY X

(28)

for any X, Y ∈ (D⊥ ). Lemma 1.11 ([27]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then we have ( M, ˜ −P(∇ˆ U X ) = A˜ F X U + t h(U, X)

(29)

for any X ∈ (D⊥ ) and U ∈ T M. Proof From (4), we have ∇˜ U J X = J ∇˜ U X for any X ∈ (D⊥ ) and U ∈ T M. Using (18)-(a) and (8), we obtain ˜ X) . ∇˜ U F X = J (∇ˆ U X ) + J h(U, Hence, it follows that ˜ ˜ − A˜ F X U + ∇˜ U⊥ F X = P(∇ˆ U X ) + F(∇ˆ U X ) + t h(U, X ) + f h(U, X) . Taking the tangential part of this equation we find (29).



Theorem 1.12 ([27]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the totally real distribution D⊥ is integrable. ( M, Proof With the help of (18)-(a) and (18)-(b), for any X, Y ∈ (D⊥ ) and U ∈ T M, we have ˜ X ), Y ) from Lemma 1.11. 0 = g(−P(∇ˆ U X ), Y ) = g( A˜ F X U, Y ) + g(t h(U, After some calculation, we find g( A˜ F X Y, U ) = g( A˜ FY X, U ). It follows that A˜ F X Y =  A˜ FY X. Thus our assertion follows from Corollary 1.10.

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Lemma 1.13 ([14]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then we have ( M,   g(∇ X Y, V ) = − sec2 θg A J Y P V − A F P V Y − 21 ω(F P V )Y, X − 21 ω(V )g(X, Y ) ,

(30) and  g(∇U V, X ) = sec θg A J X P V − A F P V X + 2

 1 ω(J X )P V, 2

U

− 21 ω(X )g(U, V )

for X, Y ∈ (D⊥ ) and U, V ∈ (Dθ ).

(31)

¯ J, g˜ = e−σ g) is a Kaehler manProof Let X, Y ∈ (D⊥ ) and V ∈ (Dθ ), since ( M, ifold, using (4), (9), (11) and (12), we have ˜ ∇˜ X Y, V ) = g( ˜ ∇˜ X J Y, J V ) g( ˜ ∇ˆ X Y, V ) =g( =g( ˜ ∇˜ X J Y, P V ) + g( ˜ ∇˜ X J Y, F V ) ˜ ∇˜ X Y, t F V ) − g( ˜ ∇˜ X Y, n F V ). = − g( ˜ A˜ J Y P V, X ) − g( Here, using (16), we obtain g( ˜ ∇ˆ X Y, V ) = − g( ˜ A˜ J Y P V, X ) + sin2 θg( ˜ ∇ˆ X Y, V ) + g( ˜ ∇˜ X Y, F P V ) ˜ A˜ F P V Y, X ) + sin2 θg( ˜ ∇ˆ X Y, V ). = − g( ˜ A˜ J Y X, P V ) + g( Hence, we get g( ˜ ∇ˆ X Y, V ) = − sec2 θ{g( ˜ A˜ J Y P V − A˜ F P V Y, X )} . Now, by using (2), (20) and (22), we find that the assertion (30). The other assertion (31) can be obtained by a similar method.  Lemma 1.14 ([14]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the slant distribution Dθ on M is integrable if and only if ( M, g(A J X P V − A F P V X, U ) = g(A J X PU − A F PU X, V ) − ω(J X )g(P V, U ) (32) for X ∈ (D⊥ ) and U, V ∈ (Dθ ). ¯ J, ω, g). Proof Let M be a proper hemi-slant submanifold of a l.c.K. manifold ( M, Then slant distribution D θ is integrable if and only if g([U, V ], X ) = 0 for all X ∈  (D⊥ ) and U, V ∈ (Dθ ). Thus, the assertion (32) follows from (31). Next, we give a necessary and sufficient condition for a proper hemi-slant of submanifold to be a is a Riemannian product.

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Lemma 1.15 ([27]) Let M be a proper hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the slant distribution Dθ defines a totally geodesic foliation on ( M, M if and only if g(A F X W + 21 ω(F X )W, Z ) = g(A F W X, Z )

(33)

for all Z , W ∈ (D θ ) and X ∈ (D⊥ ). Lemma 1.16 ([27]) Let M be a proper hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the totally real distribution D⊥ defines a totally geodesic foliation ( M, on M if and only if g(A F X Z , Y ) = g(A F Z X + 21 ω(F Z )X, Y )

(34)

for all X, Y ∈ (D⊥ ) and Z ∈ (Dθ ). By Lemmas 1.15 and 1.16, we get the following result. Theorem 1.17 ([27]) Let M be a proper hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then M is a locally Riemannian product manifold M = MD⊥ × MDθ ( M, if and only if (35) A F X Z + 21 ω(F X )Z = A F Z X + 21 ω(F Z )X for X ∈ (D⊥ ) and Z ∈ (Dθ ), where MD⊥ is a totally real and MDθ is a slant ¯ submanifold of M.

1.5 Hemi-slant Submanifolds with Parallel Canonical Structures In this subsection, we recall the recent results for hemi-slant submanifolds of a l.c.K. manifold with parallel canonical projection structures on the tangent bundle of the submanifold. ¯ J, ω, g). For the endomorphism Let M be a submanifold of a l.c.K. manifold ( M, P : T M → T M we put (∇˜ U P)V = ∇ˆ U P V − P ∇ˆ U V

(36)

for any U, V ∈ T M. We say that P is parallel if (∇˜ U P) = 0 for any U ∈ T M. From (4), (8) and (9), we have ˜ P V ) − A˜ F V U + ∇˜ U⊥ F V ∇ˆ U P V + h(U, ˜ ˜ = P ∇ˆ U V + F ∇ˆ U V + t h(U, V ) + n h(U, V)

(37)

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Hence, we obtain

˜ V ) + A˜ F V U. (∇˜ U P)V = t h(U,

Thus, for any U, V, W ∈ T M, we get g((∇˜ U P)V, W ) = g( A˜ F V W − A˜ F W V, U ) .

(38)

The following results on hemi-slant submanifolds with parallel canonical structures of a l.c.K. manifold were also proved in [27]. ¯ J, ω, g). Then Lemma 1.18 ([27]) Let M be a submanifold of a l.c.K. manifold ( M, P is parallel if and only if A FU V − A F V U =

1 {ω(F V )U − ω(FU )V } 2

for any U, V ∈ T M. Theorem 1.19 ([27]) Let M be a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then P is parallel, i.e., ∇˜ P ≡ 0, if and only if the totally real dis( M, tribution D⊥ is autoparallel. Theorem 1.20 ([27]) Let M be a totally umbilical hemi-slant submanifold of a l.c.K. ¯ J, ω, g). If P is parallel and dim(D⊥ ) ≥ 2, then the mean curvature manifold ( M, vector field H˜ of M belongs to distribution D. ¯ J, ω, g). Then Lemma 1.21 ([27]) Let M be a submanifold of a l.c.K. manifold ( M, F is parallel if and only if 1 Anξ U + Aξ PU = − {ω(nξ)U + ω(ξ)PU } 2 for any U ∈ T M and ξ ∈ T ⊥ M. Theorem 1.22 ([27]) Let M be a proper hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). If F is parallel, then M is mixed geodesic. ( M,

2 Warped Product Hemi-slant Submanifolds in Locally Conformal Kaehler Manifolds In this section, we study warped product hemi-slant submanifolds of the form M ⊥ × f M θ with warping function f on M ⊥ , where M ⊥ is a totally real and M θ is a slant submanifold of a locally or globally conformal Kaehler manifold. We give a necessarry and sufficient condition for a hemi-slant submanifold of a locally conformal Kaehler manifold to be a locally warped product submanifold. Then we establish

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a general inequality for warped product mixed geodesic hemi-slant submanifolds and get some results for such submanifolds by using the equality sign of the general inequality.

2.1 Warped Products Let (M1 , g1 ) and (M2 , g2 ) be any Riemannian manifolds, and let f is a positive smooth function defined on M1 . Also, π1 and π2 are canonical projections of M1 × M2 onto M1 and M2 , respectively. Then the warped product manifold [4] M1 × f M2 is the product manifold M¯ = M1 × M2 equipped with metric g defined by g = π1∗ (g1 ) + ( f ◦ π1 )2 π2∗ (g2 ) ,

(39)

where πi∗ (gi ) is the pullback of gi via πi for i = 1, 2. The function f is called a warping function of the warped product M1 × f M2 . If f is a constant, then we get direct product manifold [10]. A warped product manifold is said to non-trivial if f is not a constant. Let M1 × f M2 be a warped product manifold with the Levi-Civita connection ∇¯ which is calculated with respect to the metric g given in (39). Also, ∇ i denote the Levi-Civita connection of Mi for i ∈ {1, 2}. By usual convenience, we denote the set of lifts of vector fields on Mi by L(Mi ) and use the same notation for a vector field and for its lift. On the other hand, π1 is an isometry and π2 is a (positive) homothety, so they preserve the Levi-Civita connection. Thus, there is no confusion using the same notation for a connection on Mi and for its pullback via πi . Then, we have ∇ X Y = ∇ X1 Y ,

(40)

∇ X V = ∇V X = X (ln f ◦ π1 )V ,

(41)

∇U V = ∇U2 V − g(U, V )∇(ln f ◦ π1 )

(42)

for X, Y ∈ L(M1 ) and U, V ∈ L(M2 ). The manifold (M2 , g2 ) is called a fiber of the warped product and the manifold (M1 , g1 ) is called a base manifold of M1 × f M2 . As well known, the base manifold is totally geodesic and the fiber is spherical in M1 × f M2 . More details on warped products we refer to the book [10]. Remark 2.1 Throughout this secion, we will use the same symbol for the warping function f and its pullback f ◦ π1 , i.e., we will put f = f ◦ π1 .

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2.2 Warped Product Hemi-slant Submanifolds of a l.c.K. Manifold In this subsection, we give a characterization for a warped product hemi-slant submanifold in the form M ⊥ × f M θ with warping function f ∈ C ∞ (M ⊥ ) of a l.c.K. ¯ J, ω, g). We first give an (non-trivial) example of such a submanifold. manifold ( M, Example 2.2 Let M be a proper hemi-slant submanifold of a g.c.K. manifold as in Example 1.7 with eσ = y52 and y5  = 0. Then we can see that the totally real distribution D⊥ is totally geodesic and the slant distribution Dθ is integrable. Let denote the integral submanifolds of D⊥ and Dθ by M ⊥ and M θ , respectively. Let g⊥ and gθ be the induced metrics on M ⊥ and M θ with respect to the Kaehler metric g0 , respectively. We choose the conformal Riemann metric g¯⊥ = (x)2 g⊥ on M ⊥ . Since y5 = x on M, then the induced metric of M from the conformal Kaehler metric g is given by ds 2 = x 2 d x 2 + x 2 (du 2 + dv 2 ) = x 2 g⊥ + x 2 gθ = g¯⊥ + x 2 gθ . Thus, M ⊥ × f M θ is a warped product of (M ⊥ , g¯⊥ ) and (M θ , gθ ). So, it s a non-trivial warped product proper hemi-slant submanifold in g.c.K. manifold (R6 , J, g) with warping function f = x. Moreover, the Lee form of (R6 , J, g) is ω=

2 dx . x

Consequently, the Lee vector field is   2 1 ∂ B= 2 x x ∂x which is tangent to M ⊥ . Lemma 2.3 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of ¯ J, ω, g). Then, for all X ∈ L(M ⊥ ), we have a l.c.K. manifold ( M, ω(X ) = 23 X (ln f ) .

(43)

Proof Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of a l.c.K. ¯ J, ω, g) and U, V ∈ L(M θ ) and X ∈ L(M ⊥ ). Then, we have manifold ( M,

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3d(X, U, V ) = X (U, V ) + U (V, X ) + V (X, U ) − ([X, U ], V ) − ([U, V ], X ) − ([V, X ], U ) = X g(U, P V ), since [X, V ] = [X, U ] = 0 from (41) and [U, V ] = ∇Uθ V − ∇Vθ U ∈ (T M θ ) from (42). After some calculation in view of (41), we obtain 3d(X, U, V ) = 2X (ln f )g(U, P V ) .

(44)

On the other hand, we have d(X, U, V ) = ω ∧ (X, U, V ) = ω(X )(U, V ) + ω(U )(V, X ) + ω(V )(X, U ) = ω(X )g(U, P V ) . from (1). Namely, d(X, U, V ) = ω(X )g(U, P V ) . Thus, the assertion follows from (44) to (45).

(45) 

Theorem 2.4 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold ¯ J, ω, g). Then M is a locally direct product submanifold in of a l.c.K. manifold ( M, the form M ⊥ × M θ if and only if the Lee vector field B is normal to M ⊥ . Proof Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of a l.c.K. ¯ J, ω, g). If M is a locally direct product submanifold in the form manifold ( M, ⊥ θ M × M , then for any X ∈ L(M ⊥ ), X (ln f )=0, since f is a constant. From (43), we find g(B, X ) = 0. So, the Lee vector field B is normal to M ⊥ . Conversely, if the Lee vector field B is normal to M ⊥ , we have g(B, X ) = 0. Then, we get X (ln f ) = 0 for any X ∈ L(M ⊥ ) from (43). So f = c, where c is a constant. Then the induced metric tensor g M of M has the form g M = g⊥ ⊕ g˜θ , where  g˜θ = c2 gθ . Thus, M = M ⊥ × M θ is a locally direct product. Remark 2.5 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of ¯ J, ω, g). Here, if we choose θ = 0, then we get warped product a l.c.K. manifold ( M, CR-submanifold in the form M1 = M ⊥ × f M T , where M T is a holomorphic sub¯ J, ω, g) studied by Bonanzinga and Matsumoto manifold of the l.c.K. manifold ( M, in [5]. Thus, by Theorem 2.4, we get Theorem 2.2 of [5]. Lemma 2.6 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of ¯ J, ω, g). Then, for all V ∈ L(M θ ), we have a l.c.K. manifold ( M, ω(V ) = 0 .

(46)

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Proof Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of a l.c.K. ¯ J, ω, g). Then we have dω = 0 from (1). Hence, using the exterior manifold ( M, differentiation formula (see, [32, p. 17]), we obtain 0 = dω(V, X ) = V ω(X ) − X ω(V ) − ω([V, X ]) for V ∈ L(M θ ) and X ∈ L(M ⊥ ). Hence, it follows that V ω(X ) = X ω(V ) ,

(47)

since [V, X ] = 0. Here, using (43), (41) and (2), we have 3 V ω(X ) = V [X (ln f )] = V [g(X, ∇ ln f )] 2 = g(∇V X, ∇ ln f ) + g(X, ∇V ∇ ln f ) = g(X (ln f )V, ∇ ln f ) + g(X, ∇ ln f (ln f )V ) = 0. Hence, we obtain V ω(X ) = 0 .

(48)

On the other hand, using (41) and (2), we have X ω(V ) = X g(B, V ) = X g(B θ , V ) = g(∇ X B θ , V ) + g(B θ , ∇ X V ) = g(X (ln f )B θ , V ) + g(B θ , X (ln f )V ) = 2ω(V )X (ln f ) . Namely, we have X ω(V ) = 2ω(V )X (ln f ) . Now, using (47)–(49), we get (46), since f is not a constant.

(49) 

Theorem 2.7 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold ¯ J, ω, g). Then M is a locally direct product submanifold in of a l.c.K. manifold ( M, ⊥ θ the form M × M if and only if the Lee vector field B is normal to M. Proof Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of a l.c.K. ¯ J, ω, g). If M is a locally direct product submanifold in the form manifold ( M, ⊥ θ M × M , then B is normal to M ⊥ from Theorem 2.4. On the other hand, for any V ∈ L(M θ ), ω(V ) = 0 from (46). Since g(B, V ) = 0, B is normal to M θ . So, the Lee vector field B is normal to M. Conversely, if the Lee vector field B is normal to M, we have g(B, V ) = 0 and g(B, X ) = 0, for any X ∈ L(M ⊥ ) and V ∈ L(M θ ). Then, we get X (ln f ) = 0 from (43). So, f is a constant, say f = c. Then the induced metric tensor g M of M has the form g M = g⊥ ⊕ c2 gθ , where c is constant. Thus, M = M ⊥ × M θ is a locally direct product. 

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By using (41), (30) and (46), we deduce the following result. Lemma 2.8 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of ¯ J, ω, g). Then we have a l.c.K. manifold ( M, g(A J X P V − A F P V X, Y ) = 21 ω(F P V )g(X, Y )

(50)

for any X, Y ∈ L(M ⊥ ) and V ∈ L(M θ ). By using (42), (31) and (43), we deduce the following result. Lemma 2.9 Let M = M ⊥ × f M θ be a warped product hemi-slant submanifold of ¯ J, ω, g). Then we have a l.c.K. manifold ( M, g(A J X P V − A F P V X, U ) = − cos2 θ ω(X )g(V, U ) − 21 ω(J X )g(P V, U )

(51)

for any X ∈ L(M ⊥ ) and U, V ∈ L(M θ ). Now, we recall the following fact to prove the main theorem of this section. Remark 2.10 ([16]) (cf. [11], Remark 2.1) Let M be a pseudo-Riemannian manifold with pseudo-Riemann metric g and call (D1 , D2 ) the canonical foliations. Suppose that D1 and D2 are orthogonal with respect to g. Then (M,g) is a warped product M1 × f M2 if and only if D1 is totally geodesic and D2 is spherical, where M1 (resp. M2 ) is the integral manifold of D1 (resp. D2 )). We now are ready to prove main theorem. ¯ J, ω, g). Theorem 2.11 Let M be a hemi-slant submanifold of a l.c.K. manifold ( M, Then M is a locally warped product submanifold if and only if its shape operator A satisfies the following equation AJ X P V − AF PV X =

1 1 ω(F P V )X − cos2 θ ω(X )V − ω(J X )P V 2 2

(52)

for X ∈ (D⊥ ) and V ∈ (Dθ ). ¯ J, ω, g) of Proof Let M be a warped product submanifold of a l.c.K. manifold ( M, type M ⊥ × f M θ . For any X ∈ L(M ⊥ ) and V ∈ L(M θ ), we write  AJ X P V − AF PV X =

AJ X P V − AF PV X

⊥

 + AJ X P V − AF PV X

θ

, (53)

⊥  is the tangent part of A J X P V − A F P V X to M ⊥ and where A J X P V − A F P V X θ  A J X P V − A F P V X is the tangent part of A J X P V − A F P V X to M θ . Hence, for any Y ∈ L(M ⊥ ), using (50), we have

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 g(A J X P V − A F P V X, Y ) = g

 1 ω(F P V )X, Y . 2

Since Y ∈ L(M ⊥ ) is arbitrary and the metric g is Riemann, it follows that  AJ X P V − AF PV X

⊥

= 21 ω(F P V )X .

(54)

Similarly, for any U ∈ L(M θ ), using (51), we have 

 1 g(A J X P V − A F P V X, U ) = g − cos θ ω(X )V − ω(J X )P V, U . 2 2

Since U ∈ L(M θ ) is arbitrary and the metric g is Riemann, it follows that θ

 AJ X P V − AF PV X

= − cos2 θ ω(X )V − 21 ω(J X )P V .

(55)

Thus, by (53) ∼ (55), we get (52). Conversely, suppose that M is a hemi-slant submanifold of a l.c.K. manifold ¯ J, ω, g) such that (52) holds. Then, for any X ∈ (D⊥ ) and U, V ∈ (Dθ ), ( M, using (52), we deduce that (32). Thus, by Lemma 1.14, the slant distribution Dθ is integrable. On the other hand, from Theorem 1.12, we already know that the totally real distribution D⊥ is integrable. Let M ⊥ and M θ be the integral manifolds of D⊥ and Dθ , respectively and let denote by h ⊥ and h θ the second fundamental forms of M ⊥ and M θ in M, respectively. Then, for any X, Y ∈ (D⊥ ) and V ∈ (Dθ ), using (5), we have g(h ⊥ (X, Y ), V ) = g(∇ X Y, V ) . Here, if we use (30) and (52), we find g(h ⊥ (X, Y ), V ) = 0 . Hence we conclude that h ⊥ (X, Y ) = 0. Thus D⊥ is totally geodesic. On the other hand, for any X ∈ (D⊥ ) and U, V ∈ (Dθ ), using (5), we have g(h θ (U, V ), X ) = g(∇U V, X ) . Here, if we use (31) and (52), we find g(h θ (U, V ), X ) = − 23 ω(X )g(U, V ) .

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After some calculation, we obtain g(h θ (U, V ), X ) = g(−g(U, V ) 23 B ⊥ , X ) . Hence, we conclude that h θ (U, V ) = −g(U, V ) 23 B ⊥ . It means that Dθ is totally umbilic with the mean curvature vector field − 23 B ⊥ . What’s left to show that − 23 B ⊥ is parallel. We have to satisfy g(∇V (− 23 B ⊥ ), X )=0, for X ∈ (D⊥ ) and V ∈ (Dθ ). Since   3 3 3 g(∇V − B ⊥ ), X = − g(∇V B ⊥ , X ) = − g(B ⊥ (ln f )V, X ) = 0, 2 2 2 then Dθ is spherical since it is also totally umbilic. Thus, by Remark 2.10, M is a  locally warped product of type M ⊥ × f M θ .

2.3 An Inequality for Warped Product Mixed Geodesic Hemi-slant Submanifolds In this subsection, we shall establish an inequality for the squared norm of the second fundamental form of a warped product mixed geodesic hemi-slant submanifold in the form M ⊥ × f M θ , where M ⊥ is a totally real and M θ is a slant submanifold of a ¯ J, ω, g). l.c.K. manifold ( M, Lemma 2.12 Let M be a warped product hemi-slant submanifold in the form M ⊥ × f M θ , where M ⊥ is a totally real and M θ is a slant submanifold of a l.c.K. ¯ J, ω, g). Then, we have manifold ( M, g(h(X, Y ), F V ) = g(h(X, V ), J Y ) − 21 ω(F V )g(X, Y ) ,

(56)

g(h(U, V ), J X ) = g(h(U, X ), F V ) + 23 X (ln f )g(P V, U ) − 21 ω(J X )g(U, V ) , (57) for X, Y ∈ L(M ⊥ ) and U, V ∈ L(M θ ). Proof Let X, Y ∈ L(M ⊥ ) and U, V ∈ L(M θ ), by interchanging V with P V in (50) and (51), respectively, and using (7) and (14), we get (56) and (57) respectively.  Remark 2.13 We say that a hemi-slant submanifold M is mixed geodesic, if h(X, V ) = 0 for X ∈ (D⊥ ) and V ∈ (Dθ ). By Remark 2.13 together with (2), we have that:

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Corollary 2.14 Let M = M ⊥ × f M θ be a warped product mixed geodesic hemi¯ J, ω, g). If the Lee vector field B is tangent slant submanifold of a l.c.K. manifold ( M, to M ⊥ , then the Eqs. (56) and (57) become g(h(X, Y ), F V ) = 0 ,

(58)

g(h(U, V ), J X ) = 23 X (ln f )g(P V, U ) ,

(59)

respectively, where X, Y ∈ L(M ⊥ ) and U, V ∈ L(M θ ). Let M = M ⊥ × f M θ be a (m 1 + m 2 )-dimensional warped product hemi-slant sub¯ J, ω, g). We choose a canonical orthonormal manifold of a l.c.K. manifold ( M, basis {e1 , ..., em 1 , e¯1 , ..., e¯m 2 , J e1 , ..., J em 1 , e1∗ , ..., em∗ 2 , eˆ1 , ..., eˆl } of M¯ such that {e1 , ..., em 1 } is an orthonormal basis of D⊥ , {e¯1 , ..., e¯m 2 } is an orthonormal basis of Dθ , {J e1 , ..., J em 1 } is an orthonormal basis of J D⊥ , {e1∗ , ..., em∗ 2 } is an orthonormal basis of FDθ and {eˆ1 , ..., eˆl } is an orthonormal basis of D. Here, m 1 = dim(D⊥ ), m 2 = dim(Dθ ) and l = dim(D). Remark 2.15 In view of (15), we can observe that {secθ P e¯1 , ..., secθ P e¯m 2 } is also an orthonormal basis of Dθ and {cscθF e¯1 , ..., cscθF e¯m 2 } is also an orthonormal basis of FDθ , where θ is the slant angle of Dθ . Theorem 2.16 Let M = M ⊥ × f M θ be a warped product mixed geodesic hemi¯ J, ω, g) such that the Lee vector field B slant submanifold of a l.c.K. manifold ( M, ⊥ is tangent to M . Then the squared norm of the second fundamental form h of M satisfies h2 ≥ m 2 (m 2 − 1) cos2 θB ⊥ 2 , (60) where m 2 = dim(M θ ). Proof By the hypothesis, the squared norm of the second fundamental form h can be written as h2 = h(D⊥ , D⊥ )2 + h(Dθ , Dθ )2 . In view of decomposition (13), which can be explicitly written as follows: h2 =

m1 

g(h(ei , e j ), J ek )2 +

m1 m2  

g(h(e¯a , e¯b ), J ei )2 +

a,b=1 i=1

+

m 1 +m 2

g(h(ei , e j ), ea∗ )2

i, j=1 a=1

i, j,k=1

+

m1  m2 

l 

r,s=1 t=1

m1  a,b,c=1

g(h(e˜r , e˜s ), eˆt )2 .

g(h(e¯a , e¯b ), ec∗ )2

(61)

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where the set {e˜r }1≤r≤(m 1 +m 2 ) is an orthonormal basis of M. Hence, we get h2 ≥

m2 m1  

g(h(ei , e j ), ea∗ )2 +

i, j=1 a=1

m1 m2  

g(h(e¯a , e¯b ), J ei )2 .

a,b=1 i=1

By Remark 2.15, we arrive h2 ≥

m2 m1  

g(h(ei , e j ), cscθF e¯a )2 +

i, j=1 a=1

m1 m2  

g(h(e¯a , e¯b ), J ei )2 .

a,b=1 i=1

Now, using (58) and (59), we obtain h2 ≥

m2  m1 4  (ei (ln f ))2 g 2 (P e¯a , e¯b ). 9 a,b=1 i=1

Here, for a, b ∈ {1, 2, ..., m 2 }, we have  g(P e¯a , e¯b ) =

cosθ if a  = b, 0 if a = b,

since Dθ is a slant distribution with slant angle θ. Thus, by direct calculation, we obtain the following inequality. h2 ≥

4 m 2 (m 2 − 1) cos2 θ∇ ln f 2 . 9

(62)

On the other hand, by (43), we conclude that B⊥ =

2 ∇(ln f ). 3

Hence, using (63) in (62), we get the inequality (60).

(63) 

Theorem 2.17 Let M = M ⊥ × f M θ be a warped product mixed geodesic hemi¯ J, ω, g) such that the Lee vector field B is slant submanifold of a l.c.K. manifold ( M, tangent to M ⊥ . If the invariant subnormal bundle D = {0}, then the equality sign of (60) holds if and only if A J X Y ∈ L(M θ ) and A FU V ∈ L(M ⊥ ), where X, Y ∈ L(M ⊥ ) and U, V ∈ L(M θ ). Proof Under the given hypothesis, we see that the equality sign of (60) holds if and only if g(h(D⊥ , D⊥ ), J D⊥ ) = 0 and g(h(Dθ , Dθ ), FDθ ) = 0 from (61). These are equivalent to

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g(h(Y, Z ), J X ) = 0

and

g(h(V, W ), FU ) = 0

for X, Y, Z ∈ L(M ⊥ ) and U, V, W ∈ L(M θ ). But, with the help of (7), we know these conditions hold if and only if A J X Y ∈ L(M θ )

and

A FU V ∈ L(M ⊥ ) . 

Theorem 2.18 Let M = M ⊥ × f M θ be a non-trivial warped product mixed ¯ J, ω, g) such that the Lee geodesic hemi-slant submanifold of a l.c.K. manifold ( M, vector field B is tangent to M ⊥ and the invariant subnormal bundle D = {0}. If the equality sign of (60) holds, then M θ is also totally umbilic in the ambient manifold ¯ M. ¯ Then, for a ∈ Proof Let h¯ θ denote the second fundamental form of M θ in M. {1, ..., m 2 }, we have h¯ θ (e¯a , e¯a ) = h θ (e¯a , e¯a ) + h(e¯a , e¯a ) ,

(64)

where {e¯1 , ..., e¯m 2 } is an orthonormal basis of M θ and h θ is the second fundamental ¯ Since M = form of M θ in M, and h is the second fundamental form of M in M. ⊥ θ M × f M is a non-trivial warped product, we see that h θ (e¯a , e¯a ) = −g(U, V )∇(ln f )  = 0 from (42). On the other hand, we know h(Dθ , Dθ ) ⊆ J D⊥ from Theorem 2.17. Thus, we have m1  h(e¯a , e¯a ) = g(h(e¯a , e¯a ), J ei )J ei , i=1

where {e1 , ..., em 1 } is an orthonormal basis of M ⊥ . Here, for each a ∈ {1, ..., m 2 } and i ∈ {1, ..., m 1 }, using (59), we find g(h(e¯a , e¯a ), J ei ) = 23 ei (ln f )g(P e¯a , e¯a ) = 0 , since g(P e¯a , e¯a ) = 0. Which means that h(e¯a , e¯a ) = 0 for each a ∈ {1, ..., m 2 }. It follows that

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h¯ θ (e¯a , e¯a ) = h θ (e¯a , e¯a ) ¯ since it is totally umbilic in M. from (64). Thus, M θ is totally umbilic in M,



Remark 2.19 In the case of θ = 0, i.e., in the case of warped product CRsubmanifold M ⊥ × f M T , where M T is a holomorphic submanifold, Theorem 2.18 is still true without any restrictions, see Proposition 2.1 of [5].

3 Semi-slant Submanifolds of an Almost Hermitian Manifold The other subclass of bi-slant submanifolds is the semi-slant submanifolds which were first defined by Papaghiuc [22]. A semi-slant submanifold is a natural generalization of both slant and CR-submanifold and a special case of generic submanifold which was defined by Ronsse [23]. Since then many geometers have studied semi-slant submanifolds of manifolds equipped with different kind of structures. For example; see [6, 17]. ¯ J, g) is A semi-slant submanifold [22] M of an almost Hermitian manifold ( M, a submanifold which admits two orthogonal complementary distributions D and Dθ such that D is holomorphic, i.e., J D = D and Dθ is slant with slant angle θ  = 0. We note that both holomorphic and slant submanifolds are even dimesional in almost Hermitian structures. Now, we suppose the dimension of distribution D (resp.Dθ ) is m 1 = 2n 1 (resp. m 2 = 2n 2 ). Then we easily see the following particular cases. (a) If m 2 = 0, then M is a holomorphic submanifold [32]. (b) If m 1 = 0 and θ = π2 , then M is a totally real submanifold [32]. (c) If m 1 m 2  = 0 and θ = π2 , then M is a proper C R submanifold [3]. (d) If m 1 = 0 and θ  = π2 , then M is a proper slant submanifold [8]. We say that a semi-slant submanifold is proper if m 1 m 2  = 0 and θ  = π2 [22]. Now, let M be a semi-slant submanifold of an almost Hermitian manifold ¯ J, g). Then the tangent bundle T M and the normal bundle T ⊥ M of M are ( M, decomposed as T M = D ⊕ Dθ (65) and

T ⊥ M = FDθ ⊕ μ

(66)

where μ is the orthogonal complementary distribution of FDθ in T ⊥ M and μ is invariant subbundle of T ⊥ M with respect to J. Moreover, using (11)–(12) and the almost Hermitian structure, we have

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(a) PD = D, (b) FD = {0}, (c) PDθ ⊆ Dθ , (d) t (T ⊥ M) = Dθ .

(67)

Remark 3.1 Comparing the definitions of hemi-slant and semi-slant submanifolds, we observe that there exists no inclusion relation between the notion of hemi-slant submanifold and the notion of semi-slant submanifold in proper case.

3.1 Semi-slant Submanifolds of a Locally Conformal Kaehler Manifold We first give an example. Example 3.2 Let (R6 , J, g) be a g.c.K. manifold as in Example 1.7 and let M be a submanifold of (R6 , J, g) defined by y1 = x , y2 = y , y3 = u + v , y4 = −u + v , y5 = u , y6 = 0 , where x, y, u, v  = 0. Then, the local orthonormal frame field of the tangent bundle T M of M is given by     1 1 X = ∂1 , Y = ∂2 , U = √ ∂3 − ∂4 + ∂5 , V = √ ∂3 + ∂4 , 3 2 where ∂i = ∂∂yi for i ∈ {1, 2, ..., 6}. Then D = span{X, Y } is a holomorphic and Dθ = span{U, V } is a (proper) slant distribution with the slant angle θ = cos−1 ( √26 ). Thus, M is a proper semi-slant submanifold of (R6 , J, g). The following results on semi-slant submanifolds of a l.c.K. manifold were proved in [28]. ¯ J, ω, g). Lemma 3.3 Let M be a semi-slant submanifold of a l.c.K. manifold ( M, Then the holomorphic distribution D is integrable if and only if h(X, J Y ) − h(J X, Y ) = g(J X, Y )B N

(68)

for any X, Y ∈ D. ¯ J, ω, g). Corollary 3.4 Let M be a semi-slant submanifold of a l.c.K. manifold ( M, with integrable holomorphic distribution D of dimension greater than 4. If the leaves of D are totally geodesic in M, then B ∈ T M.

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¯ J, ω, g) Corollary 3.5 Let M be a semi-slant submanifold of a l.c.K. manifold ( M, with integrable holomorphic distribution D. If the leaves of D are totally umbilical in M, then 1 H = − BN, 2 where H is the mean curvature vector field of the leaves of D in M. Lemma 3.6 ([15]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then we have ( M,    1 g(∇ X Y, U ) = csc2 θ g A FU J Y − A F PU Y, X + ω(FU )g(J Y, X ) 2  1 1 − ω(F PU )g(X, Y ) − ω(U )g(X, Y ) 2 2

(69)

and 



g(∇U V, X ) = − csc θg A F V J X − A F P V X, U 2

− 21 ω(X )g(U, V ) ,

(70)

for X, Y ∈ (D) and U, V ∈ (Dθ ). ¯ J, ω, g˜ = e−σ g) is a Kaehler Proof Let X ∈ (D) and U, V ∈ (Dθ ), since ( M, manifold, using (4), (9), (11) and (14), we have ˜ ∇˜ U V, X ) = g( ˜ ∇˜ U J V, J X ) g( ˜ ∇ˆ U V, X ) =g( =g( ˜ ∇˜ U P V, J X ) + g( ˜ ∇˜ U F V, J X ) ˜ A˜ F V J X, U ) = − g( ˜ ∇˜ U J P V, X ) − g( ˜ ∇˜ U F P V, X ) − g( ˜ A˜ F V J X, U ) = − g( ˜ ∇˜ U P 2 V, X ) − g( = cos2 θg( ˜ ∇ˆ U V, X ) + g( ˜ A˜ F P V X, U ) − g( ˜ A˜ F V J X, U ). Hence, it follows that   2 ˜ ˜ ˆ g( ˜ ∇U V, X ) = − csc θ g( ˜ A F V J X − A F P V X, U ) . Now, by using (2), (20) and (22), we find that the assertion (70). The other assertion (69) can be obtained by a similar method.  We have an alternative to Lemma 3.3 in proper case. Lemma 3.7 ([15]) Let M be a proper semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the holomorphic distribution D is integrable if and only if ( M,

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g(A FU J Y, X ) − g(A FU J X, Y ) = ω(FU )g(J X, Y )

(71)

for X, Y ∈ (D) and U ∈ (Dθ ). ¯ J, ω, g). Proof Let M be a proper semi-slant submanifold of a l.c.K. manifold ( M, Then the holomorphic distribution D is integrable if and only if g([X, Y ], U ) = 0 for all X, Y ∈ (D) and U ∈ (Dθ ). Thus, the assertion (71) comes from (69).  Lemma 3.8 ([15]) Let M be a proper semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the slant distribution Dθ is integrable if and only if ( M, g(A F V J X − A F P V X, U ) = g(A FU J X − A F PU X, V )

(72)

for X ∈ (D) and U, V ∈ (Dθ ). ¯ J, ω, g). Proof Let M be a proper semi-slant submanifold of a l.c.K. manifold ( M, θ Then the slant distribution D is integrable if and only if g([U, V ], X ) = 0 for all  X ∈ (D) and U, V ∈ (Dθ ). Thus, the assertion (72) follows from (70). By using (69), we have the following result. Lemma 3.9 ([15]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the holomorphic distribution D is totally geodesic if and only ( M, if g( A F V J Y − A F P V Y, X ) =

1 2



  sin2 θω(V ) + ω(F P V ) g(Y, X ) − ω(F V )g(J Y, X )

for X, Y ∈ (D) and V ∈ (Dθ ).

(73)

Now, we are going to study the problem when a semi-slant submanifold of a l.c.K. manifold is a Riemannian product of a holomorphic and a slant submanifold. We first give an alternative result to Lemma 3.9. Lemma 3.10 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the holomorphic distribution D defines a totally geodesic foliation ( M, on M if and only if     1 1 g An F Z Y + ω(n F Z )Y, X = −g A F Z PY + ω(F Z )PY, X 2 2 for all X, Y ∈ (D) and Z ∈ (Dθ ). By Corollary 3.4 and Lemma 3.10 with the help of (2) and (7), we have that: Corollary 3.11 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). If the holomorphic distribution D is integrable and its leaves are totally ( M, geodesic in M, then g(h(D, D), n FDθ ) = −g(h(PD, D), FDθ ).

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Lemma 3.12 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then the slant distribution Dθ defines a totally geodesic foliation on ( M, M if and only if 

1 g An F W X + ω(n F W )X, Z 2



1 = −g(A F W P X + ω(F W )P X, Z ) 2

for all Z , W ∈ (Dθ ) and X ∈ (D). By Lemmas 3.10 and 3.12 we have the following result. Theorem 3.13 [28] Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). Then M is a locally Riemannian product manifold M = MD × MDθ ( M, if and only if 1 1 An F Z X + ω(n F Z )X = −{A F Z P X + ω(F Z )P X } 2 2 for X ∈ (D) and Z ∈ (Dθ ), where MD is integral manifold of the holomorphic distribution D and MDθ is integral manifold of the slant distribution Dθ .

3.2 Semi-slant Submanifolds with Parallel Canonical Structures In this subsection, we recall the recent results for semi-slant submanifolds of a l.c.K. manifold with parallel canonical projection structures on the tangent bundle of the submanifold. ¯ J, ω, g). Then Lemma 3.14 ([28]) Let M be a submanifold of a l.c.K. manifold ( M, P is parallel if and only if A FU V − A F V U =

1 {ω(F V )U − ω(FU )V } 2

for any U, V ∈ T M. ¯ J, ω, g). If Corollary 3.15 ([28]) Let M be a submanifold of a l.c.K. manifold ( M, P is parallel, then g(h(D, Dθ ), FDθ ) = 0. Corollary 3.16 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). If P is parallel, then the holomorphic distribution D is integrable. ( M, Theorem 3.17 ([28]) Let M be a totally umbilical semi-slant submanifold of a l.c.K. ¯ J, ω, g). If P is parallel, then the mean curvature vector field H˜ of M manifold ( M, belongs to distribution μ.

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We remark that Theorem 3.17 is an analog of Proposition 7.1 [29] concerning generic submanifolds (in the sense of Ronsse [23]). ¯ J, ω, g). Then Lemma 3.18 ([28]) Let M be a submanifold of a l.c.K. manifold ( M, F is parallel if and only if 1 Anξ U + Aξ PU = − {ω(nξ)U + ω(ξ)PU } 2 for any U ∈ T M and ξ ∈ T ⊥ M. Theorem 3.19 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g). If F is parallel, then the holomorphic distribution D is integrable ( M, and its leaves are totally geodesic in M. Lemma 3.20 ([28]) Let M be a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g) with holomorphic distribution D of dimension greater than 4. If F ( M, is parallel, then Anξ U = −Aξ PU (74) for any U ∈ T M and ξ ∈ T ⊥ M.

4 Warped Product Semi-slant Submanifolds of a l.c.K. Manifold In this section, we study the warped product semi-slant submanifolds of the form M T × f M θ with warping function f on M T , where M T is a holomorphic and M θ is a slant submanifold of a locally or globally conformal Kaehler manifold. We give a necessary and sufficient condition for a semi-slant submanifold of a locally conformal Kaehler manifold to be a locally warped product submanifold. Then we establish a general inequality for warped product semi-slant submanifolds. The equality case is also discussed. We first give a (non-trivial) example of such a submanifold. Example 4.1 Let M be a proper semi-slant submanifold of a g.c.K. manifold as in Example 3.2 with eσ = (y1 y2 )2 and y1 , y2  = 0. One can see that the holomorphic distribution D is totally geodesic and the slant distribution Dθ is integrable. Let denote the integral submanifolds of D and Dθ by M T and M θ , respectively. Let g T and gθ be the induced metrics on M T and M θ with respect to the Kaehler metric, respectively. We choose the conformal Riemann metric g¯ T = (x y)2 g T on M T . Since x = y1 and y = y2 on M, the induced metric of M from the conformal Kaehler metric g is

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ds 2 = (x y)2 (d x 2 + dy 2 ) + (x y)2 (du 2 + dv 2 ) = (x y)2 gT + (x y)2 gθ = g¯T + (x y)2 g¯θ . Thus, M T × f M θ is a warped product of (M T , g¯T ) and (M θ , gθ ). So, it is a (nontrivial) warped product proper semi-slant submanifold in g.c.K. manifold (R6 , J, g) with warping function f = x y. Moreover, the Lee form of (R6 , J, g) is  1 1 ω = 2 d x + dy . x y 

Consequently, the Lee vector field is   1 ∂ 1 ∂ 2 + B= (x y)2 x ∂x y ∂y which is tangent to M T . Lemma 4.2 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then, for all V ∈ (T M θ ), we have a l.c.K. manifold ( M, ω(V ) = 0 .

(75)

Proof Let M = M T × f M θ be a warped product semi-slant submanifold of a l.c.K. ¯ J, ω, g) and V ∈ (T M θ ) and X, Y ∈ (T M T ). Then using the extemanifold ( M, rior differentiation formula, we have 3d(V, X, Y ) = V (X, Y ) + X (Y, V ) + Y (V, X ) − ([V, X ], Y ) − ([X, Y ], V ) − ([Y, V ], X ) = V g(X, J Y ) − X g(J Y, V ) + Y g(V, J X ) − g([V, X ], J Y ) + g(J [X, Y ], V ) − g([Y, V ], J X ). Here, we know g(J Y, V ) = g(V, J X ) = 0, since M is a semi-slant submanifold. Also, by (41), we have [V, X ] = [Y, V ] = 0 and by (40), we have [X, Y ] = ∇ XT Y − ∇YT X . So J [X, Y ] ∈ (T M T ). Thus, we obtain 3d(V, X, Y ) = V g(X, J Y ) = g(∇V X, J Y ) + g(∇V Y, J X ). Again, using (41), we find 3d(V, X, Y ) = X (ln f )g(V, J Y ) + Y (ln f )g(V, J X ) = 0.So, d(V, X, Y ) = 0. On the other hand, from (1) we have

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d(V, X, Y ) = ω ∧ (V, X, Y ) = ω(V )(X, Y ) + ω(X )(Y, V ) + ω(Y )(V, X ) = ω(V )g(X, J Y ) . Since g is non-degenerate, it follows that ω(V ) = 0.



θ

Remark 4.3 Let M × f M be a warped product semi-slant submanifold of a l.c.K. ¯ J, ω, g). Here, if we choose θ = π , then we get warped product CRmanifold ( M, 2 submanifold in the form M2 = M T × f M ⊥ studied by Bonanzinga and Matsumoto in [5]. Thus, Proposition 3.1 of [5] deduce from Lemma 4.2. T

Lemma 4.4 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then, for all X ∈ (T M T ), we have a l.c.K. manifold ( M, ω(X ) = 23 X (ln f ) .

(76)

Proof Let M = M T × f M θ be a warped product semi-slant submanifold of a l.c.K. ¯ J, ω, g) and U, V ∈ (T M θ ) and X ∈ (T M T ). Then, we have manifold ( M, 3d(X, U, V ) = X (U, V ) + U (V, X ) + V (X, U ) − ([X, U ], V ) − ([U, V ], X ) − ([V, X ], U ) = X g(U, P V ), since [X, V ] = [X, U ] = 0 from (41) and [U, V ] = ∇Uθ V − ∇Vθ U ∈ (T M θ ) from (42). After some calculation in view of (41), we obtain 3d(X, U, V ) = 2X (ln f )g(U, P V ) .

(77)

On the other hand, we have d(X, U, V ) = ω ∧ (X, U, V ) = ω(X )(U, V ) + ω(U )(V, X ) + ω(V )(X, U ) = ω(X )g(U, P V ) . from (1). Namely, d(X, U, V ) = ω(X )g(U, P V ) . Thus, the assertion follows from (77) and (78).

(78) 

Remark 4.5 In Kaehlerian case, we have ω = 0. Then it follows that f is a constant from (76). Thus, we get Theorem 3.2 of [25]. Theorem 4.6 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then M is a locally direct product submanifold in the a l.c.K. manifold ( M, T θ form M × M if and only if the Lee vector field B is normal to M T .

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Proof Let M = M T × f M θ be a warped product semi-slant submanifold of a l.c.K. ¯ J, ω, g). If M is a locally direct product submanifold in the form manifold ( M, M T × M θ , then for any X ∈ L(M T ), X (ln f )=0, since f is a constant. From (76), we find g(B, X ) = 0. So, the Lee vector field B is normal to M T . Conversely, if the Lee vector field B is normal to M T , we have g(B, X ) = 0. Then, we get X (ln f ) = 0 for any X ∈ L(M T ) from (76). So f = c, where c is a constant. Then the induced metric tensor g M of M has the form g M = gT ⊕ g˜θ , where  g˜θ = c2 gθ . Thus, M = M T × M θ is a locally direct product. Theorem 4.7 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then M is a locally direct product submanifold in the a l.c.K. manifold ( M, form M T × M θ if and only if the Lee vector field B is normal to M. Proof Let M = M T × f M θ be a warped product semi-slant submanifold of a l.c.K. ¯ J, ω, g). If M is a locally direct product submanifold in the form manifold ( M, T θ M × M , then B is normal to M T from Theorem 4.6. On the other hand, for any V ∈ L(M θ ), ω(V ) = 0 from (75). Since g(B, V ) = 0, B is normal to M θ . So, the Lee vector field B is normal to M. Conversely, if the Lee vector field B is normal to M, we have g(B, V ) = 0 and g(B, X ) = 0, for any X ∈ L(M T ) and V ∈ L(M θ ). Then, we get X (ln f ) = 0 from (76). So, f is a constant, say f = c. Then the induced metric tensor g M of M has the form g M = gT ⊕ c2 gθ , where c is constant. Thus, M = M T × M θ is a locally direct product.  By using (69) and (75), we deduce the following result. Lemma 4.8 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then we have a l.c.K. manifold ( M, 

 g A F V J X − A F P V X, Y

 =

1 2

 ω(F P V )g(X, Y ) − ω(F V )g(J X, Y )

(79)

for any X, Y ∈ L(M T ) and V ∈ L(M θ ). By using (71) and (76), we deduce the following result. Lemma 4.9 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g). Then we have a l.c.K. manifold ( M,   g A F V J X − A F P V X, U = sin2 θω(X )g(U, V ). for any X ∈ L(M T ) and U, V ∈ L(M θ ). We now are ready to prove main theorem of this section.

(80)

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¯ J, ω, g). Theorem 4.10 Let M be a semi-slant submanifold of a l.c.K. manifold ( M, Then M is a locally warped product submanifold if and only if its shape operator A satisfies the following equation  AFV J X − AF PV X =

 ω(F P V )X − ω(F V )J X

1 2

(81)

+ sin θω(X )V 2

for X ∈ (D) and V ∈ (D θ ). ¯ J, ω, g) of Proof Let M be a warped product submanifold of a l.c.K. manifold ( M, type M T × f M θ . For any X ∈ L(M T ) and V ∈ L(M θ ), we write  AFV J X − AF PV X =

T AFV J X − AF PV X



θ

+ AFV J X − AF PV X

, (82)

 T where A F V J X − A F P V X is the tangent part of A F V J X − A F P V X to M T and θ  A F V J X − A F P V X is the tangent part of A F V J X − A F P V X to M θ . Hence, for any Y ∈ L(M T ), using (79), we have       1 ω(F P V )X − ω(F V )J X , Y g A F V J X − A F P V X, Y = g 2 Since Y ∈ L(M ⊥ ) is arbitrary and the metric g is Riemann, it follows that  AFV J X − AF PV X =

1 2

 ω(F P V )X − ω(F V )J X .

(83)

Similarly, for any U ∈ L(M θ ), using (80), we have   g A F V J X − A F P V X, U = g(sin2 θω(X )V, U ). Since U ∈ L(M θ ) is arbitrary and the metric g is Riemann, it follows that A F V J X − A F P V X = sin2 θω(X )V .

(84)

Thus, by (82)–(84), we get (81). Conversely, suppose that M is a semi-slant submanifold of a l.c.K. manifold ¯ J, ω, g) such that (81) holds. Then, for any X ∈ (D) and U, V ∈ (Dθ ), using ( M, (75) and (81), we satisfy that (73). Thus, by Lemma 3.9, the holomorphic distribu-

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tion D is totally geodesic. On the other hand, using (76) and (81), we satisfy (72). Thus, by Lemma 3.8, the slant distribution Dθ is integrable. Let M T and M θ be the integral manifolds of D and Dθ , respectively and let denote by h T and h θ the second fundamental forms of M T and M θ in M, respectively. Then, for any X, Y ∈ (D) and V ∈ (Dθ ), using (5), we have g(h T (X, Y ), V ) = g(∇ X Y, V ) . Here, if we use (69) and (81), we find g(h T (X, Y ), V ) = 0 . Hence we conclude that h T (X, Y ) = 0. Thus M T is a totally geodesic submanifold in M. On the other hand, for any X ∈ (D) and U, V ∈ (Dθ ), using (5), we have g(h θ (U, V ), X ) = g(∇U V, X ) . Here, if we use (70) and (81), we find g(h θ (U, V ), X ) = − 23 ω(X )g(U, V ) . After some calculation, we obtain g(h θ (U, V ), X ) = g(− 23 B T g(U, V ), X ) . Hence, we conclude that h θ (U, V ) = −g(U, V ) 23 B T . It means that M θ is totally umbilic submanifold in M with the mean curvature vector field − 23 B T . What’s left to show that − 23 B T is parallel. We have to satisfy g(∇V (− 23 B T ), X )=0 , for X ∈ (D) and V ∈ (Dθ ). Since     3 3 g ∇V − B T , X = − g(B T (ln f )V, X ) = 0, 2 2 then M θ is spherical since it is also totally umbilic. Consequently, Dθ is spherical.  Thus, by Remark 2.10, M is a locally warped product of type M T × f M θ .

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4.1 An Inequality for Warped Product Semi-slant Submanifolds In this subsection, we shall establish an inequality for the squared norm of the second fundamental form of a warped product semi-slant submanifold in the form M T × f M θ , where M T is a holomorphic and M θ is a slant submanifold of a l.c.K. manifold ¯ J, ω, g). ( M, Lemma 4.11 Let M = M T × f M θ be a warped product semi-slant submanifold of ¯ J, ω, g) and h be the second fundamental form of M in M. ¯ a l.c.K. manifold ( M, Then we have (85) g(h(X, Y ), F V ) = − 21 g(X, Y )ω(F V ) , g(h(X, U ), F V ) = −ω(J X )g(U, V ) − ω(X )g(U, P V ) ,

(86)

where X, Y ∈ (T M T ) and U, V ∈ (T M θ ). Proof Let M = M T × f M θ be a warped product semi-slant submanifold of a ¯ J, ω, g) and let X, Y ∈ (T M T ) and V ∈ (T M θ ), since l.c.K. manifold ( M, −σ ¯ ( M, J, ω, g˜ = e g) is a Kaehler manifold, using (8), (11) and (4), we have ˜ g( ˜ h(X, Y ), F V ) = g( ˜ ∇˜ X Y, F V ) = g( ˜ ∇˜ X Y, J V ) − g( ˜ ∇˜ X Y, P V ) = − g( ˜ ∇˜ X J Y, V ) − g( ˜ ∇ˆ X Y, P V ) ˜ ∇ˆ X Y, P V ) . = − g( ˜ ∇ˆ X J Y, V ) − g( Now, using (20), (21) and (75), we get (85). Next, let X, Y ∈ (T M T ) and V ∈ ¯ J, ω, g˜ = e−σ g) is a Kaehler manifold, using (8), (11) and (4), (T M θ ). Since ( M, we have ˜ g( ˜ h(X, U ), F V ) = g( ˜ ∇˜ U X, F V ) = g( ˜ ∇˜ U X, J V ) − g( ˜ ∇˜ U X, P V ) = − g( ˜ ∇˜ U J X, V ) − g( ˜ ∇˜ U X, P V ) ˜ ∇ˆ U X, P V ) . = − g( ˜ ∇ˆ U J X, V ) − g( Now, using (41), (20), (21), (75) and (76), we get (86).



Let M = M T × f M θ be a warped product semi-slant submanifold of a l.c.K. mani¯ J, ω, g). We choose a canonical orthonormal basis {e1 , ..., em 1 , e¯1 , ..., e¯m 2 , fold ( M, e1∗ , ..., em∗ 2 , eˆ1 , ..., eˆl } of M¯ such that {e1 , ..., em 1 } is an orthonormal basis of D, {e¯1 , ..., e¯m 2 } is an orthonormal basis of Dθ , {e1∗ , ..., em∗ 2 } is an orthonormal basis

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of FDθ and {eˆ1 , ..., eˆl } is an orthonormal basis of μ. Here, m 1 = dim(D), m 2 = dim(Dθ ) and l = dim(μ). Remark 4.12 Since D is a holomorphic distribution, {J e1 , ..., J em 1 } is also an orthonormal basis of D. Moreover, by (15), we observe that {a¯ 1 = secθ P e¯2 , a¯ 2 = − secθ P e¯1 , ..., a¯ 2n 2 −1 = secθ P e¯2n 2 , a¯ 2n 2 = − secθ P e¯2n 2 −1 } is also an orthonormal basis of Dθ and {cscθF e¯1 , ..., cscθF e¯m 2 } is also an orthonormal basis of FDθ , where θ is the slant angle of Dθ and m 2 = 2n 2 = dim(M θ ). Theorem 4.13 Let M = M T × f M θ be a warped product semi-slant submanifold ¯ J, ω, g). Then we have a l.c.K. manifold ( M, (i) the squared norm of the second fundamental form h of M satisfies h2 ≥

  1 θ m 1 B F D 2 + m 2 1 + m 2 cot 2 θ B T 2 , 4

(87)

θ

where m 1 = dim(M T ), m 2 = dim(M θ ) and B F D is tangential part of B to FDθ . (ii) If the equality sign of (87) holds identically, then M θ is also umbilic in the ambient ¯ manifold M. Proof The squared norm of the second fundamental form h can be written as h2 = h(D, D)2 + h(D, Dθ )2 + h(Dθ , Dθ )2 . In view of decomposition (66), which can be explicitly written as follows: h = 2

m1  m2 

g(h(er , es ), ei∗ )2

+

r,s=1 i=1

+

m1  l 

m2  m1 

g(h(er , e¯i ), e∗j )2

i, j=1 r=1

g(h(er , es ), eˆt )2 +

r,s=1 t=1

m1  m2  l 

g(h(er , e¯i ), eˆt )2

r=1 i=1 t=1

+ h(Dθ , Dθ )2 . Hence, we have h2 ≥

m1  m2 

g(h(er , es ), ei∗ )2 +

r,s=1 i=1

m2  m1 

g(h(e¯i , er ), e∗j )2 .

i, j=1 r =1

By Remark 4.12, we obtain h2 ≥

m1  m2  r,s=1 i=1

g(h(er , es ), ei∗ )2 +

m1 m2   i, j=1 r=1

g(h(e¯i , er ), cscθF e¯ j )2 .

(88)

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Now, using (85) and (86), we get h2 ≥

m1  m2 1  g 2 (er , es )ω 2 (ei∗ ) 4 r,s=1 i=1  m1  m2   2 2 2 2 2 + csc θ ω (J er )g (e¯i , e¯ j ) + ω (er )g (e¯i , P e¯ j ) i, j=1 r=1

+ 2 csc2 θ

m1 m2  

ω(J er )g(e¯i , e¯ j )ω(er )g(e¯i , P e¯ j ).

i, j=1 r=1

Using (2), we obtain h2 ≥

m1  m2 1  g 2 (er , es )g 2 (B, ei∗ ) 4 r,s=1 i=1  m2  m1   + csc2 θ g 2 (B, J er )g 2 (e¯i , e¯ j ) + g 2 (B, er )g 2 (e¯i , P e¯ j ) i, j=1 r=1

+ 2 csc2 θ

m1 m2  

g(B, J er )g(e¯i , e¯ j )g(B, er )g(e¯i , P e¯ j ).

i, j=1 r=1

Here, the term m2  m1 

g(B, J er )g(e¯i , e¯ j )g(B, er )g(e¯i , P e¯ j )

i, j=1 r=1

=

m2  m1 

g(B, J er )g(B, er )g(e¯i , e¯ j )g(e¯i , P e¯ j )

i, j=1 r=1

=−

m1 m2  

g(J B, er )g(B, er )g(e¯i , e¯ j )g(e¯i , P e¯ j )

i, j=1 r=1

= −g(J B T , B T )

m2 

g(e¯i , e¯ j )g(e¯i , P e¯ j ) = 0,

i, j=1

since g(J B T , B T ) = 0. Thus, we arrive m1  m2 1  h ≥ g 2 (er , es )g 2 (B, ei∗ ) 4 r,s=1 i=1  m1  m2   + csc2 θ g 2 (B, J er )g 2 (e¯i , e¯ j ) + g 2 (B, er )g 2 (e¯i , P e¯ j ) . 2

i, j=1 r=1

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Here, for i, j ∈ {1, 2, ..., m 2 }, we have  g(e¯i , P e¯ j ) =

cosθ if i  = j, 0 if i = j,

since Dθ is a slant distribution with slant angle θ. m2  Consequently, g 2 (e¯i , P e¯ j ) = m 2 (m 2 − 1) cos2 θ. Thus, by direct calculation, i, j=1

we obtain the following inequality. h ≥ 2

θ 1 m B F D 2 4 1

  T 2 2 T 2 + csc θ m 2 B  + m 2 (m 2 − 1) cos θB  . 2

Rearrange the last inequality, we get the inequality (87). If the equality sign of (87) holds identically, then we have h(Dθ , Dθ ) = 0 from (88). Namely, h vanishes on Dθ . ¯  Since Dθ is a umbilic distribution on M, it follows that M θ is umbilic in M.

4.2 Warped Product Semi-slant Submanifolds in l.c.K-Space Forms In this subsection, we recall the Matsumoto’s remarkable results for warped product semi-slant submanifolds in l.c.K.-space forms placed in [19, 20]. Matsumoto [19, 20] also studied the warped product semi-slant submanifolds in l.c.K. manifolds (especially, in l.c.K.-space forms) by choosing the induced warped product metric g as : 2 (89) g = e( f ◦π2 ) π1∗ (gD ) + π2∗ (gDθ ) , where gD (resp. gDθ ) is the induced metric on MD (resp. MDθ ) and MD is a holomor¯ J, ω, g). Matsumoto phic and MDθ is a slant submanifold of a l.c.K. manifold ( M, [19, 20] denoted by MD ⊗ f MDθ the warped product semi-slant submanifolds with the induced metric given by (89). ¯ J, ω, g) is called a l.c.K.-space form [18] if it has a conA l.c.K. manifold ( M, stant holomorphic sectional curvature. Then, the Riemannian curvature tensor R¯ ¯ J, ω, g) with the constant holomorphic with respect to g of a l.c.K.-space form ( M, sectional curvature c is given by

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 ¯ 4 R(X, Y, Z , W ) =c g(X, W )g(Y, Z ) − g(X, Z )g(Y, W ) + g(J X, W )g(J Y, Z )  −g(J X, Z )g(J Y, W ) − 2g(J X, Y )g(J Z , W ) (90)  + 3 (X, W )g(Y, Z ) − (X, Z )g(Y, W ) + g(X, W )(Y, Z )  −g(X, Z )(Y, W ) ¯ ¯ ¯ −(X, W )g(J Y, Z ) + (X, Z )g(J Y, W ) − g(J X, W )(Y, Z) ¯ +g(J X, Z )(Y, W )   ¯ ¯ , W) +2 (X, Y )g(J Z , W ) + g(J X, Y )(Z (91) ¯ where  and  ¯ defined by for any X, Y, Z , W ∈ T M,

and

(X, Y ) = −(∇¯ X ω)Y − ω(X )ω(Y ) + 21 ω2 g(X, Y ) ,

(92)

¯ (X, Y ) = (J X, Y )

(93)

¯ where ω is the length of the Lee form ω. for any X, Y ∈ T M, The Codazzi equation and the Ricci equation are respectively given by ¯ R(X, Y, Z , ξ) = g(∇¯ X (h(Y, Z )) − ∇¯ Y (h(X, Z )), ξ)

(94)

¯ R(X, Y, ξ1 , ξ2 ) = R ⊥ (X, Y, ξ1 , ξ2 ) − g([ Aξ1 , Aξ2 ]X, Y ),

(95)

¯ where R ⊥ is the normal curvature for all X, Y, Z , W ∈ T M¯ and ξ, ξ1 , ξ2 ∈ T ⊥ M, tensor, and ∇¯ X (h(Y, Z )) = ∇ X⊥ (h(Y, Z )) − h(∇ X Y, Z ) − h(Y, ∇ X Z ). ¯ = 0 identiThe second fundamental form h is called parallel if it satisfies ∇h cally. A warped product semi-slant submanifold is called normally flat, if R ⊥ = 0 identically.

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Let M = MD ⊗ f MDθ be a warped product semi-slant submanifold with distri¯ J, ω, g) with constant holomorphic butions D and Dθ of a l.c.K.-space form ( M, sectional curvature c. Let dimD = 2k1 , dimDθ = k2 , dim M = m, dim D¯ = 2s and ¯ Then, the generalized adapted frame in M¯ can be choosen as folm¯ = dim M. lows [20]: {e1 , e2 , . . . , ek1 , e1∗ , e2∗ , . . . , ek∗1 } is an orthonormal frame of D, where ei∗ = J ei for i ∈ {1, 2, . . . , k1 }, {e2k1 +1 , e2k1 +2 , . . . , e2k1 +k2 } is an orthonormal frame of Dθ such that the vectors Fe2k1 +1 , Fe2k1 +2 , . . . , Fe2k1 +k2 are orthogonal in FDθ , ∗ ∗ ∗ , em+k , . . . , em+k } is an orthonormal {em+k2 +1 , em+k2 +2 , . . . , em+k2 +s , em+k 2 +1 2 +2 2 +s Fe2k1 +a ∗ ¯ frame of D, where e = J em+k2 +a for a ∈ {1, 2, . . . , s} and e∗ = m+k2 +a

2k1 +a

for a ∈ {1, 2, . . . , k2 }.

Fe2k1 +a 

In view of above background, Matsumoto [19] obtained the following results. Theorem 4.14 Let M = MD ⊗ f MDθ be a warped product semi-slant submanifold ¯ J, ω, g) with constant with the induced metric given by (89) of a l.c.K.-space form ( M, holomorphic sectional curvature c. Then the mean curvature vector field H of M satisfies  4mH  +8k1 f 2

2

θ

(2k1 − 1) f ∇ log f  + 2(2 + f ) 2

2

2

2k 1 +k2

 (ea log f )

2

(96)

a=2k1 +1

+(m 2 + 2m − 3k2 )c + 3c

2k 1 +k2

2

(Pba )2 − 4(e f τD + τDθ )

b,a=2k1 +1 2k 1 +k2

+16k1 f 2

∇eθa ∇eθa log f + 6(m − 2)

b,a=2k1 +1

+6

2k 1 +k2 a=2k1 +1

aa − 6

m 

aa

a=1 2k 1 +k2

(J eb , ea )Pba ≥ 0,

b,a=2k1 +1

where 2k1 = dim(MD ), k2 = dim(MDθ ), m = 2k1 + k2 = dim(M) and τD (resp. τDθ ) denotes the scalar curvatures with respect to gD (resp. gDθ ). Corollary 4.15 Under the same condition with Theorem 4.14, the equality case of (96) is that the submanifold M = MD ⊗ f MDθ is locally totally geodesic and the warping function f satifies

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8k1 f 2 (2k1 − 1) f 2 ∇ θ log f 2 + 2(2 + f 2 )

2k 1 +k2

(ea log f )2

a=2k1 +1

+(m 2 + 2m − 3k2 )c + 3c

2k 1 +k2

2

2 Pba − 4(e f τD + τDθ )

b,a=2k1 +1

+16k1 f

2k 1 +k2

2

∇eθa ∇eθa

log f + 6(m − 2)

b,a=2k1 +1

−6

2k 1 +k2

m 

aa + 6

2k 1 +k2

aa

a=2k1 +1

a=1

(J eb , ea )Pba = 0

b,a=2k1 +1

and 2k 1 +k2

(m 2 + 2m − 3k2 )c + 3c

2

(Pba )2 − 4(e f τD + τDθ )

b,a=2k1 +1 2k 1 +k2

+16k1 f 2

∇eθa ∇eθa log f + 6(m − 2)

m 

b,a=2k1 +1

+6

2k 1 +k2

2k 1 +k2

aa − 6

a=2k1 +1

aa

a=1

(J eb , ea )Pba ≤ 0.

b,a=2k1 +1

Matsumoto also proved the following results in [20]. Theorem 4.16 Let M = MD ⊗ f MDθ be a warped product semi-slant submanifold with the induced metric given by (89) and with parallel second fundamental form h ¯ J, ω, g) with constant holomorphic sectional curvature c. of a l.c.K.-space form ( M, If n 1 and 2n 2 are bigger than 1, then the tensor field  B A is written by the following equation for a certain function α, ⎛

B A

 ji ⎜  j ∗i ⎜ ⎜ bi =⎜ ⎜ b∗ i ⎜ ⎝ ri s ∗ i ⎛

αδ ji ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜ ⎝0 0

 ji ∗  j ∗i ∗ bi ∗ b∗ i ∗ ri ∗ s ∗ i ∗ 0 αδ ji 0 0 0 0

 ja  j ∗a ba b∗ a ra s ∗ a

0 0 ba 0 0 0

 ja ∗  j ∗ a∗ ba ∗ b∗ a ∗ ra ∗ s ∗ a ∗

0 0 0 b∗ a ∗ sa ∗ s ∗ a ∗

 jr  j ∗r br b∗ r sr s ∗ r

0 0 0 b∗ r sr s ∗ r

⎞  jr ∗ Pj ∗r ∗ ⎟ ⎟ br ∗ ⎟ ⎟ Pb∗ r ∗ ⎟ ⎟ sr ∗ ⎠ Ps ∗ r ∗

⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟, b∗ r ∗ ⎟ ⎟ sr ∗ ⎠ sr

(97)

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where j, i ∈ {1, 2, . . . , k1 }, a, b ∈ {2k1 + 1, 2k1 + 2, . . . , 2k1 + k2 = m} and s, r ∈ ¯ {m + k2 + 1, m + k2 + 2, . . . , m}. Theorem 4.17 Let M = MD ⊗ f MDθ be a warped product semi-slant submanifold ¯ J, ω, g) with constant with the induced metric given by (89) of a l.c.K.-space form ( M, holomorphic sectional curvature c. With respect to the generalized adapted frame, the tensor field P, F, t and n defined in (11) and (12) satisfy the following equations, ⎞ ⎛ ⎞ Pni 1 + j Pai P ji 0 −δ ij 0 n +i n +i ⎝ P j 1 Pn 1+ j Pan 1 +i ⎠ = ⎝ δ ij 0 0 ⎠ , 1 0 0 Pba P ja Pna1 + j Pba

(98)

⎞ ⎛ ⎞ F js F ja 0 0 ⎝ Fna + j Fns + j ⎠ = ⎝ 0 0 ⎠ , 1 1 Fba 0 Fba Fbs

(99)

⎛

⎛



tai tan 1 +i tac tsi tsn 1 +i tsa 

n ac n as n as n rs



 =



 =

 000 , 000

00 0 nrs

(100)

 ,

(101)

where j, i ∈ {1, 2, . . . , k1 }, a, b, c ∈ {2k1 + 1, 2k1 + 2, . . . , 2k1 + k2 = m} and ¯ s, r ∈ {m + k2 + 1, m + k2 + 2, . . . , m}. Theorem 4.18 Let M = MD ⊗ f MDθ be a warped product semi-slant submanifold ¯ J, ω, g) with constant with the induced metric given by (89) of a l.c.K.-space form ( M, holomorphic sectional curvature c. If M is nomally flat, then the shape operators Aξ satisfy g([Ab∗ , Aa ∗ ]e j , ei ) g([Aa ∗ , Ar ]e j , ei ) g([As , Ar ]e j , ei ) 4g([Ab∗ , Aa ∗ ]ei , ec ) 4g([Aa ∗ , Ar ]ei , eb ) 2g([As , Ar ]ei , ea ) 4g([Ab∗ , Aa ∗ ]ed , ec ) g([Aa ∗ , Ar ]ec , eb ) 2g([As , Ar ]eb , ea )

= = = = = = =

0, 0, 0, −  Fec  (i ∗ a ∗ δcb − i ∗ b∗ δca ), −  Feb  i ∗ r δba , i ∗ a g(J es , er ), −c  Fec  Fed  (δda δcb − δdb δca ) +  Fec  {(Pde ea ∗ + Fde e∗ a ∗ )δcb − (Pde eb∗ + Fde e∗ b∗ )δca } −  Fed  {c∗ b∗ δda − (Pce ea ∗ + Fce e∗ a ∗ )δcb }, = 0, = c Pba g(J er , es ) − Fbe e∗ a g(J es , er ) − (J es , er )Pba ,

where j, i ∈ {1, 2, . . . , k1 }, a, b, c, d ∈ {2k1 + 1, 2k1 + 2, . . . , 2k1 + k2 = m} and ¯ s, r ∈ {m + k2 + 1, m + k2 + 2, . . . , m}.

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Slant Submanifolds and Their Warped Products in Locally Product Riemannian Manifolds Siraj Uddin

2020 AMS Mathematics Subject Classification: 53C15 · 53C40 · 53C42 · 53C25 · 53B25

1 Introduction The geometry of slant submanifolds was initiated by Chen [15, 16] as a natural generalization of both holomorphic and totally real submanifolds. Since then, many geometers studied these submanifolds. A. Lotta defined and studied slant submanifolds in contact geometry [31, 32]. Papaghiuc [35] introduced semi-slant submanifolds. Cabrerizo et al. studied slant, semi-slant, hemi-slant, and bi-slant submanifolds in contact geometry [13, 14]. Later, B. Sahin and M. Atceken studied slant, semi-slant, and bi-slant submanifolds of locally Riemannian product manifolds (for instance, see [8, 38]). In [20], B.-Y. Chen and O.J. Garay introduced the notion of pointwise slant submanifolds of almost Hermitian manifolds as a generalization of slant submanifolds which were introduced by B.-Y. Chen in [15, 16]. They have obtained several fundamental results and gave a method to construct examples of such submanifolds in Euclidean spaces. Earlier in [25], F. Etayo studied such submanifolds under the name of quasi-slant submanifolds and proved that a complete totally geodesic quasi-slant submanifold of a Kaehler manifold is a slant submanifold. Pointwise slant submanifolds of different structures on a Riemannian manifold are also studied in [26, 36]. Using the notion of pointwise slant submanifolds, Sahin [42] introduced pointwise semi-slant submanifolds of Kaehler manifolds. He used this notion to investigate the S. Uddin (B) Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_8

203

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S. Uddin

geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds and provided some examples. On the other hand, Bishop and O’Niell [12] introduced and studied warped product manifolds. At the beginning of this century, B.-Y. Chen initiated the study of warped product CR-submanifolds of Kaehler manifolds. Since then, the geometry of warped products has become an active field of research in differential geometry. For a detailed survey on warped product submanifolds, we refer to Chen’s books [19, 22] and his survey article [21]. In [39], B. Sahin proved that there do not exist warped product semi-slant submanifolds other than CR-warped product introduced by B.-Y. Chen in Kaehler manifolds [17, 18]. Recently, Sahin [42] introduced and studied warped product pointwise semi-slant submanifolds of Kaehler manifolds. He investigated that there exists a non-trivial class of warped product pointwise semi-slant submanifolds of the form MT × f Mθ of a Kaehler manifold M˜ such that MT is a holomorphic submanifold ˜ He obtained interesting results, an and Mθ is a pointwise slant submanifold of M. inequality and a characterization theorem, and provided examples of such submanifolds. Next, Sahin [41] and Atceken [7] studied warped product semi-slant submanifolds of locally product Riemannian manifolds. They proved that there are no warped product semi-slant submanifolds of the form MT × f Mθ of a locally product Riemannian manifold M˜ such that MT and Mθ are invariant and proper slant submanifolds of ˜ respectively. Then, they provided non-trivial examples and proved a characteriM, zation theorem for warped product semi-slant submanifolds of the form Mθ × f MT . Recently, we have studied these submanifolds for the inequality of the second fundamental form and provided non-trivial examples in [6]. In this chapter, we introduce pointwise slant and pointwise hemi-slant submanifolds of locally product Riemannian manifolds, and by using this notion, we investigate the geometry of warped product pointwise hemi-slant submanifolds of the ˜ where M⊥ is an antiform M⊥ × f Mθ in a locally product Riemannian manifold M, invariant submanifold and Mθ is a proper pointwise slant submanifold of M˜ with slant function θ. The sections of this chapter are organized as follows. In Sect. 2, we give preliminaries and definitions needed for this chapter. In Sect. 3, we study slant submanifolds of almost product Riemannian manifolds. The results and examples given in this section have been taken from [38]. In Sect. 4, we study warped product semi-slant submanifolds of locally product Riemannian manifolds. Theorem 5.1 and Theorem 6.4 in this section are the main results of [41], we mention these results without proof. We provide many non-trivial examples of warped product semi-slant submanifolds of the form Mθ × f MT such that MT is an invariant submanifold and Mθ is a proper slant submanifold of a locally product Riemannian manifold and establish a sharp inequality for the second fundamental form of the warped product immersion. In Sect. 7, we study warped product hemi-slant submanifolds of locally product Riemannian manifolds. We give a non-existence theorem (Theorem 7.4) of warped product hemi-slant submanifold of a locally product Riemannian manifold

Slant Submanifolds and Their Warped Products …

205

M˜ of the form M⊥ × f Mθ , where M⊥ and Mθ are anti-invariant and proper slant ˜ respectively. Further, we discuss another class of warped prodsubmanifolds of M, uct hemi-slant submanifolds of the form Mθ × f M⊥ . We give several non-trivial examples of such warped products and prove a characterization for warped product hemi-slant submanifolds (Theorem 7.9). Next, we derive a sharp inequality for the second fundamental form of warped product hemi-slant immersion (Theorem 7.10). As we have seen in Theorem 7.4 that there do not exist warped products of the form M⊥ × f Mθ in a locally product Riemannian manifold M˜ such that Mθ is a proper slant submanifold. In Sect. 8, we consider Mθ as a pointwise slant submanifold and we study warped product pointwise hemi-slant submanifolds of the form M⊥ × f Mθ in a locally product Riemannian manifold. We provide several non-trivial examples of such immersions. A characterization is proved for warped product pointwise hemislant submanifolds of a locally product Riemannian manifold. Further, we establish a sharp estimation between the squared norm of the second fundamental form h2 and the warping function of the warped product pointwise hemi-slant submanifold. Sect. 9 is devoted to the study of generic warped product submanifolds of locally product Riemannian manifolds.

2 Preliminaries Let M˜ be an m-dimensional differentiable manifold with a tensor field F of type (1, 1) such that F 2 = I and F  = ±I . Then we say that M˜ is an almost product manifold with almost product structure F. If an almost product manifold M˜ has a Riemannian metric g such that g(F X, FY ) = g(X, Y )

(1)

˜ then M˜ is called an almost product Riemannian manifold for any X, Y ∈ (T M), ˜ denotes the set all vector fields of M. ˜ Let ∇˜ denotes the Levi[67], where (T M) Civita connection on M˜ with respect to the Riemannian metric g. If (∇˜ X F)Y = 0, ˜ then M˜ is called a locally product Riemannian manifold for any X, Y ∈ (T M), [30]. Let M be a Riemannian manifold isometrically immersed in M˜ and we denote by the same symbol g the Riemannian metric induced on M. Let (T M) be the Lie algebra of vector fields in M and (T ⊥ M), the set of all vector fields normal to M. If ∇ be the induced Levi-Civita connection on M, then the Gauss and Weingarten formulas are, respectively, given by

and

∇˜ X Y = ∇ X Y + h(X, Y )

(2)

∇˜ X V = −A V X + ∇ X⊥ V

(3)

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S. Uddin

for any X, Y ∈ (T M) and V ∈ (T ⊥ M), where ∇ ⊥ is the normal connection in the normal bundle T ⊥ M and A V is the shape operator of M with respect to the normal vector V . Moreover, h : (T M) × (T M) → (T ⊥ M) is the second fundamental ˜ Furthermore, A V and h are related by form of M in M. g(h(X, Y ), V ) = g(A V X, Y )

(4)

for any X, Y ∈ (T M) and V ∈ (T ⊥ M). An n-dimensional submanifold M of an m-dimensional locally product Riemannian manifold M˜ is said to be totally umbilical n submanifold if h(X, Y ) = g(X, Y )H , h(ei , ei ) , the mean curvature vector of for any X, Y ∈ (T M), where H = n1 i=1 M. A submanifold M is said to be totally geodesic if h(X, Y ) = 0. A totally umbilical submanifold of dimension greater than or equal to 2 with non-vanishing parallel mean curvature vector is called an extrinsic sphere. Also, we set h2 =

n 

g(h(ei , e j ), h(ei , e j )), h ri j = g(h(ei , e j ), er )

(5)

i, j=1

for i, j = 1, · · · , n; r = n + 1, · · · , m, where {e1 , · · · , en } is an orthonormal basis of the tangent space T p M, for any p ∈ M. For any X tangent to M, we write F X = T X + ωX,

(6)

where T X and ωX are the tangential and normal components of F X , respectively. Similarly, for any V ∈ (T ⊥ M), we write F V = BV + C V,

(7)

where BV and C V are the tangential and normal components of F V , respectively. A submanifold M of an almost product Riemannian manifold M˜ is said to be F-invariant, if ω = 0, i.e., F X ∈ (T M), for any X ∈ (T M). On the other hand, M is said to be F-anti-invariant if T = 0, i.e., F X ∈ (T ⊥ M), for any X ∈ (T M). Moreover, from (1) and (6), we have g(T X, Y ) = g(X, T Y ), for any X, Y ∈ (T M).

(8)

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3 Slant Submanifolds of Almost Product Riemannian Manifolds In this section, we study slant submanifolds of almost product Riemannian manifolds. The results of this section are given in [38]. First, we present the following definition of a slant submanifold of an almost product Riemannian manifold following Chen’s (see, [15, 16]) definition for almost Hermitian manifolds. A submanifold M of a locally product Riemannian manifold M˜ is said to be slant (see [8, 15, 16, 38]), if for each non-zero vector X tangent to M, the angle θ (X ) between F X and T p M is a constant, i.e., it does not depend on the choice of p ∈ M and X ∈ T p M. Then, the angle θ of a slant immersion is called the slant angle of M. Thus, the F-invariant and F-anti-invariant immersions are slant immersions with slant angle θ = 0 and θ = π2 , respectively. A slant immersion which is neither F-invariant nor F-anti-invariant is called proper slant immersion. Next, we provide a useful characterization for slant submanifolds in an almost product Riemannian manifold. Theorem 3.1 ([38]) Let M be a slant submanifold of an almost product Riemannian ˜ Then, M is slant if and only if there exists a constant λ ∈ [0, 1] such manifold M. that T 2 = λI.

(9)

Note that, if θ is the slant angle of M, then λ = cos2 θ . ˜ Then, cos θ (X ) is independent of p ∈ M Proof Let M is a slant submanifold of M. and X ∈ T p M. Thus, using (1) and (6), we obtain cos θ (X ) =

g(T X, F X ) g(F T X, X ) g(T 2 X, X ) = = . |X ||T X | |X ||T X | |X ||T X |

On the other hand, we have cos θ (X ) = g(T 2 X,X ) . |X |2

|T X | , thus using (10), we derive cos2 |F X |

(10) θ (X ) =

Hence, we obtain T 2 X = λX, λ ∈ [0, 1]. Conversely, if T 2 X = λX , for any X ∈ (T M) and λ ∈ [0, 1], then in a similar  way, we find cos2 θ (X ) = λ and hence θ (X ) is constant on M. We have the following consequences of the above theorem. Lemma 3.2 ([38]) Let M be a slant submanifold of an almost product Riemannian ˜ Then, for any X, Y ∈ (T M), we have manifold M. g(T X, T Y ) = cos2 θ g(X, Y ) and

(11)

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S. Uddin

g(ωX, ωY ) = sin2 θ g(X, Y ).

(12)

Proof Substituting Y by T Y in (8), we get g(T X, T Y ) = g(X, T 2 Y ), for any X, Y ∈ (T M). Then, from (9), we get (11). Further, the proof of (12) follows from (1), (6), and (11).  It is also easy to show that a submanifold M of an almost product Riemannian manifold M˜ is slant if and only if BωX = sin2 θ X, ωT X = −CωX

(13)

for any X ∈ (T M). Let Rm = Rn 1 × Rn 2 be the Euclidean space of dimension m = n 1 + n 2 endowed with the Euclidean metric. Let F be a product structure defined on Rm = Rn 1 × Rn 2 by  F

∂ ∂ , ∂ xi ∂ x j



 =

∂ ∂ , − ∂ xi ∂x j

 , i ∈ {1, · · · , n 1 }, j ∈ {n 1 + 1, · · · , m}.

Now, we provide some non-trivial examples of slant submanifolds of almost product Riemannian manifolds. Example 3.3 ([38]) Consider in R4 = R2 × R2 the submanifold is given by the immersion x(u, v) = (u cos θ, v cos θ, u sin θ, v sin θ ) for any θ > 0. Then M is a slant plane with slant angle 2θ .   Example 3.4 ([38]) For any u, v ∈ 0, π2 and positive constant k  = 1, x(u, v) = (u, v, −k sin u, −k sin v, k cos u, k cos v) defines a proper slant surface in R6 = R2 × R4 with slant angle θ = cos−1



1−k 2 1+k 2

.

Example 3.5 ([38]) Let M be a submanifold in R4 = R2 × R2 given by the immersion x(u 1 , u 2 ) = (u 1 + u 2 , u 1 + u 2 ,

√ √ 2u 2 , 2u 1 ).

Then M is a slant submanifold with slant angle θ =

π . 3

Example 3.6 ([38]) Let us consider the almost product manifold R4 with coordinates (x1 , x2 , y1 , y2 ) and product structure  F

∂ ∂ xi

 =

∂ , F ∂ yi



∂ ∂ yi

 =

∂ . ∂ xi

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209

Let M be a submanifold given by the immersion x(u, v) = (u cos θ, u sin θ, v, 0). It is easy to see that M is a slant submanifold with slant angle θ . Example 3.7 ([38]) Consider the almost product Riemannian manifold R7 = R4 × R3 with cartesian coordinates (x1 , x2 , x3 , x4 , y1 , y2 , y3 ) and product structure F given by  F

∂ ∂ xi



∂ , F =− ∂ xi



∂ ∂yj

 =

∂ , ∂yj

1 ≤ i ≤ 4, 1 ≤ j ≤ 3.

Then a submanifold M given by √ x(u 1 , u 2 , u 3 ) = ( 2u 1 , u 2 , u 3 , u 2 + u 3 , u 1 + u 2 + u 3 , u 1 , u 2 + u 3 ) is a slant submanifold with slant angle cos−1

1 4

.

Remark 3.8 ([38]) It is known that in complex geometry, proper slant submanifolds are always even dimensional, while in contact geometry, proper slant submanifolds are always odd dimensional. However, in the product Riemannian manifolds, the situation is quite different from both geometries. For instance, in Examples 3.3–3.6, the slant submanifolds are even dimensional, while in Example 3.7, the proper slant submanifold is odd dimensional. Thus, one can conclude that there are even and odd dimensional proper slant submanifolds of almost product Riemannian manifolds. A submanifold M of a locally product Riemannian manifold M˜ is called a semiinvariant submanifold (see [11, 33, 37]) of M˜ if there exist a differentiable distribution D : p → D p ⊂ T p M such that D is invariant with respect to F and the complementary distribution D⊥ is anti-invariant with respect to F. A submanifold M of a locally product Riemannian manifold M˜ is called semislant (see [30, 35]), if it is endowed with two orthogonal distributions D and Dθ , where D is invariant with respect to F and Dθ is slant, i.e., θ (X ) is the angle between F X and Dθp is constant for any X ∈ Dθp and p ∈ M.

4 Warped Product Semi-slant Submanifolds In 1969, R. L. Bishop and B. O’Neill introduce the idea of warped product manifolds and according to them, let M1 and M2 be two Riemannian manifolds with Riemannian metrics g1 and g2 , respectively, and a positive differentiable function f on M1 . Consider the product manifold M1 × M2 with its projections π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2 . Then their warped product manifold M = M1 × f M2 is

210

S. Uddin

the Riemannian manifold M1 × M2 = (M1 × M2 , g) equipped with the Riemannian structure such that g(X, Y ) = g1 (π1 X, π1 Y ) + ( f ◦ π1 )2 g2 (π2 X, π2 Y ) for any vector field X, Y tangent to M, where  is the symbol for the tangent maps. A warped product manifold M = M1 × f M2 is said to be trivial or simply a Riemannian product manifold if the warping function f is constant. The following result of [12] is useful in our further study. Lemma 4.1 ([12]) Let M = M1 × f M2 be a warped product manifold with the warping function f . Then, we have: (i) ∇ X Y ∈ (T M1 ), ∀ X, Y ∈ (T M1 ) (ii) ∇ Z X = ∇ X Z = (X ln f )Z , ∀ X ∈ (T M1 ), Z ∈ (T M2 )  ln f, ∀ Z , W ∈ (T M2 ) (iii) ∇ Z W = ∇ Z W − g(Z , W )∇ where ∇ and ∇ denote the Levi-Civita connections on M and M2 , respectively. ˜ the gradient For a differentiable function f on an m-dimensional manifold M, ˜ As a consequence,  f of f is defined as g(∇  f, X ) = X ( f ), for any X tangent to M. ∇ we have  f 2 = ∇

m 

(ei ( f ))2

(14)

i=1

˜ for an orthonormal frame {e1 · · · , em } on M. In the beginning of this century, B.-Y. Chen initiated the idea of warped product submanifolds in his seminal papers [17, 18]. Since then many geometers studied the warped product submanifolds for different structures on manifolds. Remark 4.2 It is evident from Lemma 4.1 that M1 is a totally geodesic submanifold and M2 is totally umbilical in M [12, 17].

5 Warped Product Semi-slant Submanifolds of the Form MT × f Mθ In this subsection we study semi-slant submanifolds in a locally product Riemannian manifold M˜ which are the warped products of the form MT × f Mθ , where MT is ˜ Throughout an invariant submanifold and Mθ is a proper slant submanifold of M. this section, we assume that the tangent bundles of MT and Mθ respectively are D and Dθ .

Slant Submanifolds and Their Warped Products …

211

Theorem 5.1 ([41]) Let M˜ be a locally product Riemannian manifold. Then there do not exist warped product semi-slant submanifolds in the form MT × f Mθ in M˜ ˜ such that MT is an invariant submanifold and Mθ is a proper slant submanifold of M. Proof For any X ∈ (D) and U, V ∈ (Dθ ), from (1), (2), (3), and Lemma 4.1 (ii), we have g(∇ X T U, V ) = g(∇˜ T U F X, F V ). Since, D and Dθ orthogonal, then we have g(∇ X T U, V ) = −g(F X, ∇˜ T U F V ). Thus using (2), (3) and (6), we derive g(∇ X T U, V ) = −g(F X, ∇T U T V ) + g(AωV T U, F X ). Then, Lemma 4.1 (ii) and (8), imply X (ln f )g(T U, V ) = F X (ln f )g(T U, T V ) + g(h(T U, F X ), ωV ). Then, from (10), we derive X (ln f )g(T U, V ) = cos2 θ F X (ln f )g(U, V ) + g(h(T U, F X ), ωV ). Interchanging X by F X and U by T U . Then, we have F X (ln f )g(U, V ) = X (ln f )g(T U, V ) + g(h(U, X ), ωV ).

(15)

On the other hand, from (2) and (6), we have g(h(U, X ), ωV ) = g(∇˜ X U, F V ) − g(∇˜ X U, T V ), for any X ∈ (D) and U, V ∈ (Dθ ). Using the locally product structure and Lemma 4.1, we find g(h(U, X ), ωV ) = g(∇˜ X FU, V ) − X (ln f )g(U, T V ). Then, from (6), (2)–(3) and Lemma 4.1 (ii), we get g(h(U, X ), ωV ) = X (ln f )g(T U, V ) − g(h(V, X ), ωU ) − X (ln f )g(U, T V ). Then, (8) implies that g(h(U, X ), ωV ) = −g(h(V, X ), ωU ), for any X ∈ (D) and U, V ∈ (Dθ ). Then, from (15) and (16), we get

(16)

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S. Uddin

F X (ln f )g(U, V ) − X (ln f )g(T U, V ) = 0.

(17)

Substituting X by F X and U by T U in (16), we derive X (ln f )g(T U, V ) − cos2 θ F X (ln f )g(U, V ) = 0.

(18)

Thus, from (17) and (18), we obtain sin2 θ F X (ln f )g(U, V ) = 0. Since Mθ is proper slant and g is Riemannian metric, then we conclude that F X (ln f ) = 0, which shows that f is constant due to MT is invariant. Thus, the proof is complete. 

6 Warped Product Semi-slant Submanifolds of the Form Mθ × f MT In this subsection, we discuss other types of warped product semi-slant submanifolds ˜ First, we have of the form Mθ × f MT in a locally product Riemannian manifold M. the following useful lemma for later use. Lemma 6.1 Let M = Mθ × f MT be a warped product semi-slant submanifold of ˜ Then: a locally product Riemannian manifold M. (i) g(h(X, U ), ωV ) = −g(h(X, V ), ωU ); (ii) g(h(X, Y ), ωU ) = −U (ln f )g(X, FY ) + T U (ln f )g(X, Y ); (iii) g(h(X, Y ), ωT U ) = −T U (ln f )g(X, FY ) + cos2 θU (ln f )g(X, Y ) for any X, Y ∈ (D) and U, V ∈ (Dθ ). Proof For any X ∈ (T MT ) and U, V ∈ (T Mθ ), we have g(h(X, U ), ωV ) = g(∇˜ X U, F V ) − g(∇˜ X U, T V ) = g((∇˜ X F)U, V ) − g(∇˜ X FU, V ) − (U ln f )g(X, T V ). The first and last terms on the right-hand side are identically zero by using the structure of a locally product Riemannian manifold and the orthogonality of vector field. Then, from (6), we obtain g(h(X, U ), ωV ) = g(∇˜ X T U, V ) + g(∇˜ X ωU, V ). By using (2)–(4) and Lemma 4.1 in above relation, we get (i). For the second part, consider X, Y ∈ (D) and U ∈ (Dθ ), then we have

Slant Submanifolds and Their Warped Products …

213

g(h(X, Y ), ωU ) = g(∇˜ X Y, ωU ) = g(∇˜ X Y, FU ) − g(∇˜ X Y, T U ) = g((F ∇˜ X Y, U ) + g(Y, ∇˜ X T U ) = g(∇˜ X FY, U ) + T U (ln f )g(X, Y ) = −g(FY, ∇˜ X U ) + T U (ln f )g(X, Y ). Using (2) and Lemma 4.1, we obtain g(h(X, Y ), ωU ) = −U (ln f )g(X, FY ) + T U (ln f )g(X, Y ), which is (ii). If we replace U by T U in (ii), and then using (9), we get (iii), which proves the lemma completely.  From the above lemma, we can easily find the following relations by interchanging X by F X and Y by FY , for any X, Y ∈ (D) in Lemma 6.1 (ii)–(iii) g(h(X, FY ), ωU ) = −U (ln f )g(X, Y ) + (T U ln f )g(X, FY ),

(19)

g(h(F X, Y ), ωU ) = −U (ln f )g(X, Y ) + (T U ln f )g(F X, Y )

(20)

and g(h(F X, Y ), ωT U ) = −T U (ln f )g(X, Y ) + cos2 θ (U ln f )g(F X, Y )

(21)

g(h(X, FY ), ωT U ) = −T U (ln f )g(X, Y ) + cos2 θ (U ln f )g(X, FY ).

(22)

Then using (1) in (19) and (20), we find g(h(X, FY ), ωU ) = g(h(F X, Y ), ωU ).

(23)

Similarly, from (21) and (22), we obtain g(h(F X, Y ), ωT U ) = g(h(X, FY ), ωT U ).

(24)

Also, if we interchange X by F X in (19) and Y by FY in (21), we arrive at g(h(F X, FY ), ωU ) = −U (ln f )g(F X, Y ) + (T U ln f )g(X, Y ), g(h(F X, FY ), ωT U ) = −(T U ln f )g(F X, Y ) + cos2 θ (U ln f )g(X, Y ).

(25) (26)

214

S. Uddin

Then from (25) and Lemma 6.1 (ii), we get g(h(F X, FY ), ωU ) = g(h(X, Y ), ωU )

(27)

and by Lemma 6.1 (iii) and (26), we derive g(h(F X, FY ), ωT U ) = g(h(X, Y ), ωT U ).

(28)

If M is a warped product semi-slant submanifold of the form M = Mθ × f MT of a locally product Riemannian manifold M˜ and if there is no ν-components in the normal bundle of M, then M is mixed totally geodesic (Proposition 4.1 [41]), i.e., h(X, U ) = 0, for any X ∈ (D) and U ∈ (Dθ ). Now, we construct the following frame field for an n-dimensional warped product semi-slant submanifold M = Mθ × f MT of an m-dimensional locally prod˜ Let us denote by D and Dθ the tangent bundles of uct Riemannian manifold M. MT and Mθ , respectively, instead of T MT and T Mθ . Also, if we consider the dim(MT ) = p and dim(Mθ ) = q, then the orthonormal frames of D and Dθ , respectively, are given by {e1 = Fe1 , · · · , ek = Fek , ek+1 = −Fek+1 , · · · , e p = −Fe p } and {e p+1 = e1∗ = sec θ T e1∗ , · · · , e p+q = eq∗ = sec θ T eq∗ }. Then the orthonormal frame fields of the normal subbundles of ωDθ and ν, respectively are {en+1 = e˜1 = csc θ ωe1∗ , · · · , en+q = e˜q = csc θ ωeq∗ } and {en+q+1 = e˜q+1 , · · · , em = e˜m−n−q }. Now, we are able to construct the following inequality with the help of the above constructed frame fields and some previous formulas which we have obtained for warped product semi-slant submanifolds of a locally product Riemannian manifold. Theorem 6.2 Let M = Mθ × f MT be a proper warped product semi-slant subman˜ where MT and Mθ are invariant ifold of a locally product Riemannian manifold M, ˜ and proper slant submanifolds of M, respectively. Then: (i) The squared norm of the second fundamental form of the warped product immersion satisfies  θ ln f 2 h2 ≥ p(csc θ − cot θ )2 ∇  θ ln f is gradient of the warping function ln f along where p = dim(MT ) and ∇ Mθ . (ii) If equality sign in (i) holds identically, then Mθ is totally geodesic in M˜ and MT ˜ Furthermore, Mθ × f MT is a mixed is a totally umbilical submanifold of M. ˜ totally geodesic submanifold of M. Proof From the definition of h, we have h2 =

n  i, j=1

g(h(ei , e j ), h(ei , e j )) =

n m   r=n+1 i, j=1

g(h(ei , e j ), er )2 .

Slant Submanifolds and Their Warped Products …

215

Then from the assumed frame fields of D and Dθ , we derive p m  

h2 =

g(h(ei , e j ), er )2 + 2

r =n+1 i, j=1 m 

+

n 

p n m   

g(h(ei , e j ), er )2

r=n+1 i=1 j= p+1

g(h(ei , e j ), er )2 .

(29)

r =n+1 i, j= p+1

After leaving the second and third term in the right hand side of (29) and using the constructed frame fields, we find p q  

h ≥ 2

p  

m−n−q

g(h(ei , e j ), csc θ ωer∗ )2

r =1 i, j=1

+

g(h(ei , e j ), e˜r )2 .

(30)

r=q+1 i, j=1

The second term on the right-hand side of the above expression has the ν-components, therefore, we shall leave this term and hence using Lemma 6.1, we get h2 ≥ csc2 θ

p q  

(er∗ ln f )2 g(ei , Fe j )2 + (T er∗ ln f )2 g(ei , e j )2

r=1 i, j=1

−

2(T er∗

 ln f ).(er∗ ln f ) g(ei , e j ) g(ei , Fe j ) .

(31)

 θ ln f ∈ (Dθ ) and er∗ = sec θ T er∗ , then from (14), we have Now, since ∇ q q   ∗ ∗  θ ln f 2 (T er ln f ).(er ln f ) = cos θ (er∗ ln f )2 = cos θ ∇ r =1

(32)

r=1

and q q    θ ln f )2 = cos2 θ ∇  θ ln f 2 . (T er∗ ln f )2 = g(T er∗ , ∇ r =1

(33)

r=1

Then, from (31)–(33), we find  2 θ  ln f 2 , h2 ≥ p csc2 θ 1 − cos θ ∇ which is inequality (i). For the equality, from the remaining terms of (29), we obtain h(D, Dθ ) = 0, and h(Dθ , Dθ ) = 0.

(34)

Also, from the remaining second term in the right hand side of (30), we observe that

216

S. Uddin

h(D, D) ⊥ ν

⇒ h(D, D) ∈ (ωDθ ).

(35)

The second condition of (34) implies that Mθ is totally geodesic in M˜ due to Mθ being totally geodesic in M [12, 17]. On the other hand, (35) implies that MT is totally umbilical in M˜ with the fact that MT is totally umbilical in M [12, 17]. Moreover, ˜ all conditions of (34) imply that M is a mixed totally geodesic submanifold of M. Hence, the proof is complete.  From the above theorem, we have the following remark. Remark 6.3 In Theorem 6.2, if we assume θ = π2 , then the warped product becomes ˜ where MT and M⊥ M = M⊥ × f MT in a locally product Riemannian manifold M, ˜ are invariant and anti-invariant submanifolds of M, respectively, which is a case of warped product semi-invariant submanifolds which have been discussed in ([9, 40]). Thus, Theorem 4.2 of [40] and Theorem 4.1 of [9] are the special cases of Theorem 6.2. Using a well known following result, B. Sahin gave a characterization for warped product semi-slant submanifolds in locally product Riemannian manifolds. Hiepko’s Theorem ([27]) Let D1 and D2 be two orthogonal distribution on a Riemannian manifold M. Suppose that D1 and D2 both are involutive such that D1 is a totally geodesic foliation and D2 is a spherical foliation. Then M is locally isometric to a non-trivial warped product M1 × f M2 , where M1 and M2 are integral manifolds of D1 and D2 , respectively. B. Sahin proved the following useful characterization for warped product semislant submanifolds of the form Mθ × f MT in a locally product Riemannian manifold ˜ respecM˜ such that MT and Mθ are invariant and proper slant submanifolds of M, tively. Theorem 6.4 ([41]) Let M be a semi-slant submanifold of a locally product Rie˜ Then M is locally a warped product manifold of the form mannian manifold M. ˜ Mθ × f MT in M such that Mθ is a proper slant submanifold and MT is an invariant submanifold of M˜ if and only if AωU Y + AωT U FY = − sin2 θ U (μ)FY,

∀ U ∈ (Dθ ), Y ∈ (D)

for some function μ on M satisfying X (μ) = 0, for X ∈ (D). Proof We skip the proof of this theorem. Detailed proof is given in [41].



Now, we provide the following non-trivial examples of warped product semi-slant submanifolds Mθ × f MT in Euclidean spaces. Example 6.5 Let us consider the almost product manifold R7 = R3 × R4 with coordinates (x1 , x2 , x3 , y1 , y2 , y3 , y4 ) and the product structure

Slant Submanifolds and Their Warped Products …

F(

217

∂ ∂ ∂ ∂ )=− , F( )= , i = 1, 2, 3 and j = 1, 2, 3, 4. ∂ xi ∂ xi ∂yj ∂yj

Let M be a submanifold of R7 given by √ f (θ, ϕ, v, u) = (u cos θ, u sin θ, u + v, 3u − v, v, u sin ϕ, u cos ϕ)  π . with u  = 0, v  = 0 and θ, ϕ ∈ 0, 2 Then the tangent space T M of M is spanned by the following vector fields ∂ ∂ ∂ ∂ + u cos θ , Z 2 = u cos ϕ − u sin ϕ ∂ x1 ∂ x2 ∂ y3 ∂ y4 ∂ ∂ ∂ Z3 = − + ∂ x3 ∂ y1 ∂ y2 √ ∂ ∂ ∂ ∂ ∂ ∂ Z 4 = cos θ + sin θ + + 3 + sin ϕ + cos ϕ . ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y3 ∂ y4 Z 1 = −u sin θ

Then with respect to the Riemannian product structure F, we get ∂ ∂ ∂ − + ∂ x3 ∂ y1 ∂ y2 √ ∂ ∂ ∂ ∂ ∂ ∂ F Z 4 = − cos θ − sin θ − + 3 + sin ϕ + cos ϕ . ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y3 ∂ y4

F Z 1 = −Z 1 , F Z 2 = Z 2 , F Z 3 = −

Then, it is easy to see that the invariant and slant distributions are spanned by D = Span{Z 1 , Z 2 } and Dθ1 = Span{Z 3 , Z 4 } with the slant angle say θ1 . Then  θ1 = arccos

g(F Z 3 , Z 3 ) F Z 3 Z 3 



 = arccos

g(F Z 4 , Z 4 ) F Z 4 Z 4 

 = arccos

  1

= 70◦ 31 . 3

Thus, M is a proper semi-slant submanifold of R7 . It is also easy to check that D and Dθ1 are integrable. If we denote the integral manifolds of D and Dθ1 by MT and Mθ1 , respectively. Then the induced metric tensor on M is given by     g = 6du 2 + 3dv 2 + u 2 dθ 2 + dϕ 2 = g Mθ1 + u 2 g MT . Thus, M is a warped product submanifold of the form M = Mθ1 × f1 MT with the warping function f 1 = u. Example 6.6 Let R4 = R2 × R2 be a locally product Riemannian manifold with the cartesian coordinates (x1 , x2 , y1 , y2 ) and the almost product structure

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S. Uddin

 F

∂ ∂ xi

 =−

∂ , F ∂ xi



∂ ∂yj

 =

∂ , 1 ≤ i, j ≤ 2. ∂yj

Consider a submanifold M of R4 defined by χ (u, θ) = (u cos θ, u sin θ, u,

 π √ 2u), u  = 0, θ ∈ 0, . 2

Then the tangent space of M is spanned by the following vectors √ ∂ ∂ ∂ ∂ + sin θ + + 2 , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ ∂ v2 = −u sin θ + u cos θ . ∂ x1 ∂ x2 v1 = cos θ

θ It is easy to find that D = Span{v2 } is an invariant distribution   π and D = Span{v1 } is a −1 1 = 3 . Thus, M is a semi-slant proper slant distribution with slant angle θ = cos 2 submanifold of R4 . Also, it is easy to see that both the distributions are integrable. If we denote the integral manifolds of D and Dθ by MT and Mθ , respectively, then the metric g of the product manifold M is given by

g = 4du 2 + u 2 dθ 2 = g Mθ + u 2 g MT . Hence, we conclude that M is a warped product semi-slant submanifold of R4 of the type Mθ × f MT with warping function f = u. Example 6.7 Let R6 = R4 × R2 be a locally product Riemannian manifold with the cartesian coordinates (x1 , x2 , x3 , x4 , y1 , y2 ) and the almost product structure  F

∂ ∂ xi

 =−

∂ , F ∂ xi



∂ ∂yj

 =

∂ , 1 ≤ i ≤ 4, 1 ≤ j ≤ 2. ∂yj

Consider a submanifold M of R6 given by the equations x1 = u cos θ, x2 = u sin θ, x3 = v cos θ, x4 = v sin θ, y1 = u + v, y2 = u − v,   such that u, v  = 0 (u  = v), θ ∈ 0, π2 . Then the tangent bundle of M is spanned by v1 , v2 and v3 , where ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + sin θ + + , v2 = cos θ + sin θ + − , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ x3 ∂ x4 ∂ y1 ∂ y2 ∂ ∂ ∂ ∂ v3 = −u sin θ + u cos θ − v sin θ + v cos θ . ∂ x1 ∂ x2 ∂ x3 ∂ x4 v1 = cos θ

θ Then it is easy to see that D = Span{v3 } is an invariant distribution   and D◦ = −1 1 = 70 31 . Span{v1 , v2 } is a proper slant distribution with slant angle θ = cos 3

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Hence, M is a semi-slant submanifold of R6 . Clearly, both the distributions are integrable. If the integral manifolds of D and Dθ are MT and Mθ , respectively, then the metric g of the product manifold M is   g = 3 du 2 + 3 dv 2 + u 2 + v 2 dθ 2 . Thus, M is a warped product√ semi-slant submanifold of R6 of the form Mθ × f MT with warping functions f = u 2 + v 2 . Example 6.8 Consider the Euclidean space R8 = R4 × R4 be a locally product Riemannian manifold with the cartesian coordinates (x1 , · · · , x4 , y1 , · · · , y4 ) and the almost product structure  F

∂ ∂ xi

 =

∂ , F ∂ xi



∂ ∂yj

 =−

∂ , 1 ≤ i, j ≤ 4. ∂yj

Let M be a submanifold of R8 defined by χ (s, t, θ, ϕ) = (s + t, s +

√ 5t, s cos θ, s sin θ, s, 2t, s cos ϕ, s sin ϕ)

  with s, t  = 0, θ, ϕ ∈ 0, π2 . Then the tangent space of M is spanned by the following vectors ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + cos θ + sin θ + + cos ϕ + sin ϕ , ∂ x1 ∂ x2 ∂ x3 ∂ x4 ∂ y1 ∂ y3 ∂ y4 √ ∂ ∂ ∂ ∂ ∂ Z2 = + 5 +2 , Z 3 = −s sin θ + s cos θ , ∂ x1 ∂ x2 ∂ y2 ∂ x3 ∂ x4 ∂ ∂ Z 4 = −s sin ϕ + s cos ϕ . ∂ y3 ∂ y4

Z1 =

θ Thus, one can see that D = Span{Z 3 , Z 4 } is an invariant distribution   and D = Span{Z 1 , Z 2 } is a proper slant distribution with slant angle θ = cos−1 15 = 78◦ 64 . Thus M is a semi-slant submanifold of R8 . It is easy to see that both the distributions are integrable. We denote the integral manifolds of D and Dθ by MT and Mθ , respectively. Then the product metric g of M is given by

g = g Mθ + s 2 g M T , where g Mθ = 5 ds 2 + 8 dt 2 and g MT = dθ 2 + dϕ 2 . Hence, M is a warped product semi-slant submanifold of R8 of the type Mθ × f MT with warping function f = s.

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S. Uddin

7 Warped Product Hemi-Slant Submanifolds In this section, we study hemi-slant submanifolds which are the warped products of the form M = M⊥ × f Mθ and M = Mθ × f M⊥ in a locally product Riemannian manifold. First, we define the hemi-slant submanifolds as follows: Definition 7.1 Let M be a submanifold of a locally product Riemannian manifold M˜ with a pair of orthogonal distributions D⊥ and Dθ . Then, M is said to be a hemi-slant submanifold of M˜ if: (i) T M = D⊥ ⊕ Dθ (ii) the distribution D⊥ is anti-invariant under F, i.e., F(D⊥ ) ⊂ T ⊥ M (iii) Dθ is a slant distribution with slant angle θ  = 0. Let us denote by m 1 and m 2 , the dimensions of D⊥ and Dθ , then M is anti-invariant if m 2 = 0 and proper slant if m 1 = 0. It is proper hemi-slant, if the slant angle is different from 0 and π/2 and m 1  = 0. Moreover, if ν is an invariant normal subbundle under F of the normal bundle T ⊥ M, then in case of a hemi-slant submanifold, the normal bundle T ⊥ M can be decomposed as T ⊥ M = FD⊥ ⊕ ωDθ ⊕ ν. A hemi-slant submanifold of a locally product Riemannian manifold is said to be mixed totally geodesic if h(U, Z ) = 0, for any U ∈ (Dθ ) and Z ∈ (D⊥ ). First, we give the following example of a proper hemi-slant submanifold. Example 7.2 Consider a submanifold M of R4 = R2 × R2 with coordinates (x1 , x2 , y1 , y2 ) and the product structure  F

∂ ∂ xi

 =

∂ , F ∂ xi



∂ ∂ yi

 =−

∂ , i = 1, 2. ∂ yi

For any θ ∈ (0, π4 ), consider a submanifold M into R4 which is given by the immersion f (u, v) = (u cos θ, v, u sin θ, v),

u, v  = 0.

Then, the tangent space T M of M is spanned by the following vector fields e1 = cos θ

∂ ∂ ∂ ∂ + sin θ , e2 = + . ∂ x1 ∂ y1 ∂ x2 ∂ y2

It is easy to see that Fe2 is orthogonal to T M, thus the anti-invariant distribution is θ1 Span{e2 } and D⊥ =  D = Span{e1 } is the slant distribution with slant angle θ1 = g(Fe1 ,e1 ) arccos Fe = 2θ and hence M is a proper hemi-slant submanifold with slant 1 e1  angle θ1 = 2θ. The detailed study of hemi-slant submanifolds of a locally product Riemannian manifold is given in [43]. Now, we have the following result for later use.

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Proposition 7.3 On a hemi-slant submanifold M of a locally product Riemannian ˜ we have manifold M, g(∇ X Y, Z ) = sec2 θ (g(A F Z T Y, X ) + g(AωT Y Z , X )) for any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ). Proof For any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ), we have g(∇ X Y, Z ) = g(∇˜ X Y, Z ) = g(F ∇˜ X Y, F Z ) = g(∇˜ X FY, F Z ) − g((∇˜ X F)Y, F Z ). Using (6) and the characteristic of locally product Riemannian structure, we get g(∇ X Y, Z ) = g(∇˜ X T Y, F Z ) + g(∇˜ X ωY, F Z ) = g(h(X, T Y ), F Z ) + g(F ∇˜ X ωY, Z ) = g(A F Z T Y, X ) + g(∇˜ X FωY, Z ) − g((∇˜ X F)ωY, Z ). Again, using (6) and the locally product Riemannian structure, we derive g(∇ X Y, Z ) = g(A F Z T Y, X ) + g(∇˜ X BωY, Z ) + g(∇˜ X CωY, Z ). Then from (13), we obtain g(∇ X Y, Z ) = g(A F Z T Y, X ) + sin2 θ g(∇˜ X Y, Z ) − g(∇˜ X ωT Y, Z ). Using (2)–(3), we arrive at cos2 θ g(∇ X Y, Z ) = g(A F Z T Y, X ) + g(AωT Y X, Z ), thus the assertion follows from the last relation.



A submanifold M of a locally product Riemannian manifold M˜ is said to be a warped product hemi-slant submanifold if M is the warped product of an anti˜ In this case, the invariant submanifold M⊥ and a proper slant submanifold Mθ of M. two possible cases of warped product hemi-slant submanifolds occurred: M⊥ × f Mθ and Mθ × f M⊥ . The first type of warped does not exist in a locally product Riemannian manifold. In [7], M. Atceken proved the following non-existence theorem for the warped product hemi-slant submanifolds of the form M⊥ × f Mθ in a locally product Riemannian manifold. Theorem 7.4 ([7]) Let M˜ be a locally product Riemannian manifold and M be a ˜ Then there exist no warped product hemi-slant submanifolds of submanifold of M. the form M = M⊥ × f Mθ in M˜ such that M⊥ is an anti-invariant submanifold and ˜ Mθ is a proper slant submanifold of M.

222

S. Uddin

Proof Here, we skip the proof of this theorem and the detailed proof is given in [7].  Now, we consider the warped product hemi-slant submanifolds of the form M = Mθ × f M⊥ , where Mθ and M⊥ are proper slant and anti-invariant submanifolds of ˜ respectively. a locally product Riemannian manifold M, A warped product hemi-slant submanifold M = Mθ × f M⊥ is called proper if ˜ respectively. M⊥ and Mθ are anti-invariant and proper slant submanifolds of M, First, we provide the following non-trivial examples of such submanifolds.   Example 7.5 For any u  = 0, v ∈ 0, π2 , consider a submanifold M of R5 = R3 × R2 with the cartesian coordinates (x1 , x2 , x3 , y1 , y2 ) and the product structure  F

∂ ∂ xi



∂ , F ∂ xi

=



∂ ∂yj

 =−

∂ , i = 1, 2, 3, ∂yj

j = 1, 2.

The submanifold M is given by the equations x1 = u cos v, x2 = u sin v, x3 = 2u, y1 = u cos v, y2 = u sin v. Then the tangent bundle T M is spanned by Z 1 and Z 2 , where Z 1 = cos v

∂ ∂ ∂ ∂ ∂ + sin v +2 + cos v + sin v , ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2

Z 2 = −u sin v

∂ ∂ ∂ ∂ + u cos v − u sin v + u cos v . ∂ x1 ∂ x2 ∂ y1 ∂ y2

It is easy to see that F Z 2 is orthogonal to T M and hence the anti-invariant distribution is D⊥ = Span{Z 2 }. Also, the slant distribution   is spanned by the vector Z 1 , i.e., Dθ = Span{Z 1 } with slant angle θ = cos−1 23 . Thus M is a hemi-slant submanifold of R5 . It is easy to check that both the distributions are integrable. We denote the integral manifolds of D⊥ and Dθ by M⊥ and Mθ , respectively. Then, the metric tensor g of the product manifold M is given by g = 6du 2 + 2u 2 dv 2 = g Mθ + 2u 2 g M⊥ . Thus, M is a non-trivial warped product hemi-slant submanifold of R5 of the form M θ ×√ 2 u M ⊥ . Example 7.6 Consider a submanifold M of R7 = R3 × R4 with coordinates (x1 , x2 , x3 , y1 , y2 , y3 , y4 ) and the product structure  F

∂ ∂ xi

 =

∂ , F ∂ xi



∂ ∂yj

 =−

∂ , i = 1, 2, 3 and j = 1, 2, 3, 4. ∂yj

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223

Let us consider the immersion f of M into R7 as follows: f (u, ϕ) = (u cos ϕ, u sin ϕ, 2u,

√

2u, −u, u sin ϕ, u cos ϕ), ϕ  = 0 u  = 0.

Then the tangent space T M of M is spanned by the following vector fields √ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + sin ϕ +2 + 2 − + sin ϕ + cos ϕ , ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2 ∂ y3 ∂ y4 ∂ ∂ ∂ ∂ Z 2 = −u sin ϕ + u cos ϕ + u cos ϕ − u sin ϕ . ∂ x1 ∂ x2 ∂ y3 ∂ y4

Z 1 = cos ϕ

Then, it is easy to see that F Z 2 is orthogonal to T M, thus the anti-invariant distribution is D⊥ =Span{Z 2 } and the slant distribution is Dθ = Span{Z 1 } with slant   g(F Z 1 ,Z 1 ) angle θ = arccos F = arccos 19 . It is easy to see that both the distribuZ 1 Z 1  tions are integrable. If the integral manifolds of D⊥ and Dθ are denoted by M⊥ and Mθ , respectively, then the induced metric tensor g M on M is given by g M = 9du 2 + 2u 2 dϕ 2 = g Mθ +

 √ 2 2u g M⊥ .

Thus, M is a warped √ product submanifold of the form M = Mθ × f1 M⊥ with warping function f 1 = 2 u. Now, we prove the following useful lemmas. Lemma 7.7 On a warped product hemi-slant submanifold M = Mθ × f M⊥ of a ˜ we have locally product Riemannian manifold M, (i) g(h(X, Y ), F Z ) = −g(h(X, Z ), ωY ) (ii) g(h(T X, Y ), F Z ) = −g(h(T X, Z ), ωY ) for any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ). Proof For any X ∈ (Dθ ) and Z ∈ (D⊥ ), we have g(h(X, Y ), F Z ) = g(∇˜ X Y, F Z ). Then from (1) and (6), we obtain g(h(X, Y ), F Z ) = g(∇˜ X T Y, Z ) + g(∇˜ X ωX, Z ). Since, Mθ is totally geodesic in M, using this fact in the above relation then from (2)–(3), we get g(h(X, Y ), F Z ) = −g(AωY X, Z ) = −g(h(X, Z ), ωY ), which is (i). If we interchange X by T X in (i), we can get (ii). Thus, the lemma is proved. 

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Lemma 7.8 Let M = Mθ × f M⊥ be a proper warped product hemi-slant subman˜ Then, we have: ifold of a locally product Riemannian manifold M. (i) g(h(Z , W ), ωX ) = −g(h(X, Z ), F W ) + T X (ln f )g(Z , W ) (ii) g(h(Z , W ), ωT X ) = −g(h(T X, Z ), F W ) + cos2 θ X (ln f )g(Z , W ) for any X ∈ (Dθ ) and Z , W ∈ (D⊥ ). Proof For any X ∈ (Dθ ) and Z , W ∈ (D⊥ ), we have g(h(Z , W ), ωX ) = g(∇˜ Z W, ωX ). From (6), we get g(h(Z , W ), ωX ) = g(∇˜ Z W, F X ) − g(∇˜ Z W, T X ) = g(F ∇˜ Z W, X ) + g(∇˜ Z T X, W ).

By Lemma 4.1, we derive g(h(Z , W ), ωX ) = g(∇˜ Z F W, X ) + T X (ln f )g(Z , W ). Using (3), we get g(h(Z , W ), ωX ) = −g(A F W Z , X ) + T X (ln f )g(Z , W ) = −g(h(X, Z ), F W ) + T X (ln f )g(Z , W ) which proves (i). If we interchange X by T X in the above relation and using (9), we can easily derive the second part, which proves the lemma completely.  Now, by using Hiepko’s Theorem, we prove the following characterization for warped product hemi-slant submanifolds M = Mθ × f M⊥ of a locally product Riemannian manifold. Theorem 7.9 Let M be a hemi-slant submnifold of a locally product Riemannian ˜ Then M is locally a mixed totally geodesic warped product submanifold manifold M. of the form M = Mθ × f M⊥ in M˜ if and only if (i) A F Z X = 0,

(ii) AωT X Z = cos2 θ X (μ)Z , ∀ X ∈ (Dθ ), Z ∈ (D⊥ ) (36)

for a smooth function μ on M with W (μ) = 0, for each W ∈ (D⊥ ). Proof If M is a mixed totally geodesic warped product submanifold of a locally ˜ then for any X ∈ (Dθ ) and Z , W ∈ (D⊥ ), product Riemannian manifold M, we have g(A F Z X, W ) = g(h(X, W ), F Z ) = 0, i.e., A F Z X has no component in D⊥ . Also, from Lemma 7.7, g(A F Z X, Y ) = 0, i.e., A F Z X has no component in Dθ . Therefore it follows that A F Z X = 0, which is first part of (36). Similarly,

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225

g(AωT X Z , Y ) = g(h(Y, Z ), ωT X ) = 0, i.e., AωT X Z has no component in Dθ for any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ). Therefore, the second part of (36) follows from Lemma 7.8 (ii). Conversely, let M be a proper hemi-slant submanifold of a locally product Riemannian manifold M˜ such that (36) holds. Then by Proposition 7.3 and the relation (36), we find g(∇ X Y, Z ) = 0, which means that the leaves of Dθ are totally geodesic in M. On the other hand, for any X ∈ (Dθ ) and Z , W ∈ (D⊥ ) we have g([Z , W ], T X ) = g(∇˜ Z W, T X ) − g(∇˜ W Z , T X ). From (6), we get g([Z , W ], T X ) = g(∇˜ Z W, F X ) − g(∇˜ Z W, ωX ) − g(∇˜ W Z , F X ) + g(∇˜ Z W, ωX ).

Using (1), we obtain g([Z , W ], T X ) = g(∇˜ Z F W, X ) − g(∇˜ Z ωX, W ) − g(∇˜ W F Z , X ) + g(∇˜ W ωX, Z ). Thus by (2)–(3), we derive g([Z , W ], T X ) = −g( A F W X, Z ) + g(AωX Z , W ) + g(A F Z X, W ) − g(AωX W, Z ).

The first and third terms of right hand side are identically zero by using (36) (i) and the second and fourth terms can be evaluated from (36) (ii) by interchanging X by T X , as follows: g([Z , W ], T X ) = T X (μ)g(Z , W ) − T X (μ)g(W, Z ) = 0, which means that, the distribution D⊥ is integrable, thus if we consider M⊥ be a leaf of D⊥ in M and σ # be a second fundamental form of M⊥ in M, then for any Z , W ∈ (D⊥ ), we have g(A F W T X, Z ) = g(h(Z , T X ), F W ) = g(∇˜ Z T X, F W ). Then by (1), we derive g(A F W T X, Z ) = g(F ∇˜ Z T X, W ). Then, using the structure equation of a locally product Riemannian manifold, we obtain g(A F W T X, Z ) = g(∇˜ Z F T X, W ). Using (6), we get

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S. Uddin

g(A F W T X, Z ) = g(∇˜ Z T 2 X, W ) + g(∇˜ Z ωT X , W ). Thus, by (9) and (2)–(3), we derive g(A F W T X, Z ) = cos2 θ g(∇˜ Z X, W ) − g(AωT X Z , W ) = − cos2 θ g(∇˜ Z W, X ) − g(AωT X Z , W ). As D⊥ is integrable thus on using (2), we get g(A F W T X, Z ) = − cos2 θ g(σ # (Z , W ), X ) − g(AωT X Z , W ). From (36) (i)–(ii), we obtain cos2 θ g(σ # (Z , W ), X ) + cos2 θ X (μ)g(Z , W ) = 0. Since M is proper slant thus, we find  X ), g(σ # (Z , W ), X ) = −X (μ)g(Z , W ) = −g(Z , W )g(∇μ,  that is M⊥ is totally umbilical in M with which means σ # (Z , W ) = −g(Z , W )∇μ ⊥  mean curvature vector H = −∇μ. We can easily see that H ⊥ is a parallel mean curvature vector corresponding to the normal connection of M⊥ into M. Hence, by a result of Hiepko [27], M is a warped product hemi-slant submanifold, which proves the theorem completely.  Now, we find a sharp estimation between the squared norm of the second fundamental form h of the warped product hemi-slant submanifold M = Mθ × f M⊥ and the warping function f . Let M = Mθ × f M⊥ be an n-dimensional warped product hemi-slant submanifold of an m-dimensional locally product Riemannian manifold M˜ such that p = dim Mθ and q = dim M⊥ , where Mθ and M⊥ are proper slant and anti-invariant sub˜ respectively. Denote the tangent bundles of Mθ and M⊥ by Dθ and manifolds of M, ⊥ D , respectively. Consider the orthonormal frame fields {e1 , · · · , eq } and {eq+1 = e1∗ = sec θ T e1∗ , · · · , en = e∗p = sec θ T e∗p } of D⊥ and Dθ , respectively. Then the orthonormal frame fields of the normal subbundles FD⊥ , ωDθ and ν respectively are {en+1 = Fe1 , · · · , en+q = Feq }, {en+q+1 = e˜1 = csc θ ωe1∗ , · · · , en+ p+q = e˜ p = csc θ ωe∗p } and {e2n+1 , · · · , em }. Theorem 7.10 Let M = Mθ × f M⊥ be a mixed totally geodesic warped product hemi-slant submanifold of a locally product Riemannian manifold M˜ such that M⊥ ˜ Then, is an anti-invariant submanifold and Mθ is a proper slant submanifold of M. we have:

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227

(i) The squared norm of the second fundamental form h of M satisfies ||h||2 ≥ q cot2 θ ||∇ θ ln f ||2 where q = dim M⊥ and ∇ θ ln f is the gradient of ln f along Mθ . (ii) If the equality sign of (i) holds identically, then Mθ is totally geodesic and M⊥ ˜ is totally umbilical in M. Proof From the definition of h, we have ||h||2 = ||h(Dθ , Dθ )||2 + ||h(D⊥ , D⊥ )||2 + 2||h(Dθ , D⊥ )||2 . Since M is mixed totally geodesic, then the second term in the above relation is identically zero, thus we find ||h||2 = ||h(D⊥ , D⊥ )||2 + ||h(Dθ , Dθ )||2 . Then from (5), we obtain ||h||2 =

q m  

g(h(ei , e j ), er )2 +

r =n+1 i, j=1

p m  

g(h(ei∗ , e∗j ), er )2 .

r=n+1 i, j=1

The above relation can be expressed in terms of the components of FD⊥ , ωDθ and ν as follows ||h|| = 2

q q  

q p  

g(h(ei , e j ), Fer ) + 2

r =1 i, j=1

+

r=1 i, j=1

q m  

g(h(ei , e j ), er ) + 2

r =2n+1 i, j=1

+

p p  

g(h(ei , e j ), csc θ ωer∗ )2

p q  

g(h(ei∗ , e∗j ), Fer )2

r=1 i, j=1

g(h(ei∗ , e∗j ), e˜r )2 +

r =1 i, j=1

p m  

g(h(ei∗ , e∗j ), er )2 .

(37)

r=2n+1 i, j=1

As we have not found any relation for g(h(ei , e j ), Fer ), for any i, j, r = 1, · · · q and g(h(ei∗ , e∗j ), e˜r ), for any i, j, r = i, · · · , p, therefore we shall leave these terms in the above relation (37). Also, the third and sixth terms have ν-components therefore we also leave these two terms, then by using Lemma 7.7 and Lemma 7.8, we derive ||h||2 ≥ csc2 θ

q p   i, j=1 r=1

(T er∗ ln f )2 g(ei , e j )2 .

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S. Uddin

From the assumed orthonormal frame fields of Dθ , we have T er∗ = cos θ er∗ , for r = 1, · · · , p. Using this fact, we find ||h||2 ≥ cot 2 θ

p q   (er∗ ln f )2 g(ei , e j )2 = q cot2 θ∇ θ ln f 2 i, j=1 r=1

which is inequality (i). If the equality holds in (i), then from the remaining fifth and sixth terms of (37), we obtain the following conditions h(Dθ , Dθ ) ⊥ ωDθ , h(Dθ , Dθ ) ⊥ ν, which means that h(Dθ , Dθ ) ⊂ FD⊥ .

(38)

h(Dθ , Dθ ) ⊥ FD⊥ .

(39)

h(Dθ , Dθ ) = 0.

(40)

Also, from Lemma 7.7, we get

From (38) and (39), we find

˜ by using the fact that Mθ is totally geodesic in M Then Mθ is totally geodesic in M, [12, 17] with (40). Also, from the remaining first and third terms in (37), we have h(D⊥ , D⊥ ) ⊥ FD⊥ , h(D⊥ , D⊥ ) ⊥ ν ⇒ h(D⊥ , D⊥ ) ⊂ ωDθ .

(41)

From Lemma 7.8 for a mixed totally geodesic warped product submanifold, we have g(h(Z , W ), ω P X ) = cos2 θ (X ln f )g(Z , W ).

(42)

Hence, M⊥ is totally umbilical in M˜ by using the fact that M⊥ is totally umbilical in M [12, 17] with (41) and (42). Thus, the proof is complete. 

8 Warped Product Pointwise Hemi-Slant Submanifolds As a generalization of slant submanifolds of almost Hermitian manifolds, Etayo [25] has introduced the notion of pointwise slant submanifolds. Later, Chen and Garay [20] studied these submanifolds and gave a method for the construction of examples of pointwise slant submanifolds of Euclidean spaces. Pointwise slant submanifolds of a locally product Riemannian manifold were studied in [26]. On the similar line of almost Hermitian manifolds, we define the pointwise slant submanifolds as follows:

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229

Definition 8.1 A submanifold M of a locally product Riemannian manifold M˜ is called pointwise slant [26], if at each point p ∈ M, the Wirtinger angle θ (X ) between F X and T p M is independent of the choice of the non-zero vector X ∈ T p M. In this case, the Wirtinger angle gives rise to a real-valued function θ : T M − {0} → R which is called the Wirtinger function or slant function of the pointwise slant submanifold. We note that a pointwise slant submanifold of a locally product Riemannian manifold is called slant, in the sense of [8, 16, 38], if its Wirtinger function θ is globally constant. We also note that every slant submanifold is a pointwise slant submanifold. Moreover, F-invariant and F-anti-invariant submanifolds introduced in [67] and in [1] are pointwise slant submanifolds with slant function θ = 0 and θ = π2 , respectively. A pointwise slant submanifold M of a locally product Riemannian manifold M˜ is called a proper pointwise slant if it is neither F-invariant nor F-anti-invariant nor θ is constant on M. On the similar line of Chen’s result (Lemma 2.1) of [20], it is easy to show that M is a pointwise slant submanifold of a locally product Riemannian manifold M˜ if and only if T 2 = (cos2 θ )I,

(43)

for some real-valued function θ defined on M, where I denotes the identity transformation of the tangent bundle T M of M. The following relations are the consequences of (43), which is given as g(T X, T Y ) = cos2 θ g(X, Y ),

(44)

g(ωX, ωY ) = sin2 θ g(X, Y )

(45)

for any X, Y ∈ (T M). Another important relation for a pointwise slant submanifold of a locally product Riemannian manifold is obtained by using (6), (7) and (43) as BωX = sin2 θ X,

CωX = −ωT X

(46)

for any X ∈ (T M).

8.1 Pointwise Hemi-Slant Submanifolds of Locally Product Riemannian Manifolds In this subsection, we define and study pointwise hemi-slant submanifolds of a locally product Riemannian manifold. We give examples of pointwise hemi-slant submanifolds and investigate the geometry of the leaves of distributions.

230

S. Uddin

Definition 8.2 Let M˜ be a locally product Riemannian manifold and M a real sub˜ Then, we say that M is a pointwise hemi-slant submanifold if there manifold of M. exists a pair of orthogonal distributions D⊥ and Dθ on M such that: (i) The tangent space T M admits the orthogonal direct decomposition T M = D⊥ ⊕ Dθ . (ii) The distribution D⊥ is F-anti-invariant, i.e., F(D⊥ ) ⊂ T ⊥ M. (iii) The distribution Dθ is pointwise slant with slant function θ . In the above definition, the angle θ is called the slant function of the pointwise slant distribution Dθ . The anti-invariant distribution D⊥ of a pointwise hemi-slant submanifold is a pointwise slant distribution with slant function θ = π2 . If we denote the dimensions of Dθ and D⊥ by p and q, respectively, then we have the following possible cases: (i) (ii) (iii) (iv) (v)

If p = 0, then M is an anti-invariant submanifold. If q = 0, then M is a pointwise slant submanifold. If q = 0 and θ = 0, then M is an invariant submanifold. If θ is constant on M, then M is a hemi-slant submanifold with slant angle θ . If θ = 0, then M is a semi-invariant submanifold.

We note that a pointwise hemi-slant submanifold is proper if q  = 0 and θ is not a constant. Now, we construct the following examples of pointwise hemi-slant submanifolds. Example 8.3 Consider a submanifold M of R6 = R3 × R3 with cartesian coordinates (x1 , x2 , x3 , y1 , y2 , y3 ) and the product structure  F

∂ ∂ xi



∂ , F =− ∂ xi



∂ ∂yj

 =

∂ , 1 ≤ i, j ≤ 3. ∂yj

(47)

  For any θ, ϕ ∈ 0, π2 , consider a submanifold M of R6 defined as χ (θ, ϕ) = (cos θ cos ϕ, sin θ cos ϕ, sin ϕ, sin θ sin ϕ, cos θ sin ϕ, cos ϕ) such that ϕ is a real-valued function on M. Then, the tangent space T M of M is spanned by the following vector fields Z 1 = − sin θ cos ϕ

∂ ∂ ∂ ∂ + cos θ cos ϕ + cos θ sin ϕ − sin θ sin ϕ , ∂ x1 ∂ x2 ∂ y1 ∂ y2

∂ ∂ ∂ ∂ − sin θ sin ϕ + cos ϕ + sin θ cos ϕ ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ ∂ + cos θ cos ϕ − sin ϕ . ∂ y2 ∂ y3

Z 2 = − cos θ sin ϕ

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231

It is easy to see that F Z 2 is orthogonal to T M, thus the anti-invariant distribution θ1 is D⊥ = Span{Z 2 } and 1 } is a pointwise slant distribution with slant  D = Span{Z g(F Z 1 ,Z 1 ) function θ1 = arccos F Z 1 Z 1  = 2ϕ and hence M is a proper pointwise hemislant submanifold with slant function θ1 = 2ϕ. Example 8.4 Consider a submanifold M of R4 = R2 × R2 with cartesian coordinates (x1 , x2 , y1 , y2 ) and the product structure  F

∂ ∂ xi



∂ , F = ∂ xi



∂ ∂ yi

 =−

∂ , i = 1, 2. ∂ yi

For a real-valued function v and M, define an immersion φ(u, v) = (u + v, sin v, − u − v, cos v),

u, v  = 0.

Its tangent space T M is spanned by the vectors Z1 =

∂ ∂ ∂ ∂ ∂ ∂ − , Z2 = + cos v − + sin v . ∂ x1 ∂ y1 ∂ x1 ∂ x2 ∂ y1 ∂ y2

It is easy to see that F Z 1 is orthogonal to T M and hence the anti-invariant distribution is D⊥ = Span{Z 1 }and Dθ = Span{Z 2 } is a pointwise slant distribution with slant function θ = cos−1 cos32v . Since v is a real-valued function on M, then the slant function θ is not a constant and hence M is a proper pointwise hemi-slant submanifold. Now, we give the following useful lemma. Lemma 8.5 ([3]) Let M be a pointwise hemi-slant submanifold of a locally product ˜ Then: Riemannian manifold M. (i) For any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ), we have cos2 θ g(∇ X Y, Z ) = g(A F Z T Y, X ) + g(AωT Y Z , X ). (ii) For any Z , V ∈ (D⊥ ) and X ∈ (Dθ ), we have cos2 θ g(∇ Z V, X ) = −g(A F V T X, Z ) − g(AωT X V, Z ). Theorem 8.6 Let M be a proper pointwise hemi-slant submanifold of a locally ˜ Then: product Riemannian manifold M. (i) The distribution D⊥ is integrable if and only if g(h(X, V ), F Z ) = g(h(X, Z ), F V ), for any X ∈ (Dθ ) and Z , V ∈ (D⊥ ).

232

S. Uddin

(ii) The distribution Dθ is integrable if and only if g(A F Z T Y, X ) − g(A F Z T X, Y ) = g(AωT X Z , Y ) − g(AωT Y Z , X ), for any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ). Proof Using polarization identity in Lemma 8.5 (ii), we have cos2 θ g(∇V Z , X ) = −g(A F Z T X, V ) − g(AωT X Z , V ),

(48)

for Z , V ∈ (D⊥ ) and X ∈ (Dθ ). Then, Lemma 8.5 (ii), (48) and the symmetry of the shape operator imply that cos2 θ g([Z , V ], X ) = g(h(T X, V ), F Z ) − g(h(T X, Z ), F V ), which gives the assertion by interchanging X by T X and using (43). Similarly, by using the polarization identity in Lemma 8.5 (i) and the definition of Lie bracket, we obtain (ii).  Theorem 8.7 Let M be a proper pointwise hemi-slant submanifold of a locally ˜ Then: product Riemannian manifold M. (i) The anti-invariant distribution D⊥ defines a totally geodesic foliation if and only if g(h(T X, Z ), F V ) = −g(h(Z , V ), ωT X ), for any Z , V ∈ (D⊥ ) andX ∈ (Dθ ). (ii) The pointwise slant distribution Dθ defines a totally geodesic foliation if and only if g(h(X, T Y ), F V ) = −g(h(X, V ), ωT Y ), for any X, Y ∈ (Dθ ) and V ∈ (D⊥ ). Proof The proof follows from Lemma 8.5.



Thus, the following corollary is an immediate consequence of Theorem 8.6. Corollary 8.8 Let M be a proper pointwise hemi-slant submanifold of a locally ˜ Then, M is a locally Riemannian product manifold product Riemannian manifold M. M = M⊥ × Mθ if and only if AωT X V = −A F V T X, for any V ∈ (D⊥ ) and X ∈ (Dθ ).

Slant Submanifolds and Their Warped Products …

233

8.2 Warped Product Pointwise Hemi-Slant Submanifolds M⊥ × f Mθ As we have seen in Theorem 7.4 that there are no warped product hemi-slant submanifolds of a locally product Riemannian manifold M˜ of the form M⊥ × f Mθ , ˜ respecwhere M⊥ and Mθ are anti-invariant and proper slant submanifolds of M, tively. In the following examples, we can see that the warped product M⊥ × f Mθ exists only when the spherical manifold is a pointwise slant submanifold and we call such warped product, a pointwise hemi-slant warped product. Example 8.9 Consider a submanifold of R6 = R3 × R3 with the cartesian coordinates and the product structure given in Example 8.3. Then the immersed submanifold M of R6 is given by χ (u, v) = (u cos v, u sin v, v, u cos v, u sin v, −2v)   such that u ∈ R − {0} is a real-valued function on M and v ∈ 0, π2 . The tangent space of M is spanned by the following vectors ∂ ∂ ∂ ∂ + sin v + cos v + sin v , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ ∂ ∂ ∂ ∂ ∂ Z 2 = −u sin v + u cos v + − u sin v + u cos v −2 . ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2 ∂ y3 Z 1 = cos v

Thus, it is clear that F Z 1 is orthogonal to T M and hence the anti-invariant distribution θ is D⊥ = Span{Z 1 } and  3D  = Span{Z 2 } is a pointwise slant distribution with slant −1 . Thus, M is a poitwise hemi-slant submanifold of R6 . function θ = cos 5+2u 2 Also, it is easy to see that both the distributions are integrable. If we denote the integral manifolds of D⊥ and Dθ by M⊥ and Mθ , respectively, then the metric g of the product manifold M is given by  2   5 + 2u 2 g Mθ . g = 2du 2 + 5 + 2u 2 dv 2 = g M⊥ + Hence, we conclude that M is a warped product pointwise hemi-slant submanifold √ of R6 of the form M⊥ × f Mθ with the warping function f = 5 + 2u 2 . Example 8.10 Let R6 be an Euclidean space with the cartesian coordinates (x1 , x2 , x3 , y1 , y2 , y3 ) and the almost product structure  F

∂ ∂ xi



∂ , F = ∂ xi



∂ ∂yj



Consider a submanifold M of R6 defined by

=−

∂ , 1 ≤ i, j ≤ 3. ∂yj

234

S. Uddin

χ (u, v, w) = (u cos v, u sin v, w, w cos v, w sin v, −u) for non-vanishing real-valued functions u, w on M such that u  = w. Then the tangent bundle T M is spanned by Z 1 , Z 2 and Z 3 , where ∂ ∂ ∂ + sin v − , ∂ x1 ∂ x2 ∂ y3 ∂ ∂ ∂ ∂ Z 2 = −u sin v + u cos v − w sin v + w cos v , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ ∂ ∂ Z3 = + cos v + sin v . ∂ x3 ∂ y1 ∂ y2

Z 1 = cos v

It is easy to see that F Z 1 and F Z 3 are orthogonal to T M. Then D⊥ = Span{Z 1 , Z 3 } is an anti-invariant distributionand D θ = Span{Z 2 } is a pointwise slant distribution 2 2 with slant function θ = cos−1 uu 2 −w . Thus, M is a pointwise hemi-slant subman+w 2 ifold of R6 . It is easy to check that both the distributions are integrable. If we denote the integral manifolds of D⊥ and Dθ by M⊥ and Mθ , respectively, then the metric tensor g of the product manifold M is given by     g = 2du 2 + 2dw2 + u 2 + w2 dv 2 = g M⊥ + u 2 + w2 g Mθ . Hence, M is a non-trivial warped product pointwise hemi-slant submanifold of R6 √ 2 of the form M⊥ × f Mθ with the warping function f = u + w 2 . Example 8.11 Consider a submanifold of R6 = R3 × R3 with the cartesian coordinates and the product structure given in Example 8.3. Let M be a submanifold of R6 given by the equations x1 = u cosh v, x2 = u sinh v, x3 = −v, y1 = u sinh v, y2 = u cosh v, y3 =

√ 2v

  such that u, v ∈ R − {0} are real-valued functions on M and v ∈ 0, π2 . Then the tangent bundle of M is spanned by Z 1 and Z 2 , where ∂ ∂ ∂ ∂ + sinh v + sinh v + cosh v , ∂ x1 ∂ x2 ∂ y1 ∂ y2 √ ∂ ∂ ∂ ∂ ∂ ∂ Z 2 = u sinh v + u cosh v − + u cosh v + u sinh v + 2 . ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2 ∂ y3 Z 1 = cosh v

Since F Z 1 is orthogonal to T M and θ = cos

−1



1 3 + 2u 2 cosh 2v

 .

Slant Submanifolds and Their Warped Products …

235

θ Then the anti-invariant distribution is D⊥ = Span{Z 1 }, and 2 } is a  D 1 = Span{Z  −1 . Hence, M pointwise slant distribution with slant function θ = cos 3+2u 2 cosh 2v 6 is a pointwise hemi-slant submanifold of R . Clearly, both the distributions are integrable. If the integral manifolds of D⊥ and Dθ are M⊥ and Mθ , respectively, then the metric g of the product manifold M is given by

  g = 2 cosh 2v du 2 + 3 + 2u 2 cosh 2v dv 2 √ 2  2 = 2 cosh 2v g M⊥ + 3 + 2u 2 cosh 2v g Mθ . Thus, M is a warped product pointwise hemi-slant submanifold of R6 √ of the form √ 2 3 + 2u cosh 2v and f 2 = 2 cosh 2v. f 2 M⊥ × f 1 Mθ with warping functions f 1 = In fact, M is a doubly warped product submanifold of R6 with the warping functions f 1 and f 2 . Now, we investigate the geometry of the warped product pointwise hemi-slant submanifolds of the form M⊥ × f Mθ . First, we prove the following useful lemmas for later use. Lemma 8.12 ([3]) Let M = M⊥ × f Mθ be a warped product pointwise hemi-slant ˜ Then: submanifold of a locally product Riemannian manifold M. (i) g(h(Z , V ), ωX ) = −g(h(X, Z ), F V ); (ii) g(h(X, Z ), ωY ) = −g(h(Y, Z ), ωX ) for any Z , V ∈ (D⊥ ) and X, Y ∈ (Dθ ). Lemma 8.13 ([3]) Let M = M⊥ × f Mθ be a warped product pointwise hemi-slant ˜ Then: submanifold of a locally product Riemannian manifold M. (i) g(h(X, Y ), F Z ) = −Z (ln f )g(X, T Y ), (ii) g(h(T X, Y ), F Z ) = − cos2 θ Z (ln f ) g(X, Y ) − g(h(Y, Z ), ωT X ) for any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ). Proof For any X, Y ∈ (Dθ ) and Z ∈ (D⊥ ), we have g(h(X, Y ), F Z ) = g(∇˜ X Y, F Z ) = g(∇˜ X FY, Z ). Using (6), we obtain g(h(X, Y ), F Z ) = g(∇˜ X T Y, Z ) + g(∇˜ X ωY, Z ) = −g(∇˜ X Z , T Y ) − g(AωY X, Z ). Then from (4) and Lemma 4.1 (ii), we derive g(h(X, Y ), F Z ) = −Z (ln f ) g(X, T Y ) − g(h(X, Z ), ωY ). By polarization identity, we derive

(49)

236

S. Uddin

g(h(X, Y ), F Z ) = −Z (ln f ) g(Y, T X ) − g(h(Y, Z ), ωX ).

(50)

Using (1) and Lemma 8.12 (ii), we arrive at g(h(X, Y ), F Z ) = −Z (ln f ) g(X, T Y ) + g(h(X, Z ), ωY ).

(51)

Thus, from (49) and (51), we get (i). The second part of the lemma follows from (50) by interchanging X by T X and using (44). Hence, the proof is complete.  We can easily find the following relations by interchanging X by T X and Y by T Y , for any X, Y ∈ (Dθ ) in Lemma 8.13 (ii) as follows: g(h(T X, Y ), F Z ) = −Z (ln f ) cos2 θ g(X, Y ),

(52)

g(h(X, T Y ), F Z ) = −Z (ln f ) cos2 θ g(X, Y )

(53)

g(h(T X, T Y ), F Z ) = −Z (ln f ) cos2 θ g(X, T Y ).

(54)

and

Then from (49) and (54), we get g(h(T X, T Y ), F Z ) = g(h(X, Y ), F Z ). Also, from (52) and (53), we have g(h(T X, Y ), F Z ) = g(h(X, T Y ), F Z ). The following result gives a characterization of warped product pointwise hemislant submanifolds in a locally product Riemannian manifold. Theorem 8.14 Let M be a pointwise hemi-slant submanifold of a locally product ˜ Then M is locally a warped product submanifold of the Riemannian manifold M. form M⊥ × f Mθ in M˜ such that M⊥ is an anti-invariant submanifold and Mθ is a pointwise slant submanifold of M˜ if and only if AωT X V + A F V T X = −V (μ) cos2 θ X, ∀ X ∈ (Dθ ), V ∈ (D⊥ )

(55)

for some smooth function μ on M satisfying Y (μ) = 0, for any Y ∈ (Dθ ). Proof Let M = M⊥ × f Mθ be a warped product pointwise hemi-slant submanifold. Then from Lemma 8.12 (i), we have g(AωX V, Z ) = −g(A F V X, Z ), for any X ∈ (Dθ ) and Z , V ∈ (D⊥ ). Interchanging X by T X , we get g(AωT X V + A F V T X, Z ) = 0, which means that AωT X V + A F V T X has no component in T M⊥ , i.e., AωT X V + A F V T X lies in T Mθ . Using this fact with Lemma 8.13 (ii), we get (55).

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237

Conversely, if M is a pointwise hemi-slant submanifold such that (55) holds, then from Lemma 8.5 (ii), we have g(∇ Z V, X ) = − sec2 θ g(AωT X V + A F V T X, Z ). Using the hypothesis of the theorem, i.e., the relation (55) and the orthogonality of two distributions, we arrive at g(∇ Z V, X ) = 0 for any Z , V ∈ (D⊥ ) and X ∈ (Dθ ), which means that the leaves of the distribution D⊥ are totally geodesic in M. Let M⊥ be a leaf of D⊥ , thus M⊥ is totally geodesic in M. Also, from Lemma 8.5 (i), we have cos2 θ g(∇ X Y, V ) = g(A F V T Y + AωT Y V, X ). Using (55), we derive g(∇ X Y, V ) = −V (μ) g(X, Y ).

(56)

By polarization identity, we obtain g(∇Y X, V ) = −V (μ) g(X, Y ).

(57)

Subtracting (57) from (56) and using the definition of Lie bracket, we find that g([X, Y ], V ) = 0, which implies that the pointwise slant distribution Dθ is integrable. Let Mθ be the integral manifold of Dθ and σ # be the second fundamental form of Mθ in M. Then, for any X, Y ∈ (Dθ ) and V ∈ (D⊥ ), from (56), we have g(σ # (X, Y ), V ) = g(∇ X Y, V ) = −V (μ) g(X, Y ) or equivalently, we have

 g(X, Y ) σ # (X, Y ) = −∇μ

(58)

 is the gradient vector of the function μ which means that Mθ is totally where ∇μ  Furthermore, Y (μ) = umbilical in M with mean curvature vector H θ = −∇μ. θ θ 0, Y ∈ (D ) implies that H is parallel with respect to the normal connection D n of Mθ in M. Thus, Mθ is a totally umbilical submanifold with non-vanishing parallel mean curvature vector H θ . Hence, the spherical condition is also fulfilled, that is Mθ is an extrinsic sphere in M. Then, from Hiepko Theorem, M is a nontrivial warped product of the form M = M⊥ ×μ Mθ , which proves the theorem completely. 

238

S. Uddin

Remark 8.15 If we assume θ = 0 in Theorem 8.14, then the warped product pointwise hemi-slant submanifolds reduce to warped product semi-invariant submanifolds of the form M⊥ × f MT which have been discussed in [40], thus Theorem 8.14 is a generalization of Theorem 4.1 of [40]. Now, we establish the following inequality with the help of the following constructed frame fields and some previous formulas which we have obtained for warped product pointwise hemi-slant submanifolds of a locally product Riemannian manifold. We construct the following frame fields for an n = ( p + q)-dimensional warped product pointwise hemi-slant submanifold M = M⊥ × f Mθ of an m-dimensional ˜ where M⊥ and Mθ are anti-invariant and locally product Riemannian manifold M, ˜ respectively. Let us denote by D⊥ and Dθ proper pointwise slant submanifolds of M, the tangent bundles of M⊥ and Mθ , respectively. Also, if we consider the dim(M⊥ ) = q and dim(Mθ ) = p, then the orthonormal frames of D⊥ and Dθ , respectively, are given by {e1 , · · · , eq },

{eq+1 = e1∗ = sec θ T e1∗ , · · · , en = eq+ p = e∗p = sec θ T e∗p }.

Then the orthonormal frame fields of the normal subbundles of FD⊥ , ωDθ and ν, respectively, are {en+1 = Fe1 , · · · , en+q = Feq }, {en+q+1 = e˜1 = csc θ ωe1∗ , · · · , e2n = e˜ p = csc θ ωe∗p }, {en+q+ p+1 = e˜ p+1 , · · · , em = e˜m−2n }. Theorem 8.16 Let M = M⊥ × f Mθ be a proper warped product pointwise hemi˜ where M⊥ and Mθ slant submanifold of a locally product Riemannian manifold M, ˜ respectively. Then: are anti-invariant and proper pointwise slant submanifolds of M, (i) The squared norm of the second fundamental form of M satisfies  ln f 2 h2 ≥ p cos2 θ ∇

(59)

 ln f is gradient of ln f . where p = dim Mθ and ∇ (ii) If equality sign in (i) holds identically, then M⊥ and Mθ are totally geodesic and ˜ respectively. Furthermore, M⊥ × f Mθ is totally umbilical submanifolds of M, a mixed totally geodesic submanifold of M˜ Proof From the definition of h, we have h2 =

n  i, j=1

g(h(ei , e j ), h(ei , e j )) =

n m   r=n+1 i, j=1

Thus, from the frame fields of D⊥ and Dθ , we find

g(h(ei , e j ), er )2 .

Slant Submanifolds and Their Warped Products …

h2 =

q m  

239

g(h(ei , e j ), er )2 + 2

r =n+1 i, j=1

+

m 

q n m   

g(h(ei , e j ), er )2

r=n+1 i=1 j=q+1

n 

g(h(ei , e j ), er )2 .

(60)

r =n+1 i, j=q+1

Leaving the second positive term on the right-hand side of the above relation, we have q p m m     h2 ≥ g(h(ei , e j ), er )2 + g(h(ei∗ , e∗j ), er )2 . r =n+1 i, j=1

r=n+1 i, j=1

Using the frame fields of FD⊥ , ωDθ and ν, the above equation takes the form h2 =

q q  

g(h(ei , e j ), Fer )2 +

r =1 i, j=1

+

m−2n 

+





g(h(ei , e j ), e˜r )2

r=1 i, j=1

q

g(h(ei , e j ), e˜r )2 +

r = p+1 i, j=1 p

q p  

p q  

g(h(ei∗ , e∗j ), Fer )2

r=1 i, j=1

p

g(h(ei∗ , e∗j ), e˜r )2

r =1 i, j=1

m−2n 

p 

g(h(ei∗ , e∗j ), e˜r )2 .

(61)

r= p+1 i, j=1

The third and sixth terms have ν-components and we have not found any relation for these components, therefore, we can leave these two positive terms. Also, we could not find any relation for g(h(ei , e j ), Fer ), for any i, j, r = 1, · · · , q and g(h(ei∗ , e∗j ), e˜r ), for any i, j, r = 1, · · · , p, therefore, we shall leave these positive terms also. After leaving these terms on the right-hand side of (61) and using the constructed frame fields, we find h ≥ 2

q p  

g(h(ei , e j ), csc θ

ωer∗ )2

r =1 i, j=1

+

p q  

g(h(ei∗ , e∗j ), Fer )2 .

(62)

r=1 i, j=1

Again leaving the first positive term on the right-hand side of the above equation. Thus, from Lemma 8.13 (i), we derive h2 ≥

p q   

2 (er ln f ) g(ei∗ , T e∗j ) .

r=1 i, j=1

Then from the adopted frame fields of Dθ , we know that T e∗j = cos θ e∗j , using this fact in the above relation, then we have

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h2 ≥ p cos2 θ

q  (er ln f )2 . r=1

Thus, by using (14), we get  ln f 2 , h2 ≥ p cos2 θ ∇ which is inequality (i). By leaving the second term on the right-hand side of (60), we have (63) h(D⊥ , Dθ ) = 0. Also, from the remaining first and third terms of (61), we obtain h(D⊥ , D⊥ ) ⊂ ωDθ .

(64)

On the other hand, from Lemma 8.12 (i) and (63), we find that h(D⊥ , D⊥ ) ⊥ ωDθ .

(65)

Then from (64) and (65), we conclude that h(D⊥ , D⊥ ) = 0.

(66)

Also, from the remaining fifth and sixth terms on the right-hand side of (61), we find that (67) h(Dθ , Dθ ) ⊂ FD⊥ . Since M⊥ is totally geodesic in M [12, 17], using this fact with (65) we get M⊥ is ˜ On the other hand, (66) implies that Mθ is totally umbilical totally geodesic in M. ˜ in M due to Mθ being totally umbilical in M [12, 17]. Moreover, (63), (65), and ˜ Hence, the proof (66) imply that M is a mixed totally geodesic submanifold of M. is complete  From the above theorem, we have the following remark. Remark 8.17 In Theorem 8.16, if we put θ = 0, then the warped product becomes ˜ where MT and M⊥ M = M⊥ × f MT in a locally product Riemannian manifold M, ˜ respectively, which is a case of are invariant and anti-invariant submanifolds of M, warped product semi-invariant submanifolds which have been discussed in ([9, 40]). Thus, Theorem 4.2 of [40] and Theorem 4.1 of [9] are the special cases of Theorem 8.16.

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9 Generic Warped Product Submanifolds In this section, we introduce a new class of warped products, called generic warped product submanifolds. We define generic submanifolds in [62]. Definition 9.1 A submanifold M of a locally product Riemannian manifold M˜ is said to be a generic submanifold of order 1 if there exist orthogonal distributions D, D⊥ and Dθ on M such that: (i) The tangent space T M admits the orthogonal direct decomposition T M = D ⊕ D⊥ ⊕ Dθ . (ii) The distribution D is F-invariant, i.e. F(D) = D. (iii) The distribution D⊥ is F anti-invariant, i.e., FD⊥ ⊂ T ⊥ M. (iv) The distribution Dθ is proper pointwise slant with slant function θ . Now, we give the following examples of a generic submanifold of order 1 in Euclidean spaces. Example 9.2 Consider the Euclidean 7-space R7 with cartesian coordinates (x1 , x2 , x3 , y1 , y2 , y3 , z). Then, R7 is a locally product Riemannian manifold with standard Euclidean metric and the almost product structure F : R7 → R3 × R3 × R is given by  F

∂ ∂ xi

 =

∂ , F ∂ xi



∂ ∂yj

 =−

∂ , F ∂yj



∂ ∂z

 =

∂ 1 ≤ i, j ≤ 3. ∂z

If a submanifold M of R7 is defined by the immersion x(u, v, w, s) = (u + v, u − v, w, cos w, sin w, s cos s, s sin s), u  = 0, v  = 0, w  = 0;

for non-vanishing real-valued function s on M, then the tangent space of M is spanned by the following vector fields ∂ ∂ ∂ ∂ ∂ ∂ ∂ + , X2 = − , X3 = − sin w + cos w , ∂ x1 ∂ x2 ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2 ∂ ∂ X 4 = (−s sin s + cos s) + (s cos s + sin s) . ∂ y3 ∂z X1 =

Then D = Span{X 1 , X 2 } is an invariant distribution and D⊥ = Span{X 3 } is an antiproper pointwise slant distribution invariant distribution; while, Dθ = Span{X 4 } is a −1 2s sin 2s−(1−s 2 ) cos 2s with slant function θ = cos . Hence, M is a generic subman1+s 2 ifold of R7 . Example 9.3 Consider a submanifold M of a Euclidean 8-space R8 with cartesian coordinates (x1 , x2 , x3 ; y1 , y2 ; z 1 , z 2 , z 3 ). Then, R8 is an almost product Riemannian manifold with product structure F : R8 →= R3 × R2 × R3 is given by

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S. Uddin

 F

∂ ∂ xi

 =−

∂ , F ∂ xi



∂ ∂yj

 =−

∂ , F ∂ yi



∂ ∂z k

 =

∂ ; ∂z k

for 1 ≤ i, k ≤ 3, 1 ≤ j ≤ 2. If M is defined by the following immersion ψ(u, v, w) = (sinh u, cosh u, w, cos v, sin v, v, sec w, tan w), u, v  = 0, for a non-vanishing real-valued function w on M, then, the tangent space of M is spanned by the following vector fields ∂ ∂ ∂ ∂ ∂ + sinh u , X 2 = − sin v + cos v + , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂z 1 ∂ ∂ ∂ X3 = + sec w tan w + sec2 w . ∂ x3 ∂z 2 ∂z 3 X 1 = cosh u

Clearly, we find that F X 2 is orthogonal to T M, thus D = Span{X 1 } and D⊥ = Span{X 2 } are invariant and anti-invariant distributions on M, respectively. Moreθ over, D 3 } is a pointwise slant distribution with slant function θ =  2= Span{X −1 sin w(2+cos2 w) cos . Each distribution is integrable and the tangent space of M is 2− 1 sin2 2w 4

T M = D ⊕ D⊥ ⊕ Dθ , and hence M is a generic submanifold of order 1 in R8 .

Example 9.4 Let R8 = R3 × R3 × R2 be a locally product Riemannian manifold with the cartesian coordinates (x1 , x2 , x3 , y1 , y2 , y3 , z 1 , z 2 ) and the almost product structure       ∂ ∂ ∂ ∂ ∂ ∂ , F , F =− = =− F ∂ xi ∂ xi ∂yj ∂yj ∂z k ∂z k such that 1 ≤ i, j ≤ 3, 1 ≤ k ≤ 2. Consider an isometric immersion χ : M → R8 defined by χ(θ, φ, w) = (cos θ, sin θ, φ, − sin φ, cos φ, w, cos w 2 , sin w2 ), θ  = 0, φ  = 0;

for any non-vanishing real-valued function w. Then, the tangent space T M of M is spanned by the following vectors ∂ ∂ ∂ ∂ ∂ + cos θ , X2 = − cos φ − sin φ ∂ x1 ∂ x2 ∂ x3 ∂ y1 ∂ y2 ∂ ∂ ∂ X3 = − 2w sin w 2 + 2w cos w2 . ∂ y3 ∂z 1 ∂z 2

X 1 = − sin θ

Then from considered product Riemannian structure on R8 , it is easy to see that F X 2 is orthogonal to T M, and hence D⊥ = Span{X 2 } is an anti-invariant distribution and D = Span{X 1 } is an invariant distribution; while Dθ = Span{X 3 } is a proper

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pointwise slant distribution with slant function  = cos−1



1−4w 2 1+4w 2

. Thus, M is a

generic submanifold of R8 . Example 9.5 Consider the Euclidean 10-space R10 = R6 × R2 × R2 with the cartesian coordinates (x1 , · · · , x6 , y1 , y2 , z 1 , z 2 ) and the Euclidean metric tensor < . , . >. We define the almost product structure F : R10 → R6 × R2 × R2 by  F

∂ ∂ xi



∂ , F = ∂ xi



∂ ∂yj



∂ , F =− ∂yj



∂ ∂z k

 =−

∂ , ∂z k

for 1 ≤ i ≤ 6, 1 ≤ j, k ≤ 2. Then, we find that F 2 = I (F  = ±I ) and < X , FY > =< F X , Y >, for any X, Y ∈ R10 . Hence, (R10 , F, < . , . >) is an almost product Riemannian manifold. Let M be a submanifold of R10 defined by the immersion ψ : M → R10 : ψ(u, v, w, t) =(cos(u + v), sin(u + v), cos(u − v), sin(u − v), w, t, sin w, cos w, t2 t , e ), 2

such that u, v, w  = 0 (u = ±v) and a non-constant function t. Then the tangent bundle T M of M is spanned by X 1 , X 2 , X 3 and X 4 , where ∂ ∂ ∂ ∂ + cos(u + v) − sin(u − v) + cos(u − v) , ∂ x1 ∂ x2 ∂ x3 ∂ x4 ∂ ∂ ∂ ∂ + cos(u + v) + sin(u − v) − cos(u − v) , X 2 = − sin(u + v) ∂ x1 ∂ x2 ∂ x3 ∂ x4 ∂ ∂ ∂ ∂ ∂ ∂ + cos w − sin w , X4 = +t + et . X3 = ∂ x5 ∂ y1 ∂ y2 ∂ x6 ∂z 1 ∂z 2

X 1 = − sin(u + v)

It is easy to see that F X 3 ⊥ T M = Span{X1 , · · · , X4 } and hence, D⊥ = Span{X 3 } is an anti-invariant distribution and D = Span{X 1 , X 2 } is an invariant distribution. θ Moreover, D 4 } is a proper pointwise slant distribution with slant function  =2 Span{X 2t 1−t θ = cos−1 1+t 2 −e . Thus, M is a generic submanifold of R10 . +e2t Next, we study generic warped product submanifolds of locally product Riemannian manifolds. We define these submanifolds as follows: Definition 9.6 A submanifold of the form M = B × f Mθ of a locally product Riemannian manifold M˜ is said to be generic warped product submanifold of order 1 if M is a warped product of generic submanifolds. In this case, B = MT × M⊥ , a Riemannian product of semi-invariant submanifolds and Mθ is a pointwise slant ˜ respectively. submanifold as a fiber of warped product in M, From now on, for the simplicity we denote the tangent spaces of T MT , T M⊥ and T Mθ by D, D⊥ and Dθ , respectively.

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A warped product submanifold M = M1 × f M2 is said to be mixed totally geodesic if h(X, Z ) = 0, for any X ∈ (T M1 ) and Z ∈ (T M2 ). In [62], we proved the following useful results. Theorem 9.7 Let M = B × f Mθ be a generic warped product submanifold of a locally product Riemannian manifold M˜ such that B = MT × M⊥ . Then, if M is D − Dθ mixed totally geodesic then M is a Riemannian product, i.e., f is constant. Theorem 9.8 Let M = B × f Mθ be a generic warped product submanifold of a locally product Riemannian manifold M˜ such that B = MT × M⊥ and M is D⊥ − Dθ mixed totally geodesic, where MT , M⊥ and Mθ are invariant, anti-invariant and ˜ respectively. Then: proper pointwise slant submanifolds of M, (i) The squared norm of the second fundamental form h2 of M satisfies 

 ⊥ (ln f )2 + 2 (csc θ + cot θ)2 ∇  T (ln f )2 , h2 ≥ s cos2 θ ∇

(68)

 T (ln f ) and ∇  ⊥ (ln f ) are the gradient components where s = dim(Mθ ) and ∇ of ln f along MT and M⊥ , respectively. (ii) If equality sign in (i) holds identically, then MT and M⊥ are totally geodesic ˜ Furthermore, submanifolds of M˜ and Mθ is a totally umbilical submanifold of M. ˜ M never be a D − Dθ mixed totally geodesic submanifold of M. Remark 9.9 The inequality (68) is valid only when the fiber of the warped product is either anti-invariant or a proper pointwise slant submanifold. Remark 9.10 For a generic warped product submanifold M = B × f Mθ of a locally ˜ we cannot assume that M is D − Dθ mixed totally product Riemannian manifold M, geodesic in M˜ because this is the case of non-existence case of such warped products (see, Theorem 9.7). Theorem 9.8 implies that Theorem 9.11 Let M = M⊥ × f MT be a warped product semi-invariant subman˜ Then, we have: ifold of a locally product Riemannian manifold M. (i) The squared norm of the second fundamental form of M satisfies  ⊥ (ln f )2 , h2 ≥ s∇

(69)

 ⊥ (ln f ) are the gradient component of ln f along where s = dim(MT ) and ∇ M⊥ . (ii) If the equality sign in (i) holds identically, then M⊥ is a totally geodesic sub˜ Furthermore, manifold of M˜ and MT is a totally umbilical submanifold of M. ˜ M never be a mixed totally geodesic submanifold of M. Hence, we conclude that Theorem 4.1 of [9] and Theorem 4.2 of [40] are special cases of Theorem 9.8.

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Theorem 9.12 Let M = MT × f Mθ be a warped product pointwise semi-slant sub˜ Then, we have: manifold of a locally product Riemannian manifold M. (i) The squared norm of the second fundamental form of M satisfies  T ln f 2 , h2 ≥ 2s (csc θ + cot θ )2 ∇

(70)

 T ln f is gradient of ln f along MT . where s = dim(Mθ ) and ∇ (ii) If equality sign in (i) holds identically, then MT and Mθ are totally geodesic and ˜ respectively. Furthermore, M is not mixed totally umbilical submanifolds of M, ˜ totally geodesic in M. Therefore, Theorem 9.8 is a generalization of a main result obtained in [65]. Theorem 9.13 Let M = M⊥ × f Mθ be a mixed totally geodesic warped product ˜ pointwise hemi-slant submanifold of a locally product Riemannian manifold M. Then: (i) The squared norm of the second fundamental form h2 of M satisfies  ⊥ ln f 2 , h2 ≥ s cos2 θ ∇

(71)

 ⊥ ln f is gradient of ln f along M⊥ . where s = dim(Mθ ) and ∇ (ii) If equality sign in (i) holds identically, then M⊥ is totally geodesic in M˜ and Mθ ˜ is a totally umbilical submanifold of M. Which shows that Theorem 9.8 generalizes Theorem 8.16. Now, we provide the following non-trivial examples of generic warped product submanifolds in Euclidean spaces. Example 9.14 Consider the Euclidean 7-space R7 = R2 × R3 × R2 with the cartesian coordinates (x1 , x2 , y1 , y2 , y3 , z 1 , z 2 ) and the Euclidean metric tensor < . , . >. We define the almost product structure F : R7 → R2 × R3 × R2 by:  F

∂ ∂ xi



∂ , F =− ∂ xi



∂ ∂yj



∂ , F = ∂yj



∂ ∂z k

 =−

∂ ∂z k

for 1 ≤ i, k ≤ 2, 1 ≤ j ≤ 3. Clearly, we observe that F 2 = I (F  = ±I ) and < X , FY >=< F X , Y >, for any X, Y ∈ R7 . Hence, (R7 , F, < . , . >) is an almost product Riemannian manifold. Let M be a submanifold of R7 given by the immersion ψ : M → R7 ψ(u, v, w) = (u cos θ, u sin θ, u cos w, u sin w, w, v cos w, v sin w), such that u, v  = 0. Then the tangent bundle T M of M is spanned by

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∂ ∂ ∂ ∂ + sin θ + cos w + sin w , ∂ x1 ∂ x2 ∂ y1 ∂ y2 ∂ ∂ X 2 = cos w + sin w , ∂z 1 ∂z 2 ∂ ∂ ∂ ∂ ∂ + u cos w + − v sin w + v cos w . X 3 = −u sin w ∂ y1 ∂ y2 ∂ y3 ∂z 1 ∂z 2 X 1 = cos θ

Clearly, we find that F X 1 is orthogonal to T M and hence, D⊥ = Span{X 1 } is an antiinvariant distribution and D = Span{X 2 } is an invariant distribution. Furthermore, Dθ = Span{X 3 } is a proper pointwise slant distribution with slant function θ = 2 −v 2 cos−1 1+u . Thus, M is a generic submanifold of R7 . Also, it is easy to see 1+u 2 +v 2 that all the distributions are integrable. If the integral manifolds of D, D⊥ and Dθ , respectively, are MT , M⊥ and Mθ then, the metric structure g of M is given by ds 2 = 2du 2 + dv 2 + (1 + u 2 + v 2 )dw2 = g B + (1 + u 2 + v 2 )g Mθ . Hence, M = B × f Mθ is generic warped product submanifold with B = MT × M⊥ and the warping function f = (1 + u 2 + v 2 ) in R7 . Example 9.15 Let R8 be the Euclidean 8-space with the cartesian coordinates (x1 , x2 , x3 , y1 , y2 , z 1 , z 2 , z 3 ) and the Euclidean metric tensor < . , . >. If we have the almost product structure F : R8 → R3 × R2 × R3 , which satisfying  F

∂ ∂ xi

 =−

∂ , F ∂ xi



∂ ∂yj

 =−

∂ , F ∂yj



∂ ∂z k

 =

∂ , ∂z k

such that 1 ≤ i, k ≤ 3, 1 ≤ j ≤ 2 then, we have F 2 =I (F = ±I ) and < X , FY > =< F X , Y >, for any X, Y ∈ R8 . Thus, (R8 , F, < . , . >) is an almost product Riemannian manifold. Now, if a submanifold M of R8 is given by the immersion ψ : R4 → R8 ψ(u 1 , u 2 , u 3 , u 4 ) =(u 1 − u 2 , u 3 cos θ, u 3 sin θ, (u 1 + u 2 ) cos u 4 , (u 1 + u 2 ) sin u 4 , u 3 cos u 4 , u 3 sin u 4 , u 4 ) for non-vanishing functions u 1 , u 2 and u 3 on M; then the tangent bundle T M of M is given by X 1 , X 2 , X 3 and X 4 such that ∂ ∂ ∂ ∂ ∂ ∂ + cos u 4 + sin u 4 , X2 = − + cos u 4 + sin u 4 , ∂ x1 ∂ y1 ∂ y2 ∂ x1 ∂ y1 ∂ y2 ∂ ∂ ∂ ∂ X 3 = cos θ + sin θ + cos u 4 + sin u 4 , ∂ x2 ∂ x3 ∂z 1 ∂z 2 X1 =

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247

∂ ∂ ∂ + (u 1 + u 2 ) cos u 4 − u 3 sin u 4 ∂ y1 ∂ y2 ∂z 1 ∂ ∂ + u 3 cos u 4 + . ∂z 2 ∂z 3

X 4 = −(u 1 + u 2 ) sin u 4

It is easy to observe that F X 3 is perpendicular to T M and hence, D⊥ = Span{X 3 } is an anti-invariant distribution and D = Span{X 1 , X 2 } is an invariant distribution. On θ = Span{X 4 } is a pointwise slant distribution with slant function the other hand,  D 2 2 2 −1 1−u 1 −u 2 −2u 1 u 2 +u 3 θ = cos . Hence, M is a generic submanifold of R8 . It is easy 1+u 21 +u 22 +2u 1 u 2 +u 23 to see that all the distributions are integrable. Let MT , M⊥ and Mθ be the integral manifolds of D, D⊥ and Dθ , respectively. Then the metric structure g of M is given by ds 2 = 2(du 21 + du 22 ) + 2du 23 + (1 + u 21 + u 22 + 2u 1 u 2 + u 23 )du 24 = g B + (1 + u 21 + u 22 + 2u 1 u 2 + u 23 )g Mθ . Then M = B × f Mθ is a generic

warped product submanifold with B = MT × M⊥

and the warping function f =

(1 + u 21 + u 22 + 2u 1 u 2 + u 23 ) in R8 .

Acknowledgements The author would like to express his very great appreciation to Professor Mohammad Hasan Shahid, JMI, New Delhi, and anonymous referees for their critical comments and valuable suggestions, which improve the presentation of this chapter. My special thanks are extended to the Distinguished Professor Bang-Yen Chen, Michigan State University, USA, for the explorative and creative discussion during the preparation of this chapter.

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Slant Submanifolds of Quaternion Kaehler and HyperKaehler Manifolds Mohammad Hasan Shahid, Falleh Al-Solamy, and Mohammed Jamali

2000 AMS Mathematics Subject Classification: 53C15 · 53C25 · 53C40

1 Introduction The aim of this chapter is to discuss and survey briefly some results on slant submanifolds of quaternion Kaehler and hyperKaehler manifolds. For this, let M be a 4m-dimensional Riemannian manifold with metric tensor g. Then M is said to be a quaternion Kaehler manifold if there exists a three-dimensional vector bundle E consisting of tensors of type (1,1) with local basis of almost Hermitian structures J1 , J2 , J3 such that (a) J12 = −I, J22 = −I, J32 = −I J1 J2 = −J2 J1 = J3 , J2 J3 = −J3 J2 = J1 , J3 J1 = −J1 J3 = J2 where I is the identity tensor  of type (1,1) on M. (b) ∇ X Ja = 3b=1 Q ab (X )Jb , a = 1, 2, 3 for all vector fields X tangent to M, where ∇ denotes the Riemannian connection in M and Q ab are 1-forms defined locally on M such that Q ab + Q ba = 0.

M. H. Shahid (B) Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India e-mail: [email protected] F. Al-Solamy President, King Khalid University, Abha, Saudi Arabia M. Jamali Department of Mathematics, Al-Falah University, Faridabad 121004, Haryana, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_9

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Let M be a quaternion Kaehler manifold and X be a non-null vector field on M. Then 4-plane spanned by {X, J1 X, J2 X, J3 X } denoted by φ(X ) is called a quaternion 4plane. Any 2-plane in φ(X ) is called a quaternion plane. The sectional curvature of a quaternion plane is called quaternion sectional curvature. A quaternion Kaehler manifold is said to be quaternion space form if its quaternion sectional curvatures are equal to a constant c. Moreover, it is known that a quaternion Kaehler manifold M is a quaternion space form, denoted by M(c), if and only if its curvature tensor is given by [14]  c [g(Ja Y, Z )Ja X {g(Y, Z )X − g(X, Z )Y + 4 a=1 3

R(X, Y )Z =

(1)

− g(Ja X, Z )Ja Y + 2g(X, Ja Y )Ja Z ]} for all vector fields X, Y, Z tangent to M and any local basis {J1 , J2 , J3 } of E. Definition 1.1 ([1]) Let M be a Riemannian manifold isometrically immersed in a quaternion Kaehler manifold M. A distribution D :→ D p ⊆ T p M is called a quaternion distribution if we have Ja (D) ⊆ D, a = 1, 2, 3 Definition 1.2 ([1]) A submanifold M in a quaternion Kaehler manifold M is called quaternion C R-submanifold if it admits a differentiable quaternion distribution D such that its orthogonal complementary distribution D ⊥ is totally real, i.e. Ja D ⊥ p ⊆ ⊥ T p M and invariant under quaternion structure, i.e. Ja D p ⊆ D p , a = 1, 2, 3 for any p ∈ M. Here T p⊥ M denotes the normal space of M at p. A submanifold M of a quaternion Kaehler manifold M is called a quaternion submanifold if dim D ⊥ p = 0 and a totally real submanifold if dim D p = 0. A quaternion C R-submanifold is said to be proper if it is neither totally real nor quaternion. Definition 1.3 ([12]) A submanifold M of a quaternion Kaehler manifold M is said to be quaternion slant submanifold if for p ∈ M and any X ∈ T p M, the angle between Ja X, a = 1, 2, 3 and T p M is constant θ ∈ [0, π2 ] called the slant angle of quaternion submanifold M in M. A slant submanifold of a quaternion Kaehler manifold is said to be proper (or θ slant proper) if it is neither quaternion nor totally real. In particular, quaternion submanifold and totally real submanifolds of M are quaternion slant submanifolds with slant angle θ = 0 and θ = π2 respectively. Now, we discuss two examples of proper slant submanifolds [12]. Let R4m , (m > 1) be an Euclidean space, then the canonical complex structure J1 , J2 , J3 on R4m and the Hermitian metric g are given by J1 (x1 , y1 , z 1 , w1 , ...xm , ym , z m , wm ) = (−y1 , x1 , −w1 , z 1 , ... − ym , xm , −wm , z m ),

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J2 (x1 , y1 , z 1 , w1 , ...xm , ym , z m , wm ) = (−z 1 , w1 , x1 , −y1 , ... − z m , wm , xm , −ym ), J3 (x1 , y1 , z 1 , w1 , ...xm , ym , z m , wm ) = (−w1 , −z 1 , y1 , x1 , ... − wm , −z m , ym , xm ), and g((x1 , y1 , z 1 , w1 , ...xm , ym , z m , wm ), (u 1 , v1 , t1 , s1 , ...u m , vm , tm , sm )) = x1 u 1 + y1 v1 + z 1 t1 + w1 s1 , ...xm u m + ym vm + z m tm + wm sm .

Example 1.4 If m = 2, then for any θ ∈ (0, π2 ), consider a surface given by X (u, v) = (u cosθ, v, v, v, u sinθ, 0, 0, 0). Then T M is spanned by Z 1 = cosθ ∂ x1 + sinθ ∂ x5 ,

Z 2 = ∂ x2 + ∂ x3 + ∂ x4 .

Then it is easy to see that M is a slant surface of R 8 with respect to the canonical complex structures J1 , J2 and J3 with slant angle α such that cosα = √13 cosθ. Example 1.5 For any k > 0, consider the submanifold given by X (t, s) = (t + 2s, t + s, t + s, t + s, k cost, k sint, 0, 0, k coss, k sins, 0, 0). Then T M is spanned by Z 1 = ∂ x1 + ∂ x2 + ∂ x3 + ∂ x4 − k sint ∂ x5 + k cost ∂ x6 , Z 2 = 2∂ x1 + ∂ x2 + ∂ x3 + ∂ x4 − k sins ∂ x9 + k coss ∂ x10 . Here it is easy to see that M is a slant submanifold of R 12 with respect to J1 , J2 and J3 with slant angle θ = cos −1 √1+k 21√7+k 2 . Definition 1.6 ([14]) A submanifold M of a quaternion Kaehler manifold M is called a quaternion bi-slant submanifold if there exists two orthogonal distributions D1 and D2 on M such that (a) T M admits orthogonal direct decomposition T M = D1 ⊕ D2 (b) For any i = 1, 2, the distribution Di is slant distribution with slant angle θi . It is noted that if dim D1 or dim D2 becomes zero, bi-slant submanifold is a slant submanifold. Thus, a slant submanifold is particular case of bi-slant submanifolds. Papaghuic [10] introduced the notion of semi-slant submanifold using two orthogonal distributions D1 and D2 of tangent bundle T M such that D1 is invariant by Ja , a ∈ {1, 2, 3} and D2 is slant with respect to Ja with slant angle θ = 0 which we will

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use in section 3. However, Sahin [12] studied the same notion using two orthogonal distributions ν and ν ⊥ of the normal bundle T ⊥ M. Definition 1.7 ([12]) Let M be a submanifold of a quaternion Kaehler manifold M. Then M is said to be a semi-slant submanifold if there exist two orthogonal vector subbundles ν and ν ⊥ of normal bundle T ⊥ M such that (i) T M ⊥ = ν ⊕ ν ⊥ (ii) The distribution ν ⊥ is anti-invariant with respect to Ja , a = 1, 2, 3. (iii) The distribution ν is slant with respect to Ja , a = 1, 2, 3 with slant angle θ = π2 . We say that M is a proper semi-slant submanifold if θ = 0, π2 and n 1 , n 2 = 0 where dim(ν) = n 1 and dim(ν ⊥ ) = n 2 . Now, we give an example of semi-slant submanifold of a quaternion Kaehler manifold [12] Let M be a submanifold R 12 given by equations x1 = −u 1 − u 2 ,

x2 = u 1 − u 3

x3 = u 2 + u 3 ,

x4 = u 1 − u 2 + u 3

x5 = u 4 cosθ + u 8 cosθ + u 9 sinθ,

x6 = −u 4 sinθ + u 8 cotθ cosθ − u 9 tanθ sinθ

x7 = u 6 sinθ + u 9 sinθ, x9 = u 5 cosθ + u 8 cosθ, x11 = u 7 cosθ,

x8 = u 4 cosecθ − u 6 cosθ + u 9 tanθ sinθ x10 = u 5 sinθ − u 8 cotθ cosθ x12 = −u 7 sinθ

for θ ∈ (0, π2 ). Then the tangent bundle T M is spanned by Z1 = Z2 = Z3 = Z4 = Z5 =

−∂ x1 + ∂ x2 + ∂ x4 , −∂ x1 + ∂ x3 − ∂ x4 , −∂ x2 + ∂ x3 + ∂ x4 , cosθ ∂ x5 − sinθ ∂ x6 + cosecθ ∂ x8 ,

cosθ ∂ x9 + sinθ ∂ x10 , Z 6 = sinθ ∂ x7 − cosθ ∂ x8 , Z 7 = cosθ ∂ x11 − sinθ ∂ x12 , Z 8 = cosθ ∂ x5 + cotθ cosθ ∂ x6 + cosθ ∂ x9 − cotθ cosθ ∂ x10 , Z 9 = sinθ ∂ x5 − tanθ sinθ ∂ x6 + sinθ ∂ x7 + tanθ sinθ ∂ x8 .

The normal bundle T M ⊥ is spanned by

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V1 = ∂ x1 + ∂ x2 + ∂ x3 , V2 = sinθ ∂ x5 + cosθ ∂ x6 − sinθ ∂ x9 + cosθ ∂ x10 , V3 = cosθ ∂ x5 − sinθ ∂ x6 − cosθ ∂ x7 − sinθ ∂ x8 + sinθ ∂ x11 + cosθ ∂ x12 .

Then it is easy to see that J1 (V1 ) = −Z 1 , J2 (V1 ) = −Z 2 and J3 (V1 ) = −Z 3 , thus ν ⊥ = span{V1 } is an anti-invariant distribution. On the other hand, we can easily obtain that ν = span{V2 , V3 } is a slant vector subspace with slant angle θ = cos −1 ( √16 ). Thus M is a semi-slant submanifold of R 12 . Next, we recall the following lemma of Chen [4] Lemma 1.8 Let a1 , a2 ....an , b be (n+1), real numbers n ≥ 2 such that    2 n n ai = (n − 1) ai2 + b i=1

i=1

Then 2a1 a2 ≥ b with equality holding if and only if a1 + a2 = a3 = a4 = .... = an . Let M be a submanifold of quaternion Kaehler manifold M. We denote by g, the metric on M as well as on M. Let ∇ be the Levi-Civita connection on M. The Gauss and Weingarten formulas for M are given as follows: ∇ X Y = ∇ X Y + h(X, Y ),

(2)

∇ X V = −A V X + ∇ X⊥ V

(3)

for any vector fields X, Y ∈ T M and any vector field V ∈ T ⊥ M, where h, A V and ∇ ⊥ are the second fundamental form, the shape operator in the direction of V and the normal connection on the normal bundle T ⊥ M respectively. Moreover, h and A V are related by the following equation g(h(X, Y ), V ) = g(A V X, Y ).

(4)

The covariant differentiation of the second fundamental form h is given by (∇ X h)(Y, Z ) = ∇ X⊥ h(Y, Z ) − h(∇ X Y, Z ) − h(Y, ∇ X Z )

(5)

for any vector fields X, Y, Z ∈ T M. The Gauss equation for M is written as R(X, Y, Z , W ) = R(X, Y, Z , W ) + g(h(X, W ), h(Y, Z )) − g(h(X, Z ), h(Y, W )).

(6)

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Also, the mean curvature vector H ( p) at p ∈ M is defined by H ( p) =

n 1 h(ei , ei ) n i=1

(7)

where n denotes the dimension of M. If the equation h(X, Y ) = λg(X, Y )H

(8)

is satisfied for any vector fields X, Y ∈ T M, then M is called totally umbilical submanifold. In particular, if h = 0 holds identically, M is called totally geodesic submanfold. A submanifold M is called invariantly quasi-umbilical if there exist 4m − n mutually orthogonal unit normal vectors en+1 , en+2 .....e4m such that the shape operators with respect to all directions er have an eigenvalue of multiplicity n − 1 and that for each er , the distinguished eigen direction is the same [15]. We set h ri j = g(h(ei , e j ), er ), i, j ∈ {1, 2...n}, r ∈ {n + 1....4m}

(9)

and ||h||2 =

n 

g(h(ei , e j ), h(ei , e j ))

(10)

i, j=1

where {e1 , e2 ....en } and {en+1 , en+2 ....e4m } are the orthonormal basis for T M and T ⊥ M, respectively. We denote by K (π ) the sectional curvature of M associated with a plane section π ⊂ T p M, p ∈ M. If {e1 , e2 ...en } is an orthonormal basis of the tangent space T p M, the scalar curvature τ at p is defined by τ ( p) =



K (ei ∧ e j )

(11)

1≤i< j≤m

We denote (in f k)( p) = in f {K (π )/π ⊂ T p M, dim(π ) = 2}. Then the Chen’s first invariant is given by [16] δ M ( p) = τ ( p) − (in f k)( p).

(12)

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Suppose L is an r-dimensional subspace of T p M, r ≥ 2 and {e1 , e2 ....er } an orthonormal basis of L. We define scalar curvature τ (L) of r-plane section L by τ (L) =



K (ei ∧ e j ).

(13)

i< j

For any integer k ≥ 0, we also denote by S(n, k), the set of k-tupples (n 1 , n 2 ....n k ) of integers≥ 2 satisfying n 1 ≤ n, n 1 + n 2 .....n k ≤ n. We also denote by S(n), the set of unordered k-tuples with k ≥ 0 for a fixed n. The Casorati curvature C(L) of the subspace L is defined as [15] C(L) =

r 4m 1   r 2 (h ) r r=n+1 i, j=1 i j

(14)

The normalized δ-Casorati curvatures δc (n − 1) and δˆc (n − 1) are given by [15] [δc (n − 1)] p =

Cp n+1 + in f {C(L)/L is a hyperplane of Tp M} (15) 2 2n(n − 1)

and [δˆc (n − 1)] p = 2C p −

2n − 1 sup{C(L)/L is a hyperplane of Tp M}. (16) 2n

For any X ∈ T M, we put Ja X = Ta X + Fa X, a = 1, 2, 3

(17)

where Ta X and Fa X are the tangential and normal components of Ja X respectively. Similarly, for any V ∈ T ⊥ M, we have Ja V = Ba V + Ca V, a = 1, 2, 3 where Ba V and Ca V denote the tangential and normal components of Ja V We recall that relative null space of M at a point p ∈ M is defined by [14] N p = {X ∈ T p M/ h(X, Y ) = 0 Y ∈ T p M}.

(18)

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2 Ricci Curvature, Squared Mean Curvature and Shape Operator of Slant Submanifolds of Quaternion Space Forms 2.1 Ricci Curvature and Squared Mean Curvature In 1999, Chen [5] obtained a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds of real space forms. In 2003, Motsumoto, Mihai and Tawaza [8] obtained a sharp estimate of the Ricci tensor of slant submanifold M in a complex space form M(c). Afterwards, Shahid and Falleh [13] estimated Ricci curvature of θ -slant submanifold in a quaternion projective space Q P m (4c). However, Shukla and Rao discussed similar problem in quaternion space forms M(c) and obtained the following results on Ricci curvature of slant, bi-slant submanifold of quaternion space forms [14]. Theorem 2.1 Let M be an n-dimensional quaternion slant submanifolds of 4mdimensional quaternion space form M(c) of constant quaternion sectional curvature ‘c’. Then (a) For each unit vector X ∈ T p M, we have Ric(X ) ≤

1 2 {n ||H ||2 + (n − 1)c + 6ccos 2 θ}. 4

(19)

(b) If H ( p) = 0, then a unit tangent vector X at p satisfies the equality case of (19) if and only if X belongs to the relative null space N p . Proof Let e1 , e2 ......en be an orthonormal basis for T p M and {en+1 .....e4m } be the basis for T p⊥ M at any point p ∈ M such that en = X and en+1 is parallel to the mean curvature vector H ( p). Then (by (1) and (17)), Gauss equation (6) gives the Ricci tensor S(X, Y ) and scalar curvature τ as S(X, Y ) =

n 

R(ei , X, Y, ei ) =

i=1

c {(n − 1)g(X, Y ) 4

+3

3 

g(Pa X, Pa Y )} +

a=1

n 

{g(h(ei , ei ), h(X, Y ))

i=1

− g(h(ei , Y ), h(X, ei ))},

(20)

and τ=

n  j=1

S(e j , e j ) =

c {n(n − 1) + 12ncos 2 θ } + n 2 ||H ||2 − ||h||2 . 4

(21)

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We put =τ−

n2 c ||H ||2 − {n(n − 1) + 12ncos 2 θ }. 2 4

(22)

Using equation (22) in (21) it follows that n 2 ||H ||2 = 2( + ||h||2 ).

(23)

which yields  n

2 h iin+1

 n n   2 =2 + (h iin+1 )2 + (h in+1 j )

i=1

i=1

+

n 4m  



i = j

(h ri j )2 .

(24)

r=n+2 i, j=1

Next, setting a1 = h n+1 11 , a2 = (24) gives 

n−1 i=2

h iin+1 and a3 = h n+1 nn , in lemma (1.8) equation

n+1 h n+1 αα h ββ ≥  + 2

1≤α =β≤n−1

n 4m    2 (h in+1 ) + (h ri j )2 j

(25)

r =n+2 i, j=1

i< j

or,  c n2 n+1 h n+1 ||H ||2 + {n(n − 1) + 12ncos 2 θ } ≥ τ − αα h ββ 2 4 1≤α =β≤n−1 +2



2 (h in+1 j ) +

i< j

n 4m  

(h ri j )2 .

r =n+2 i, j=1

(26) On the other hand, using equation of Gauss, we have 

τ −

n+1 h n+1 αα h ββ + 2

1≤α =β≤n−1

n 4m    2 (h in+1 ) + (h ri j )2 j i< j

r =n+2 i, j=1

c = 2S(en , en ) + {(n − 1)(n − 2) + 12(n − 1)cos 2 θ } 4 4m   n+1 2 (h in ) + {(h rnn )2 +2 r=n+2

i

then the shape operator at the mean curvature

   n−1 c 9 θk ( p) − 1+ cos 2 θ In n 4 n−1

at p, where In denotes the identity map of T p M. 9 cos 2 θ ), then A H ≥ 0 at p. (ii) If θk ( p) = 4c (1 + n−1

(39)

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(iii) A unit vector X ∈ T p M satisfies    n−1 c 9 2 θk ( p) − 1+ cos θ X A H (X ) = n 4 n−1 if and only if θk ( p) = 4c (1 + M at p. (iv) The identity

9 cos 2 θ ) n−1

(40)

and X belongs to the null space N p of

   n−1 c 9 2 θk ( p) − 1+ cos θ In AH = n 4 n−1 holds at p if and only if p is totally geodesic point. Proof (i) Let us consider the orthonormal basis for T p M and T p⊥ M as in Theorem 2.6 such that the normal vector en+1 is in the direction of the mean curvature vector H . Consequently the shape operators have the forms (32) and (33). Consider the following two cases Case(I)

H = 0. In this case, it follows from Theorem 2.6 that θk ( p) =

Case(II)

c 9 (1 + cos 2 θ ) 4 n−1

and the conclusion follows. H = 0. Assume X = Z = ei and Y = W = e j in Gauss equation (6) and using (1), it follows ai a j = K i j − −

  3  c g 2 (Tb ei , e j ) 1+3 4 b=1

4m  

h rii h rj j − (h ri j )2



(41)

r=n+2

Therefore (41), gives   3 k (k − 1)c 3c   2 g (Tb e1 , ei j ) − a1 (ai2 + .....aik ) = Ric L 1 i1 i2 ...ik (e1 ) − 4 4 b=1 j=2 −

k 4m    r=n+2 j=2

which implies

h r11 h ri j i j − (h r1i j )2



(42)

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a1 (a2 + a3 + .....an ) =

 (k − 2)!(n − k)! (n − 2)! 2≤i θk ( p) − In n 4 at p, where In denotes the identity map of T p M. (ii) If θk ( p) = 4c , then A H ≥ 0 at p. (iii) A unit vector X ∈ T p M satisfies   n−1 c θk ( p) − X A H (X ) > n 4 if and only if θk ( p) = (iv) The identity

c 4

and X ∈ N p .

AH =

  n−1 c θk ( p) − In n 4

holds at p if and only if p is totally geodesic point.

3 Inequalities for Slant Submanifolds of Quaternion Space Forms 3.1 B. Y. Chen Inequality for Slant Submanifolds Oiaga and Mihai [9] obtained B. Y. Chen inequality for a θ -slant submanifold M in a complex space form M(c) of constant holomorphic sectional curvature c. Motivated by this work, Vilcu [16] obtained a Chen-like inequality for proper θ -slant submanifolds in quaternion space forms. Theorem 3.1 ([16]) Let M be a θ -slant proper submanifold of a quaternionic space form M(c). Thus, for each point p ∈ M, we have δ M ( p) ≤

  2 n−2 c n ||H ||2 + (n + 1 + 9 cos 2 θ ) 2 n−1 4

(48)

Equality in (48) holds at p ∈ M if and only if there exists an orthonormal basis {e1 , e2 ....en } of T p M and an orthonormal basis {en+1 , en+2 ....e4m } of T p⊥ M such that the shape operators Ar = Aer , take the following forms:

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⎛

An+1

C ⎜0 ⎜ ⎜. ⎜ =⎜ ⎜. ⎜. ⎜ ⎝. 0

0 0 D 0 . C+D . . . . . . 0 0

... ... ... ... ... ... ...C

267

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ . ⎟ ⎟, . ⎟ ⎟ . ⎠ +D

(49)

⎛

⎞ Cr Dr 0 . . . 0 ⎜ Dr −Cr 0 . . . 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 . . . 0⎟ ⎜ ⎟ . . . .⎟ Ar = ⎜ ⎜ . ⎟, ⎜ . ⎟ . . . . ⎜ ⎟ ⎝ . . . . .⎠ 0 0 0...0 r ∈ {n + 2, n + 3......4m}.

(50)

Proof Proceeding in a similar way as in Theorem 2.6 it follows c n 2 ||H ||2 ( p) = 2τ ( p) + ||h||2 ( p) − [n(n − 1) + 9hcos 2 θ]. 4 Putting  = 2τ ( p) −

n 2 (n − 2) c ||H ||2 − {n(n − 1) + 9n cos 2 θ } n−1 4

(51)

we have n 2 ||H ||2 = (n − 1){ + ||h||2 } or  n

2 h iin+1





= n−1

 +

n  

i=1

h iin+1

2

+

+

n 



h ri j

2

h iin+1

2

i = j

i=1 4m 

n  

 .

(52)

r=n+2 i, j=1

Now applying Chen Lemma 1.8 by taking ai = h iin+1 , ∀i ∈ {1, 2....n} and b=+

n   i = j

2 h in+1 j

+

n 4m    r=n+2 i, j=1

h ri j

2

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we arrive at n+1 h n+1 11 h 22 ≥

  n n 4m     n+1 2  r 2 1 hi j hi j + + . 2 r=n+2 i, j=1 i = j

(53)

Also from (1) and (6), it follows that   3 4m    r r  c 2 r 2 h 11 h 22 − (h 12 ) . g (Tb e1 , e2 ) + 1+3 K (π ) = 4 r=n+1 b=1 Thus, K (π ) =

4m   r r   c  h 11 h 22 − (h r12 )2 . + 1 + 9cos 2 θ + 4 r=n+1

(54)

Combining (53) and (54), we conclude that K (π ) ≥

c  (1 + 9cos 2 θ ) + . 4 2

(55)

Equality holds at p ∈ M if and only if we have the equality in all the previous inequalities, i.e. h in+1 = 0, i = j > 2 j h r1 j = h r2 j = h ri j = 0, r ≥ n + 2, i, j > 2 n+1 h n+1 1 j = h 2 j = 0,

j >2

h r11 + h r22 = 0, r ≥ n + 2 n+1 n+1 n+1 n+1 h n+1 11 + h 22 = h 33 = h 44 = ..... = h nn



3.2 Inequalities for Casorati Curvature of Slant Submanifolds The Casorati curvature inequality of a slant submanifold in a Riemannian manifold is an extrinsic invariant defined as the normalized square of the length of the second fundamental form, and this notion extends the concept of the principal direction of a hypersurface of a Riemannian manifold to a submanifold of Riemannian manifold.

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In 2011, Ghisoiu [6] obtained inequalities for Casorati curvatures of slant submanifolds in complex space forms, whereas Slesar, Sahin and Vilcu [15] generalized these inequalities in a quaternion setting proving the following result. Theorem 3.2 Let M be a θ -slant proper submanifold of a quaternion space form M(c). Then (a) The normalized δ-Casorati curvature δc (n − 1) satisfies c 9 ρ ≤ δc (n − 1) + (1 + cos 2 θ ) 4 n−1

(56)

Moreover, the equality sign holds if and only if M is an invariantly quasiumbilical submanifold with trivial normal connection in M(c) such that with respect to suitable orthonormal tangent frame {e1 , e2 ....en } and normal orthonormal frame {en+1 , .....e4m }, the shape operators Ar = Aer , r ∈ {n + 1, .....4m}, take the following forms: ⎞ ⎛ a00...0 0 ⎜0 a 0 . . . 0 0 ⎟ ⎜ ⎟ ⎜0 0 a . . . 0 0 ⎟ ⎜ ⎟ ⎜. . . ... . . ⎟ ⎜ ⎟ ,An+2 = ..... = A4m = 0. An+1 = ⎜ ⎟ ⎜. . . ... . . ⎟ ⎜. . . ... . . ⎟ ⎜ ⎟ ⎝0 0 0 . . . a 0 ⎠ 0 0 0 . . . 0 2a (b) The normalized δ-Casorati curvature δˆc (n − 1) satisfies c 9 ρ ≤ δˆc (n − 1) + (1 + cos 2 θ ) 4 n−1

(57)

Moreover, the equality sign holds if and only if M is an invariantly quasiumbilical submanifold with trivial normal connection in M(c) such that with respect to suitable orthonormal tangent frame {e1 , e2 ....en } and normal orthonormal frame {en+1 , en+2 .....e4m }, the shape operators Ar = Aer , r ∈ {n + 1, n + 2.....4m},⎛take the following forms ⎞ 2a 0 0 . . . 0 0 ⎜ 0 2a 0 . . . 0 0 ⎟ ⎜ ⎟ ⎜ . . . ... . .⎟ ⎜ ⎟ ⎟ An+1 = ⎜ ⎜ . . . . . . . . ⎟ ,An+2 = ..... = A4m = 0. ⎜ . . . ... . .⎟ ⎜ ⎟ ⎝ 0 0 0 . . . 2a 0 ⎠ 0 0 0... 0 a Proof Choosing an adopted slant frame as {e1 , e2 = secθ Ta e1 , ....e2s−1 , e2s = secθ Ta e2s−1 }, where a is 1,2 or 3 and n = 2s, then from (1) and (6) it follows

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n 2 ||H ||2 = 2τ ( p) + ||h||2 −

n 3 n(n − 1) 3c   2 g (Tb ei , e j ). c− 4 4 b=1 i, j=1

(58)

Moreover, we have g 2 (Tb ei , ei+1 ) = g 2 (Tb ei+1 , ei ) = cos 2 θ, i = 1, 3....2s − 1.

(59)

and g(Tb ei , e j ) = 0

f or (i, j) ∈ {(2l − 1, 2l), (2l, 2l − 1)/l ∈ {1, 2, 3....s}}. (60)

Combining (58), (59) and (60), it follows c 2τ ( p) = n 2 ||H ||2 − nC + [n(n − 1) + 9n cos 2 θ] 4

(61)

where C is called the Casorati curvature of the submanifold M given by C=

n 4m 1   r 2 (h ) . n r=n+1 i, j=1 i j

Let P=

1 1 c n(n − 1)C + (n + 1)C(L) − 2τ ( p) + [n(n − 1) + 9n cos 2 θ] 2 2 4

where L is the hypersurface of T p M spanned by e1 , e2 ...en−1 . Then we have P=

n n−1 4m 4m n+1   r 2 n−1   r 2 (h i j ) + (h ) 2 r=n+1 i, j=1 2(n − 1) r =n+1 i, j=1 i j

c − 2τ ( p) + [n(n − 1) + 9n cos 2 θ ]. 4

(62)

Equations (61) and (62) yield P= +

n−1 4m    n2 − n + 2 r 2 (h ii ) + (n + 1)(h rin )2 ] 2(n − 1) r =n+1 i=1 4m n−1 n    n(n + 1)  n−1 r 2 (h ri j )2 − 2 h rii h rj j + (h nn ) ]. (n − 1) 2 r =n+1 i< j=1 i< j=1

(63)

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n+1 4m 4m 4m If h c = (h n+1 11 , h 12 .....h 11 , h 12 .....h nn ) are the critical points of P, which are the solutions of the following system of linear homogeneous equations n  n(n + 1) r ∂P = − 2 h rkk = 0, h ∂h rii n − 1 ii k=1 n−1  ∂P r = (n − 1)h − 2 h rkk = 0, nn ∂h rnn k=1

2n(n + 1) r ∂P h i j = 0, r = ∂h i j n−1

(64)

∂P = 2(n + 1)h rin = 0, ∂h rin with i, j ∈ {1, 2....n − 1}, i = j and r ∈ {n + 1, n + 2......4m}. Then the equations (63) and (64) imply that P ≥ 0 and hence 2τ ( p) ≤

1 c 1 n(n − 1)C + (n + 1)C(L) + [n(n − 1) + 9n cos 2 θ] 2 2 4

which gives ρ≤

C n+1 c 9 + C(L) + [1 + cos 2 θ] 2 2n(n − 1) 4 n−1

for every tangent hyperplane L of M. If we take the infirm over all tangent hyperplanes L, we get (56). Moreover, the equality sign holds in (56) if and only if h ri j = 0,

f oralli, j ∈ {1, 2.....n} , i = j and r ∈ {n + 1, n + 2.......4m}.

(65)

and h rnn = 2h r11 = 2h r22 = ..... = 2h rn−1n−1

f orall r ∈ {n + 1, n + 2......4m}.

(66)

From (65) and (66), we conclude that the equality sign holds in (56) if and only if the submanifold M is invariantly quasi-umbilical with trivial normal connection in M(c) such that with respect to suitable orthonormal tangent and orthonormal tangent frames, the ⎛ shape operators take⎞ the form 2a 0 0 . . . 0 0 ⎜ 0 2a 0 . . . 0 0 ⎟ ⎜ ⎟ ⎜ . . . ... . .⎟ ⎜ ⎟ ⎟ An+1 = ⎜ ⎜ . . . . . . . . ⎟ , An+2 = ..... = A4m = 0. ⎜ . . . ... . .⎟ ⎜ ⎟ ⎝ 0 0 0 . . . 2a 0 ⎠ 0 0 0... 0 a

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(b) can be proved in similar way by taking 1 c Q = 2n(n − 1)C − [2n − 1(n − 1)]C(L) − 2τ ( p) + [n(n − 1) + 9n cos 2 θ]. 2 4  Remark Theorem 2.2 was later generalized in [J. W. Lee, G. E. Vilcu, Taiwan J. Math. 19, No. 3 (2015), 691–702] to the case of generalized normalized δCasorati curvatures for slant submanifolds in quaternionic space forms using a similar approach. An alternate proof of the theorem was given in [C. W. Lee, J. W. Lee, G. E. Vilcu, J. Inequal Appl 310(2015)] analyzing a suitable constrained extremum problem on submanifold.

4

4.1

Pointwise h-Semi-slant and Warped Product Submanifolds of Hyperkaehler Manifolds Pointwise h-Semi-slant Submanifolds

The notion of pointwise almost h-slant submanifolds and pointwise almost h-semislant were introduced by K. S. Park [11] as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds and pointwise semi-slant submanifolds. In this section, we report the results of Park on the integrability of distributions D1 and D2 , the conditions for such distribution to be totally geodesic foliations and the properties of nontrivial warped product of proper pointwise h-semi-slant submanifolds. Definition 4.1 ([11]) Let M be an almost quaternionic Hermitian manifold and M is a submanifold of M. The submanifold M is said to be a pointwise almost h-slant submanifold if given a point p ∈ M with a neighbourhood V, there exists an open set U ⊂ M with U ∩ M = V and a quaternionic Hermitian basis which we denote by {J1 , J2 , J3 } or {I, J, K } such that for each Ja , a ∈ {1, 2, 3} at each given point q ∈ V , the angle θa = θa (X ) between Ja X and the tangent space Tq M is constant for non-zero X ∈ Tq M. 3 a pointwise almost h-slant basis and the angle Remark 1 We call the basis {Ja }a=1 3 {θa }a=1 almost h-slant functions. Moreover if θ = θ1 = θ2 = θ3 , then we call the 3 , a pointwise submanifold M, a pointwise h-slant submanifold and the basis {Ja }a=1 h-slant basis. The angle θ is now called h-slant function.

Definition 4.2 Let M be an almost quaternionic Hermitian manifold and M a submanifold of M. Then submanifold M is said to be a pointwise almost h semi-slant

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submanifold if given a point p ∈ M with a neighbourhood V, there exists are open set U ⊂ M with U ∩ M = V and a quaternionic Hermitian basis {J1 , J2 , J3 } such that for each Ja , a ∈ {1, 2, 3} there is a distribution D1a ⊂ T M on V such that T M = D1a ⊕ D2a and Ja (D1a ) = D1a , a ∈ {1, 2, 3} and at each given point q ∈ V , the angle between θa = θa (X ) between Ja X and the space (D2a )q is constant for non-zero X ∈ (D2a )q , where D2a is the orthogonal complement of D1a in TM. We call such a basis {J1 , J2 , J3 }, a pointwise almost h-semi-slant basis and the angle {θ1 , θ2 , θ3 }, almost h-semi-slant functions. Remark (a) If D1 = D11 = D12 = D13 , then the submanifold M is called a pointwise h-semi-slant submanifold, the basis {J1 , J2 , J3 } is called pointwise h-semi-slant basis and the angles {θ1 , θ2 , θ3 } h-semi-slant functions. (b) if θ1 = θ2 = θ3 = θ , then we call such submanifolds, pointwise strictly h-semislant submanifold, the basis {J1 , J2 , J3 } is called pointwise strictly h-semi-slant basis and the angle θ is called h-semi-slant function. (c) if M is a pointwise almost h-semi-slant submanifold of an almost quaternionic Hermitian manifold M with θ1 = θ2 = θ1 = π2 then M is an called pointwise almost h-semi-invariant submanifold and the basis {J1 , J2 , J3 }, a pointwise almost h-semi-invariant basis. Similar is the case of pointwise h-semi-slant submanifolds. (d) If M is a pointwise h-semi-slant submanifold of an almost quaternionic Hermitian manifold M with D1 = 0 and D2 = T M, then the submanifold M is called pointwise almost h-semi-slant submanifold of M. For any X ∈ (T M) and R ∈ {I, J, K }, we have X = PR X + Q R X

(67)

where PR X ∈ ((D1R ) and Q R X ∈ (D2R ). For X ∈ T M and R ∈ {I, J, K }, we have X = φR X + ωR X where φ R X ∈ T M and ω R X ∈ T ⊥ M. For Z ∈ T ⊥ M and R ∈ {I, J, K }, we have R Z = BR Z + C R Z where B R Z ∈ T M and C R Z ∈ T ⊥ M. Then given R ∈ {I, J, K } T ⊥ M = ω R D2R ⊕ μ R

(68)

where μ R is the orthogonal complement of ω R D2R in T ⊥ M and is J R -invariant.

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It is easy to observe the following sets of equations φ R D1R = D1R ; ω R D1R = 0, φ R D2R ⊂ D2R ,

B R (φ ⊥ M) = D2R

φ 2R + B R ω R = −I ; C R2 + ω R B R = −I ω R φ R + C R ω R = 0;

B R C R + φ R B R = 0.

(69) (70) (71)

For X, Y ∈ T M, R ∈ {I, J, K } we define (∇ X φ R )Y = ∇ X (φ R Y ) − φ R ∇ X Y

(72)

(∇ X⊥ ω R )Y = ∇ X⊥ (ω R Y ) − ω R ∇ X Y.

(73)

Now, we may state the following Lemma 4.3 Let M be a pointwise almost h-semi-slant submanifold of a hyperKaehler manifold M such that {J1 , J2 , J3 } is a pointwise almost h-semi-slant basis on M. Then we have (1) (∇ X φ R )Y = Aω RY X + B R h(X, Y ) (∇ X⊥ ω R )Y = −h(X, φ R Y ) + C R h(X, Y ) for any X, Y ∈ (T M) and R ∈ {J1 , J2 , J3 }. (2) −φ R A Z X + B R ∇ X⊥ Z = ∇ X (B R Z ) − AC R Z X −ω R A Z X + C R ∇ X⊥ Z = h(X, B R Z ) − ∇ X⊥ (C R Z ) for any X ∈ (φ M), Z ∈ (T ⊥ M) and R ∈ {J1 , J2 , J3 }. Proposition 4.4 Let M be a pointwise almost h-semi-slant submanifold of an almost Quaternion Hermitian manifold M. Then we have φ 2R X = −cos 2 θ R X, X ∈ (D2R ) and R ∈ {J1 , J2 , J3 }

(74)

for {J1 , J2 , J3 }, a pointwise almost h-semi-slant basis with the almost h-semi-slant functions {θ1 , θ2 , θ3 }. We now have the following theorems giving the equivalent conditions for the integrability of distributions D1 and D2 [11]. Theorem 4.5 Let M be a pointwise h-semi-slant submanifold of a hyperKaehler manifold M such that {I, J, K } is a pointwise h-semi-slant basis. Then the following conditions are equivalent

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(a) the complex distribution D1 is integrable. (b) h(X, φ I Y ) − h(Y, φ I X ) + ∇ X⊥ (ω I Y ) − ∇Y⊥ (ω I X ) = 0 Q(∇ X (φ I Y ) − ∇Y (φ I X ) + Aω I X Y − Aω I Y X ) = 0 (c) h(X, φ J Y ) − h(Y, φ J X ) + ∇ X⊥ (ω J Y ) − ∇Y⊥ (ω J X ) = 0 Q(∇ X (φ J Y ) − ∇Y (φ J X ) + Aω J X Y − Aω J Y X ) = 0 (d) h(X, φ K Y ) − h(Y, φ K X ) + ∇ X⊥ (ω K Y ) − ∇Y⊥ (ω K X ) = 0 Q(∇ X (φ K Y ) − ∇Y (φ K X ) + Aω K X Y − Aω K Y X ) = 0 for any X, Y ∈ (D1 ). Proof Let X, Y ∈ (D1 ) and R ∈ {I, J, K }. Then, R[X, Y ] = R{∇ X Y − ∇ Y X } = ∇ X (RY ) − ∇ Y (R X ) = ∇ X (φ R Y ) + h(X, φ R Y ) − Aω R Y X + ∇ X⊥ (ω R Y ) − ∇Y (φ R X ) − h(Y, φ R X ) + Aω R X Y − ∇Y⊥ (ω R X ) Since D1 is R-invariant, it follows from the above equation that (a) ⇐⇒ (b), (a) ⇐⇒ (c), (a) ⇐⇒ (d).  Theorem 4.6 ([11]) Let M be a pointwise h-semi-slant submanifold of a hyperKaehler manifold M such that {I, J, K } is a pointwise h-semi-slant basis. Then, the following conditions are equivalent (a) (b) (c) (d)

The slant distribution D2 is integrable. PI (∇ X (φ I Y ) − ∇Y (φ I X ) + Aω I X Y − Aω I Y X ) = 0 PJ (∇ X (φ J Y ) − ∇Y (φ J X ) + Aω J X Y − Aω J Y X ) = 0 PK (∇ X (φ K Y ) − ∇Y (φ K X ) + Aω K X Y − Aω K Y X ) = 0

for any X, Y ∈ (D2 ) and I, J, K ∈ {1, 2, 3}. Proof For X, Y ∈ (D2 ), Z ∈ (D1 ) and R ∈ {I, J, K }, and M being hyperKaehler, we have g([X, Y ], R Z ) = −g(R[X, Y ], Z ) = −g(∇ X (RY ) − ∇ Y (R X ), Z ) = −g(∇ X (φ R Y ) − Aω R Y X − ∇Y (φ R X ) + Aω R X Y, Z ) As [X, Y ] ∈ (φ M) and D1 is R-invariant, we have (a) ⇐⇒ (b), (a) ⇐⇒ (c) , (a) ⇐⇒ (d).  We now discuss two theorems which give equivalent conditions for D1 and D2 to be totally geodesic [11] .

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Theorem 4.7 Let M be a pointwise h-semi-slant submanifold of a hyperKahler manifold M such that {I, J, K } is a pointwise h-semi-slant basis and all the h-semi-slant functions {θ I , θ J , θ K } are nowhere zero. Then, the following conditions are equivalent (a) (b) (c) (d)

The complex distribution D1 defines a totally geodesic foliation. g(h(X, Y ), ω I φ I Z ) = g(h(X, I Y ), ω I Z ) for X, Y ∈ (D1 ), Z ∈ (D2 ). for X, Y ∈ (D1 ), Z ∈ (D2 ). g(h(X, Y ), ω J φ J Z ) = g(h(X, J Y ), ω J Z ) g(h(X, Y ), ω K φ K Z ) = g(h(X, K Y ), ω K Z ) for X, Y ∈ (D1 ), Z ∈ (D2 ).

Proof Given X, Y ∈ (D1 ), Z ∈ (D2 ) and R ∈ {I, J, K }, we have g(∇ X Y, Z ) = g(∇ X RY, R Z ) = g(∇ X Y, Rφ R Z ) + g(∇ X RY, ω R Z ) = cos 2 θ R g(∇ X Y, Z ) − g(h(X, Y ), ω R φ R Z ) + g(h(X, RY ), ω R Z ) This gives sin 2 θ R g(∇ X Y, Z ) = −g(h(X, Y ), ω R φ R Z ) + g(h(X, RY ), ω R Z ) Therefore, we have (a) ⇐⇒ (b), (a) ⇐⇒ (c), (a) ⇐⇒ (d).



Theorem 4.8 Let M be a pointwise h-semi-slant submanifold of a hyperKaehler manifold Msuch that {I, J, K } is a pointwise h-semi-slant basis and all the h-semislant functions θ I , θ J , θ K are no where zero. Then, the following conditions are equivalent (a) The slant distribution D2 defines a totally geodesic foliation. (b) g(ω I φ I W, h(Z , X )) = g(ω I W, h(Z , I X )) (c) g(ω J φ J W, h(Z , X )) = g(ω J W, h(Z , J X )) (d) g(ω K φ K W, h(Z , X )) = g(ω K W, h(Z , K X )) for X ∈ (D1 ) and Z, W ∈ (D2 ). Proof For X ∈ (D1 ), Z , W ∈ (D2 ) and a ∈ {1, 2, 3}, we have g(∇ Z W, X ) = −g(W, ∇ Z X ) = −g(RW, ∇ Z R X ) = −g(φ R W + ω R W, ∇ Z R X ) = g(φ 2R W + ω R φ R W, ∇ Z X ) − g(ω R W, ∇ Z R X ) = −cos 2 θa g(W, ∇ Z X ) + g(ω R φ R W, h(Z , X )) − g(ω R W, h(Z , R X ))

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so that sin 2 θ R .g(∇ Z W, X ) = g(ω R φ R W, h(Z , X )) − g(ω R W, h(Z , R X )) Hence (a) ⇐⇒ (b),(a) ⇐⇒ (c), (a) ⇐⇒ (d).



4.2 Warped Product Submanifolds In this final section, first we discuss a non-existence theorem and then an example of a nontrivial warped product submanifolds of a hyperKaehler manifold [11]. Theorem 4.9 Let (M, I, J, K ) be hyperKaehler manifold. Then for R ∈ {I, J, K } there do not exist any nontrivial warped product submanifold M = B × f F of a Kaehler manifold (M, R, g) such that B is a proper pointwise slant submanifold of (M, R, g) and F is a holomorphic submanifold of a (M, R, g). Proof For Z ∈ T M, we have R Z = φ Z + ωZ where φ Z ∈ T M and ωZ ∈ T ⊥ M. Now using ∇ X Y = ∇Y X = (X ln f )Y and for X, Y ∈ T F and φ 2 V = −cos 2 V , V ∈ T B, then for a proper semi-slant function θ on M, we have V (ln f )g(X, Y ) = −g(∇ X (φ 2 V + ωφV ), Y ) − g(AωV X, RY ) = cos 2 θg(∇ X V, Y ) + g(h(X, Y ), ωφV ) − g(h(X, RY ), ωV ). Thus sin 2 θ V (ln f )g(X, Y ) = g(h(X, Y ), ωφV ) − g(h(X, RY ), ωV ) Interchanging X and Y in above equation, we have sin 2 θ V (ln f )g(X, Y ) = g(h(X, Y ), ωφV ) − g(h(Y, R X ), ωV ) Comparing the above two equations, we get

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g(h(X, RY ), ωV ) = g(h(Y, R X ), ωV ) But g(h(X, RY), ωV) = g(∇ X RY, RV − φV ) = g(∇ X Y, V ) + g(RY, ∇ X φV ) = −V (ln f )g(X, Y ) + φV (ln f )g(X, RY ). Hence, φV(lnf)g(X, RY) = 0. Replacing X by R X and V by φV , we have cos 2 θ V (ln f )g(X, Y ) = 0 which shows V (ln f ) = 0 so that f is constant.



Example 4.10 Let M = {(x1 , x2 , x3 , x4 , u, v)/0 < xi < 1, 1 ≤ i ≤ 4, 0 < u, v
, D2 =< Y1 , Y2 >, and the h-semi-slant functions θ I , θ J , θ K are given by cosθ I =

| − a11 a22 + a21 a12 − a31 a42 + a41 a32 | ,  1 + 4k=1 xk2

cosθ J =

| − a11 a32 + a21 a42 + a31 a12 − a41 a22 | ,  1 + 4k=1 xk2

cosθ K =

| − a11 a42 − a21 a32 + a31 a22 + a41 a12 | .  1 + 4k=1 xk2

Obviously, the distributions D1 and D2 are integrable so that we may denote by B and F the integral manifolds of D1 and D2 , respectively. Then we can easily check that M = (M, g) is a nontrivial warped product Riemannian submanifold of R 20 such that

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M = B × F, g = 2(d x12 + d x22 + d x32 + d x42 ) + (1 +

4 

xk2 )(du 2 + dv 2 ),

k=1

  the warping function f = (1 + 4k=1 xk2 ). Therefore, M is a nontrivial warped product proper pointwise h-semi-slant submanifold of (R 20 , I, J, K ). Acknowledgements The authors are very thankful to Prof. K. S. Park and the referees for providing constructive comments and valuable suggestions.

References 1. Barros, M., Chen, B.Y., Urbano, F.: Quaternion CR-submanifolds of Quaternion manifolds. Kodai Math. J. 4(3), 399–417 (1981) 2. Bejancu, A.: CR-submanifold of a Kaehler manifold I, II. Proc. Am. Math. Soc. 69, 135–142 (1978); Trans. Am. Math. Soc. 250, 333–345 (1979) 3. Chen, B.Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Louven (1990) 4. Chen, B.Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60(6), 568–578 (1993) 5. Chen, B.Y.: Relation between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasg. Math. J. 41, 33–41 (1999) 6. Ghisoiu, V.: Inequlities for Casorati curvature of slant submanifolds in complex space forms. In: Proceedings of the RIGA, pp. 145–150. Bucharest (2011) 7. Ishihara, S.: Quaternion Kaehlerian manifolds. J. Diff. Geom. 9, 483–500 (1979) 8. Matsumoto, K., Mihai, I., Tazawa, Y.: Ricci tensor of slant submanifolds in complex space forms. Kodai Math. J. 26, 85–94 (2003) 9. Oiaga, A., Mihai, I.: B. Y. Chen inequalities for slant submanifolds in complex space forms. Demonst. Math. (1999) 10. Papaghuic, N.: Semi-slant submanifolds of a Kaehler manifold. An. Stiint. Univ. Al. I. Cuza Iasi Sect. I a Math. 40(1), 55–61 (1994) 11. Park, K.S.: Pointwise almost h-semi slant submanifolds. Int. J. Math. 26(12), 1550099 (2015) 12. Sahin, B.: Slant submanifolds of Quaternion Kaehler manifolds. Commun. Korean Math. Soc. 22(1), 123–135 (2007) 13. Shahid, M.H., Al-Solamy, F.: Ricci tensor of slant submanifolds in Quaternion projective space. C. R. Math. Acad. Sci. Paris 349, 571–573 (2011) 14. Shukla, S.S., Rao, P.K.: Ricci curvature of Quaternion slant submanifolds in Quaternion space forms. Acta Math. Acad. Paedagogical Nyiregyh 28, 69–81 (2012) 15. Slesar, V., Sahin, B., Vilcu, G.E.: Inequalities for the Casorati curvatures of slant submanifolds in Quaternion space forms. J. Ineq. App. 123 (2014) 16. Vilcu, G.E.: B. Y. Chen inequalities for slant submanifolds in Quaternion space form. Turk J. Math. 34, 115–128 (2010) 17. Vilcu, G.E.: Slant submanifolds of Quaternion space forms. Publ. Math. Debr. 81(3–4), 397– 413 (2012) 18. Yano, K., Kon, M.: Structures on Manifolds. World Scientific, Singapore (1984)

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds Ion Mihai, Aliya Naaz Siddiqui, and Mohammad Hasan Shahid

1 Introduction In 1990, B.-Y. Chen defined the slant submanifolds in complex manifolds as a natural generalization of holomorphic and totally real submanifolds [9]. Examples of slant submanifolds in C2 and C4 were given by Chen and Tazawa [20, 21]. This study was extended by Lotta [28] in almost contact geometry and further investigated by Cabrerizo et al. in 2000. The theory of slant submanifolds became a very rich area of research for geometers. Slant submanifolds have been studied in different kinds of structures of almost Hermitian manifolds by several geometers. In 1901, the notion of totally geodesic submanifolds was introduced by J. Hadamard. He defined (totally) geodesic submanifolds of a Riemannian manifold as submanifolds such that each geodesic of them is a geodesic of the ambient space. This condition is equivalent to the vanishing on the second fundamental form on the submanifolds. Totally geodesic submanifolds are the simplest and the most fundamental submanifolds of Riemannian manifolds. It is known that there do not exist totally geodesic proper slant immersions in non-trivial complex space forms, by the equation of Codazzi. So it is important to study whether there exist minimal proper slant submanifolds in non-trivial complex space forms. Chen and Tazawa I. Mihai (B) Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest 010014, Romania e-mail: [email protected] A. Naaz Siddiqui M.M. Engineering College, Maharishi Markandeshwar (Deemed to be) University, Mullana-Ambala, Haryana 133207, India M. Hasan Shahid Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia, New Delhi 110025, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_10

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[22] proved that there are not minimal proper slant surfaces in CP2 and CH2 . Then it seems that the slant immersion has some interesting properties. Now, we recall some necessary facts and formulae from the theory of Kähler manifolds and their submanifolds. Furthermore, we give some basic definitions. Let M be a complex m-dimensional Kähler manifold, that is, M is endowed with an almost complex structure J and with a J -Hermitian metric g, we have J 2 = −I,

(1)

g(J X, J Y ) = g(X, Y ),

(2)

∇ J = 0,

(3)

for X, Y ∈ (T M), and

where ∇ is the Levi-Civita connection of g, and the covariant derivative of the complex structure J is defined as (∇ X J )Y = ∇ X J Y − J ∇ X Y, for any X, Y ∈ (T M). It follows that J is integrable. Here, (T M) denotes the Lie algebra of vector fields and on M. If the ambient manifold M is of constant holomorphic sectional curvature c, then M is called a complex space form and is denoted by M(c). Thus, the Riemannian curvature tensor R of M(c) is given as [47] R(X, Y, Z , W ) =

 c g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) 4 +g(J Y, Z )g(J X, W ) − g(J X, Z )g(J Y, W )  +2g(X, J Y )g(J Z , W ) ,

(4)

for any X, Y, Z , W ∈ (T M). A complete simply connected complex space form M(c) is holomorphically isometric to the complex Euclidean m-space Cm , the complex projective m-space CPm (4c), or the complex hyperbolic m-space CHm (4c), according to c = 0, c > 0, or c < 0, respectively. If an almost complex structure J satisfies (∇ X J )Y + (∇ Y J )X = 0, for any X, Y ∈ (T M), then the manifold is called a nearly Kähler manifold.

(5)

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Now, we will consider a class of almost Hermitian manifolds, called RKmanifolds, which contains the nearly Kähler manifolds. Gray [25] introduced the notion of constant type for a nearly Kähler manifold, which led to the definition of R K -manifolds. An RK-manifold M is an almost Hermitian manifold for which the curvature tensor R is J -invariant, that is, R(J X, J Y, J Z , J W ) = R(X, Y, Z , W ),

(6)

for any X, Y, Z , W ∈ (T M). An almost Hermitian manifold M is said to have (pointwise) constant type if for each p ∈ M and for any X, Y, Z ∈ (T M) such that g(X, Y ) = g(X, Z ) = g(X, J Y ) = g(X, J Z ) = 0, g(Y, Y ) = 1 = g(Z , Z ), we have R(X, Y, X, Y ) − R(X, Y, J X, J Y ) = R(X, Z , X, Z ) − R(X, Z , J X, J Z ). An R K -manifold M has (pointwise) constant type if and only if there is a differentiable function α on M such that R(X, Y, X, Y ) − R(X, Y, J X, J Y ) = α{g(X, X )g(Y, Y ) − g 2 (X, Y ) − g 2 (X, J Y )},

for any X, Y ∈ (T M). Furthermore, M has global constant type if α is constant. The function α is called the constant type of M. An R K -manifold of constant holomorphic sectional curvature c and constant type α is called a generalized complex space form, denoted by M(c, α). The curvature tensor R of M(c, α) has the following expression.  c + 3α R(X, Y, Z , W ) = [g(X, Z )g(Y, W ) − g(X, W )g(Y, Z ) 4  c−α + g(J X, Z )g(J Y, W ) − g(J X, W )g(J Y, Z ) 4  +2g(X, J Y )g(Z , J W ) ,

(7)

for any X, Y, Z , W ∈ (T M). If c = α, then M(c, α) is a space of constant curvature. A complex space form M(c) (that is, a Kähler manifold of constant holomorphic sectional curvature c) belongs to the class of almost Hermitian manifold M(c, α) (with constant type zero).

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An almost Hermitian manifold M is called a generalized complex space form M( f 1 , f 2 ) if its Riemannian curvature tensor R satisfies 



R(X, Y, Z , W ) = f 1 g(Y, Z )g(X, W ) − g(X, Z )g(Y, W )  + f 2 g(J Y, Z )g(J X, W ) − g(J X, Z )g(J Y, W )  +2g(X, J Y )g(J Z , W ) ,

(8)

for any X, Y, Z , W ∈ (T M), where f 1 and f 2 are smooth functions on M. Remark 1.1 We have the inclusion relation M(c) ⊂ M(c, α) ⊂ M( f 1 , f 2 ). Recall that a Hermitian m-manifold (M J, g) is called a locally conformal Kähler manifold if there exist an open cover {U i}i∈I of M and a family { f i }i∈I of smooth functions f i : Ui −→ R such that each local metric gi = e−2 fi g|Ui is Kählerian. A differentiable map ψ : M −→ M of a differentiable manifold M into another differentiable manifold M is called an immersion if the differential dψ : T M −→ T M is injective. Now, let M be an n-dimensional Riemannian manifold isometrically immersed in a Kähler manifold M of real dimension 2m with almost complex structure J . The Riemannian metric on M and M is denoted by the same symbol g. Let (T M) and (T ⊥ M) denote the Lie algebra of vector fields and set of all normal vector fields on M, respectively. The operator of covariant differentiation with respect to the Levi-Civita connection on M and M is denoted by ∇ and ∇, respectively. The Gauss and Weingarten formulae are, respectively, given as ∇ X Y = ∇ X Y + h(X, Y ),

(9)

∇ X N = −AN (X ) + ∇ X⊥ N,

(10)

and

for any X, Y ∈ (T M) and N ∈ (T ⊥ M). Here, h is the second fundamental form, A is the shape operator of M, and ∇ ⊥ is the operator of covariant differentiation with respect to the linear connection induced in the normal bundle T ⊥ M of M. The second fundamental form and the shape operator are related by the following equation. g(h(X, Y ), N) = g(AN (X ), Y ),

(11)

for any X, Y ∈ (T M) and N ∈ (T ⊥ M). Denote by R and R the curvature tensor of M and M, respectively. Then the equation of Gauss is given by

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

R(X, Y, Z , W ) = R(X, Y, Z , W ) − g(Ah(Y,Z ) X, W ) +g(Ah(X,Z ) Y, W ),

285

(12)

for any X, Y, Z , W ∈ (T M). Let p ∈ M and {e1 , . . . , en } be an orthonormal basis of T p M and {en+1 , . . . , e2m } be an orthonormal basis of T p⊥ M. The mean curvature vector H of the submanifold M at p is given by the following relation: H ( p) =

n 1 h(ei , ei ). n i=1

Also, we set h iaj = g(h(ei , e j ), ea ), i, j ∈ {1, . . . , n}, a ∈ {n + 1, . . . , 2m}, and  h2 = i,n j=1 g(h(ei , e j ), h(ei , e j )). The mean curvature vector is said to be parallel in the normal bundle if ∇ ⊥ H = 0 holds identically. Definition 1.2 A submanifold M of a Kähler manifold M is said to be [8] (1) totally umbilical if its second fundamental form satisfies h(X, Y ) = g(X, Y )H , for any X, Y ∈ (T M). (2) totally geodesic if h(X, Y ) = 0, for any X, Y ∈ (T M). (3) minimal if H = 0, that is, trace(h) ≡ 0. For any X ∈ (T M), we put J X = P X + F X,

(13)

where P X and F X denote the tangential and normal components of J X , respectively. Then P is an endomorphism of T M, and F is the normal bundle valued 1-form on T M. In the same way, for any vector field N ∈ (T ⊥ M), we put J N = tN + f N,

(14)

where tN and f N denote the tangential and normal components of J N, respectively. It is easy to see that P and f are skew-symmetric and g(F X, N) = −g(X, tN),

(15)

for any X ∈ (T M) and N ∈ (T ⊥ M). Moreover, the covariant derivatives of the tangential and normal components of (13) and (14) are given by [47] ⎧ ⎪ (∇ X P)Y = ∇ X PY − P∇ X Y, ⎪ ⎪ ⎪ ⎨ (∇ F)Y = ∇ ⊥ FY − F∇ Y, X X X ⎪ (∇ X t)N = ∇ X tN − t∇ X⊥ N, ⎪ ⎪ ⎪ ⎩ (∇ X f )N = ∇ X⊥ f N − f ∇ X⊥ N,

(16)

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for any X, Y ∈ (T M) and N ∈ (T ⊥ M). By using (12), we can write the Riemannian curvature tensor R of M as [47]  c R(X, Y, Z , W ) = g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) 4 +g(PY, Z )g(P X, W ) − g(P X, Z )g(PY, W )  +2g(X, PY )g(P Z , W ) + g(A h(Y,Z ) X, W ) −g(Ah(X,Z ) Y, W ),

(17)

for any X, Y, Z ∈ (T M). Let M be a submanifold in a generalized complex space form; then the Gauss equation becomes   R(X, Y, Z , W ) = f 1 g(Y, Z )g(X, W ) − g(X, Z )g(Y, W )  + f 2 g(PY, Z )g(P X, W ) − g(P X, Z )g(PY, W )  +2g(X, PY )g(P Z , W ) + g(Ah(Y,Z ) X, W ) −g(Ah(X,Z ) Y, W ),

(18)

for any X, Y, Z ∈ (T M). A distribution D on a manifold M is called autoparallel if ∇ X Y ∈ (D) for any X, Y ∈ (D) and called parallel if ∇U X ∈ (D) for any X ∈ (D) and U ∈ (T M). If a distribution D on M is autoparallel, then it is clearly integrable, and by the Gauss formula, D is totally geodesic in M . If D is parallel, then the orthogonal complementary distribution D⊥ is also parallel, which implies that D is parallel if and only if D⊥ is parallel. In this case, M is locally the product of the leaves of D and D⊥ . For two distributions D1 and D2 on a submanifold M of M, we say that M is (D1 , D2 )-mixed totally geodesic if for all X ∈ (D1 ) and Y ∈ (D2 ), we have h(X, Y ) = 0 [30, 46]. There are certain important classes of submanifolds: totally real submanifolds, holomorphic submanifolds, and C R-submanifolds (a generalization of totally real and holomorphic submanifolds). Both totally real submanifolds and holomorphic submanifolds were generalized by B.-Y. Chen as slant submanifolds [9]. Definition 1.3 ([8]) Let M be a Riemannian submanifold of an almost Hermitian manifold (M, J, g). Then M is said to be a slant submanifold if for each p ∈ M and

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non-zero X p ∈ T p M, the angle θ (X p ) between J X p and the tangent space T p M is constant. It is easy to see that a submanifold M of (M, J, g) is a slant submanifold if and only if P 2 = −λI,

(19)

for some real number λ ∈ [0, 1], where I denotes the identity transformation of the tangent bundle T M of M. A slant submanifold is totally real if its slant angle is equal to π/2. A slant submanifold is said to be proper if it is neither complex nor totally real. Let M be an n-dimensional proper slant submanifold of a 2n-dimensional almost Hermitian manifold M, p ∈ M,  ⊂ T p M a 2-plane section, and {e1 , e2 , . . . , en } an orthonormal basis of tangent space T p M such that e1 , e2 ∈ . An orthonormal basis {e1∗ , e2∗ , . . . , en∗ } of the normal space T p⊥ M is defined by ek∗ = csc θ Fek , k = 1, . . . , n.

(20)

Chen [8] introduced the notion of a Kählerian slant submanifold in a Kähler manifold as a proper slant submanifold such that the canonical endomorphism P (defined above) is parallel, that is, ∇ P = 0. In fact, a Kählerian slant submanifold is a Kähler manifold with respect to the induced metric and the almost complex structure J = (sec θ )P, where θ is the slant angle. For slant submanifolds, the following facts are known [8]. ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

P 2 X = − cos2 θ X, t F X = − sin2 θ X, f F X = −F P X, ⎪ ⎪ g(P X, PY ) = cos2 θ g(X, Y ), ⎪ ⎪ ⎩ g(F X, FY ) = sin2 θ g(X, Y ),

(21)

for any X, Y ∈ (T M), where θ is the slant angle of M in M. Let D be a distribution on M. Then it is known [6] that D is slant if and only if there exists a constant λ ∈ [−1, 0] such that (P T )2 X = λX,

(22)

for X ∈ (D), where T denotes the orthogonal projection on D. Moreover, in this case, λ = − cos2 θ .

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There are some other important classes of submanifolds which are determined by the behavior of the tangent bundle of the submanifold under the action of an almost complex structure J of M [42]. (1) A submanifold M of M is called a C R-submanifold [3] of M if there exists a holomorphic differentiable distribution D on M whose orthogonal complementary distribution D⊥ is totally real. (2) A submanifold M of M is called a semi-slant submanifold [38] of M if there exists a pair of complementary orthogonal distributions D and Dθ such that D is holomorphic and Dθ is proper slant. (3) A submanifold M of M is called a hemi-slant submanifold [5] of M if there exists a pair of complementary orthogonal distributions D⊥ and Dθ such that D⊥ is totally real and Dθ is proper slant. (4) A submanifold M of M is called a bi-slant submanifold [5] of M if there exists a pair of complementary orthogonal distributions Dθ1 and Dθ2 such that Dθ1 is proper slant with slant angle θ1 and Dθ2 is proper slant with slant angle θ2 . For any non-zero vector X ∈ T p M, p ∈ M, the angle θ (X ) between J X and the tangent space T p M is called the Wirtinger angle of X . The Wirtinger angle gives rise to a real-valued function θ : T p M −→ R, called the Wirtinger function, defined on the set T p M consisting of all non-zero tangent vectors on M. Etayo studied pointwise slant submanifolds under the name of quasi-slant submanifolds in [24]. Definition 1.4 An immersion ψ : M −→ M of a manifold M into an almost Hermitian manifold (M, J, g) is called pointwise slant if, at each given point p ∈ M, the Wirtinger angle θ (X ) is independent of the choice of the non-zero tangent vector X ∈ T p M. In this case, θ can be regarded as a function on M, which is called the slant function of the pointwise slant submanifold. Remark 1.5 It was proved in [24] that a complete, totally geodesic quasi-slant submanifold of a Kähler manifold is a slant submanifold. We point that every 2dimensional submanifold (or simply, a surface) in (M, J, g) is always a pointwise slant. Definition 1.6 A point p in a pointwise slant submanifold is called a totally real point if its slant function θ satisfies cos θ = 0 at p. Similarly, a point p is called a complex point if its slant function satisfies sin θ = 0 at p. A pointwise slant submanifold M in an almost Hermitian manifold (M, J, g) is called totally real if every point of M is a totally real point, otherwise, it is known as the pointwise proper slant. A totally real submanifold M in (M, J, g) is called Lagrangian if dimM = dimC M. Remark 1.7 A pointwise slant submanifold M becomes slant if its slant function θ is globally constant, that is, θ is also independent of the choice of the point on M. In this case, the constant θ is called the slant angle of the slant submanifold.

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2 An Integral Formula of Simons’ Type with a Slant Factor This section deals with the construction of Simons’ type formula. The main result is printed out in the subsection. We quote some basic lemmas without their proofs. Lemma 2.1 ([8]) Let M be a proper slant submanifold of a Kähler manifold M. Then M is Kählerian slant if and only if A F X (Y ) = A FY (X ),

(23)

for any X, Y ∈ (T M). An immediate consequence of above Lemma 2.1. Lemma 2.2 ([42]) Let M be a Kählerian slant submanifold of a Kähler manifold M. Then we have AN P + P AN = 0,

(24)

for any N ∈ (T ⊥ M). Lemma 2.3 ([8]) Let M be a submanifold of a Kähler manifold M. Then ∇ F = 0 if and only if A f N (X ) = −AN (P X ),

(25)

for any N ∈ (T ⊥ M) and X ∈ (T M). For the Laplacian of the second fundamental form A of an n-dimensional minimal submanifold M in an m-dimensional Riemannian manifold M, the following Simons’ type formula is well known [43].

− A ◦ A + R(A) + R  , ∇ 2 A = −A ◦ A

and A are defined as follows: where the operators A

= t A ◦ A and A = A

m−n 

(26)

(ad Aa )(ad Aa ),

a=1

for a normal frame {ea }, a = 1, . . . , m − n, and Aa = Aea . Here R(A) and R  are given by the following.

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g(R(A)N (X ), Y ) =

n  

2g(R(ei , Y )h(X, ei ), N) + 2g(R(ei , X )h(Y, ei ), N)

i=1

−g(AN (X ), R(ei , Y )ei ) − g(AN (Y ), R(ei , X )ei ) +g(R(ei , h(X, Y ))ei , N) − 2g(AN (ei ), R(ei , X )Y ) , (27) and N g(R  (X ), Y )

=

n  

g((∇ X R)(ei , Y )ei , N)

(28)

i=1

+g((∇ ei R)(ei , X )Y, N) , for any X, Y ∈ (T M) and N ∈ (T ⊥ M).

2.1 Kählerian Slant Submanifolds in Complex Space Forms In this section, first, we have the following results on Kählerian slant submanifolds in complex space forms. Lemma 2.4 ([26]) Let ψ : M → M(4c) be an n-dimensional proper θ -slant immersion in a complex space form with constant holomorphic sectional curvature 4c. For any N1 , N2 ∈ (T ⊥ M), we have g( f N1 , f N2 ) = g(N1 , N2 ) cos2 θ.

(29)

We recall that for a submanifold M in a Riemannian manifold, the first normal space I m h p and relative null space K er h p of M at a point p are defined, respectively, by [26] I m h p = sp{h(X, Y )| X, Y ∈ T p M}, and K er h p = {X ∈ T p M|h(X, Y ) = 0, for all Y ∈ T p M}. It is easily seen that the first normal space I m h p and the relative null space K er h p of a Kählerian slant submanifold M in a complex space form M(4c) are related by (I m h p )⊥ = F(K er h p ). Lemma 2.5 ([26]) If M is a Kählerian slant immersion in M(4c), then t (I m h p )⊥ = K er h p ,

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where (I m h p )⊥ is the complementary orthogonal subspace of the first normal space in T p M. Theorem 2.6 ([26]) Let ψ : M → M(4c) be an n-dimensional Kählerian slant immersion in a complex space form with constant holomorphic sectional curvature 4c. If the first normal bundle is not full, then c = 0. Proof Since the first normal bundle of M is not full, there exists a unit normal vector N1 ∈ (T M) such that g(h(X, Y ), N1 ) = 0, for any X, Y ∈ (T M). So, we can choose an orthonormal basis {e1 , . . . , en } of T p M and {en+1 , . . . , e2n } of T p⊥ M, p ∈ M such that n e2i = sec θ Pei , i = 1, . . . , , 2 en+i = csc θ Fei , i = 1, . . . , n, e2n = N1 . On the other hand, by Lemma 2.5, te2n ∈ K er h p . Thus, we obtain en ∈ K er h p , which implies that h(ei , en ) = 0, i = 1, . . . , n. Now, let Pen = μ1 e1 + · · · + μn−1 en−1 , then we get (R(e j , en )en )⊥ = 3c sin θ μ j e2n . Thus, by the equation of Codazzi, we have μ j c = 0, j = 1, . . . , n − 1. By the assumption that M is a proper slant submanifold, we get c = 0.  In 1981, A. Bejancu, M. Kon, and K. Yano derived Simons’ type formula for C R-submanifolds of a complex space form [4]. Many authors derived Simons’ type formula for different submanifolds in different ambient spaces. Using Lemmas 2.1, 2.2, and 2.3 and Eqs. (26), (27), and (28), we prove the following proposition to get the required Simons’ type formula for an n-dimensional minimal Kählerian slant submanifold of a complex space form M(c) which plays an important role in working out the main result. Proposition 2.7 ([42]) Let M be an n-dimensional minimal Kählerian slant submanifold of a complex space form M(c). Then 

+ A ◦ A, A) + c n + 4 − 3 sin2 θ ||A||2 . g(∇ 2 A, A) = −g(A ◦ A 4

N

(30)

Proof Since M(c) is locally symmetric, then (28) implies that g(R  (X ), Y ) = 0. Now, we compute g(R(A)N (X ), Y ) by using (4) in (27); then a straightforward computation gives (see [4])

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g(R(A)N (X ), Y ) =

 c ng(AN (X ), Y ) − 2g(A FY (X ), tN) − 2g(A F X (Y ), tN) 4 −4g(A f N (X ), PY ) − 4g(A f N (Y ), P X ) + 3g(P X, P AN (Y )) +3g(PY, P AN (X )) − 6g(AN (P X ), PY ) n   g(A Fei (ei ), X )g(FY, N) + g(A Fei (ei ), Y )g(F X, N) −2 i=1

  3 + g(A Fei (X ), Y )g(Fei , N) . 2

(31)

By using the above Lemmas and (21), equation (31) becomes g(R(A)N (X ), Y ) =

  c n + 12 cos2 θ g(AN (X ), Y ) − 4g(A FY (X ), tN) 4 +4g(AN (P X ), PY ) + 4g(AN (PY ), P X )  2 −3 sin θ g(AN (X ), Y ) .

Further calculations reduce the above relation to g(R(A)N (X ), Y ) =

 c n + 4 − 7 sin2 θ g(AN (X ), Y ) 4 −cg(A FY (X ), tN).

(32)

With the help of (32) and (26), one has 

+ A ◦ A, A) + c n + 4 − 7 sin2 θ g(AN X, Y ) g(∇ 2 A(X ), Y ) = −g(A ◦ A 4

−cg(A FY X, tN). Putting X = ei and Y = AN (ei ) and summing for i = 1, . . . , n, we get 

+ A ◦ A, A) + c n + 4 − 7 sin2 θ ||A||2 g(∇ 2 A, A) = −g(A ◦ A 4

n  −c g(A Fei AN (ei ), tN). i=1

Again, by considering the Lemmas and (21), we arrive at 

+ A ◦ A, A) + c n + 4 − 7 sin2 θ ||A||2 g(∇ 2 A, A) = −g(A ◦ A 4

+c sin2 θ ||A||2 ,

(33)

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293

and hence the desired result (30) follows immediately from (33). This completes the proof of the Proposition.  For a bi-slant submanifold M of dimension n = 2n 1 + 2n 2 in a complex space form M(c), we have T M = D θ1 ⊕ D θ2 , then ||P|| = 2

n 

g 2 (Pei , e j ) = 2(n 1 cos2 θ1 + n 2 cos2 θ2 ),

i, j=1

where dim(Dθ1 ) = 2n 1 and dim(Dθ2 ) = 2n 2 . Theorem 2.8 ([41]) Let M be an n-dimensional minimal bi-slant submanifold of a complex space form M(c) of dimension 2n. If ∇ P = 0, then   ∼ g(∇ 2 A, A) = −g A ◦ A + A ◦ A, A ∼

+

c n + 1 + 6(n 1 cos2 θ1 + n 2 cos2 θ2 ) ||A||2 . 4

(34)

The following is Simons’ type formula for hemi-slant and semi-slant submanifolds in M(c). Corollary 2.9 ([41]) Let M be an n-dimensional minimal hemi-slant submanifold in a complex space form M(c) of dimension 2n. If ∇ P = 0, then 



∼

g(∇ A, A) = −g A ◦ A + A ◦ A, A + 2

∼

c n + 1 + 6n 2 cos2 θ ||A||2 . 4

(35)

Corollary 2.10 ([41]) Let M be an n-dimensional minimal semi-slant submanifold in a complex space form M(c) of dimension 2n. If ∇ P = 0, then 



∼

g(∇ A, A) = −g A ◦ A + A ◦ A, A + 2

∼

c n + 1 + 6(n 1 + n 2 cos2 θ ) ||A||2 . 4 (36)

Now, we give a classification of Kählerian slant submanifolds of a complex space form by using Simons’ type formula for the second fundamental form (30). Theorem 2.11 ([42]) Let M be an n-dimensional compact minimal Kählerian slant submanifold in a complex space form M(c). Then any of the following can hold. (1) M is totally geodesic, (2) M is S1 × S1 and n = 2 for θ = π2 and c = 4, 

(3) ||A||2 > c 1  n + 4 − 3 sin2 θ everywhere on M. 4 2− n

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Proof From Lemma 5.3.1 of Simons [43], we get



+ A ◦ A, A) ≤ 2 − 1 ||A||4 . g(A ◦ A n

(37)

Since M is compact, we have 

 g(∇ 2 A, A) = − M

g(∇ A, ∇ A).

(38)

M

Thus, from Eqs. (37) and (38) and Proposition 2.7, we see that 



0≤

g(∇ A, ∇ A) ≤ M

   1 c 2 − ||A||2 − n + 4 − 3 sin2 θ ||A||2 . n 4 M

This is known as an “integral formula of Simons’ type”. Suppose that

 c  n + 4 − 3 sin2 θ ||A||2 ≤ 1 4 2− n everywhere on M. By combining last two inequalities, we find ∇ A = 0 and hence ∇||A||2 is constant, that is, either ||A||2 = 0, or

 c  n + 4 − 3 sin2 θ . ||A||2 = 1 4 2− n

(39)

From A = 0, we conclude the statement (1), that is, M is totally geodesic. Now, if we consider (39) and put θ = π2 , c = 4, the equation reduces to ||A||2 =

n+1 2 − n1

which implies the statement (2), M = S1 × S1 and n = 2 [29]. Except for these possibilities,

 c  n + 4 − 3 sin2 θ ||A||2 > 1 4 2− n everywhere on M, which is the statement (3). This completes the proof of the Theorem.  Example 1 ([42]) Consider the flat Clifford torus M = S1 × S1 in R4 defined by  1 r (u, v) = √ cos u, sin u, cos v, sin v . 2

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It is minimal in S3 ⊂ R4 and hence austere. The cone over M   1 C(M) = √ (w cos u, w sin u, w cos v, w sin v) : u, v, w ∈ R 2 is easily proved to be austere in R4 . On the other hand, we have e1 = (− sin u, cos u, 0, 0), e2 = (0, 0, − sin v, cos v). If the almost complex structure J is defined as J (a, b, c, d) = (−b, a, −d, c) on R4 , then M is a totally real surface in R4 as θ = π2 . Moreover, the length of the second fundamental form of a 2-dimensional minimal submanifold M in R4 is ||A||2 = 2 (see [47]). From this, we conclude that the statement (2) of Theorem 2.11 holds. Lemma 2.12 ([41]) Let M be an n-dimensional minimal bi-slant submanifold of a complex space form M(c) of dimension 2n such that c > 0. If ∇ P = 0, then  g(∇ A, A) ≥ 2

  1 c 2 2 2 −2 + ||A|| + n + 1 + 6(n 1 cos θ1 + n 2 cos θ2 ) ||A||2 . n 4 (40)

At this stage, we are ready to state the following result by using the developed inequality (40) and well-known Hopf’s Lemma. Theorem 2.13 ([41]) Let M be an n-dimensional compact minimal bi-slant submanifold of a complex space form M(c) of dimension 2n such that c > 0 and ∇ P = 0. If the second fundamental form A satisfies ||A||2 ≤

 c  n + 1 + 6(n 1 cos2 θ1 + n 2 cos2 θ2 ) , 1 4 2− n

then either M is totally geodesic or ||A||2 =

 c  n + 1 + 6(n 1 cos2 θ1 + n 2 cos2 θ2 ) . 1 4 2− n

Since we know that bi-slant submanifolds generalize the hemi-slant and semislant submanifolds, therefore, we can easily state the following theorem by using Corollary 2.9. Theorem 2.14 ([41]) Let M be an n-dimensional compact minimal hemi-slant submanifold of a complex space form M(c) of dimension 2n such that c > 0 and ∇ P = 0. If the second fundamental form A satisfies ||A||2 ≤

 c  n + 1 + 6n 2 cos2 θ , 1 4 2− n

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then either M is totally geodesic or ||A||2 =

 c  n + 1 + 6n 2 cos2 θ . 1 4 2− n

The following theorem can be stated by using Corollary 2.10: Theorem 2.15 ([41]) Let M be an n-dimensional compact minimal semi-slant submanifold of a complex space form M(c) of dimension 2n such that c > 0 and ∇ P = 0. If the second fundamental form A satisfies ||A||2 ≤

 c  n + 1 + 6(n 1 + n 2 cos2 θ ) , 1 4 2− n

then either M is totally geodesic or ||A||2 =

 c  n + 1 + 6(n 1 + n 2 cos2 θ ) . 1 4 2− n

Moreover, we quote some results which are based on Theorem 2.11. As the ambient manifold is a complex space form M(c), we can take a complex projective space CP2n with constant holomorphic sectional curvature 4 of real dimension 2n. Then, we have the following result. Corollary 2.16 ([42]) Let M be an n-dimensional complete, compact, and minimal Kählerian slant submanifold of CP2n . If ||A||2
1 4 2− n

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297

everywhere on M, then the scalar curvature of M has the lower bound, that is, τ≤

c n (n + 1) . 2(2 − n1 )

2.2 Some Examples Here, we construct some examples of slant submanifolds in an almost complex manifold inspired by Chen [8]. Let (R2m , J ) be the Euclidean 2m-space endowed with the Euclidean metric g, where J is a standard almost complex structure on R2m . The Euclidean metric g is defined by 

g (x1 , . . . , xm , y1 , . . . , ym ), (z 1 , . . . , z m , w1 , . . . , wm ) = x1 z 1 + · · · + xm z m + y1 w1 + · · · + ym wm . Example 2 ([42]) Consider a 2-dimensional submanifold M in (R4 , J ) given by r (u, s) = (−s sin u, sin s, s cos u, cos s). Then at any point p ∈ M, we have ⎡

−s cos u ⎢ 0 ⎢ dr p = ⎣ −s sin u 0

⎤ − sin u cos s ⎥ ⎥. cos u ⎦ − sin s

Let {e1 , e2 } be a local orthonormal frame on M. e1 = (− cos u, 0, − sin u, 0), (− sin u, cos s, cos u, − sin s) e2 = . √ 2 Since J (a, b, c, d) = (−c, −d, a, b) satisfies J 2 = −I , then we have J e1 = (sin u, 0, − cos u, 0), (− cos u, sin s, − sin u, cos s) J e2 = . √ 2 We see that   g(J ei , e j ) = √1 , i, j = 1, 2. 2

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Hence, M is a proper slant surface in R4 with the slant angle θ =

π . 4

Example 3 ([42]) Consider a 2-dimensional submanifold M in (R4 , J ) given by r (u, v) = (u sin β1 , v cos β2 , u cos β1 , v sin β2 ), where  β1 and β2 are constant. Then M is a slant surface with the slant angle θ =  arccos  sin(β1 + β2 ) , where J (a, b, c, d) = (−b, a, −d, c). Example 4 ([42]) Consider a 2-dimensional submanifold M in (R4 , J ) given by r (u, v) = (u + v, u + v, u, v).

 Then M is a proper slant surface with the slant angle θ = arccos √13 , where J (a, b, c, d) = (−b, a, −d, c). If the almost complex structure J is defined as J (a, b, c, d) = (−c, −d, a, b) on R4 , then M is a totally real surface in R4 as θ = π2 . Example 5 ([42]) Consider a 2-dimensional submanifold M in (R4 , J ) given by r (u, v) = (u, u, v cos t, v sin t), where  t is any constant. Then M is a slant surface with the slant angle θ =  t arccos  cos√t+sin , where J (a, b, c, d) = (c, d, −a, −b). √ 2

Example 6 Consider a 2-dimensional submanifold M in (R8 , J ) given by r (u, v, w, z) = (u, v, k sin w, k sin z, k w, k z, k cos w, k cos z). Then M is a Kählerian slant submanifold with the slant angle π4 , for any positive number k. It is an example of Kählerian slant submanifolds given by B.-Y. Chen. Example 7 If M is a slant submanifold in an almost Hermitian manifold M, then M × R is a slant submanifold in the almost contact metric manifold M × R with the usual product structure [9]. Remark 2.19 It is known that in complex geometry, proper slant submanifolds are always even-dimensional, while in contact geometry, proper slant submanifolds are always odd-dimensional.

3 Some Geometric Inequalities for Slant Submanifolds B.-Y. Chen recalled this as one of the basic problems in submanifold theory.

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299

Find simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. It is well known that Riemannian invariants play the most fundamental role in Riemannian geometry. The Riemannian invariants provide the intrinsic characteristics of Riemannian manifolds which affect the behavior in general of the Riemannian manifold. In this section, first, we recall a string of Riemannian invariants on a Riemannian manifold. Let M be a Riemannian manifold. Denote by K (π ) the sectional curvature of M associated with a plane section π ⊂ T p M, p ∈ M. For any orthonormal basis {e1 , e2 , . . . , en } of the tangent space T p M, the scalar curvature τ at p is defined by [36] τ ( p) =



K (ei ∧ e j ).

(41)

i< j

We denote by (inf K )( p) = inf{K (π ) : π ⊂ T p M, dim(π ) = 2},

(42)

the first Chen invariant is defined as [36] δ M ( p) = τ ( p) − (inf K )( p).

(43)

Let  be a subspace of T p M of dimension r > 2 and {e1 , e2 , . . . , er } an orthonormal basis of . We define the scalar curvature τ () of the r -plane section  by τ () =



K (ei ∧ e j ), i, j = 1, . . . , r.

(44)

i< j

Given an orthonormal basis {e1 , e2 , . . . , en } of the tangent space T p M, we simply denote by τ1,...,r the scalar curvature of the r -plane section spanned by {e1 , . . . , er }. For an integer k > 0, we denote by S(n, k) the finite set which consists of k-tuples (n 1 , n 2 , . . . , n k ) of integers ≥ 2 satisfying n j < n and n 1 + · · · + n k < n. Denote by S(n) the set of k-tuples with k > 0 for a fixed n. The Riemannian invariant is defined for each k-tuples (n 1 , . . . , n k ) ∈ S(n) as [36] δ(n 1 , . . . , n k ) = τ ( p) − inf{τ (1 ) + · · · + τ (k )},

(45)

where 1 , . . . , k run over all k mutually orthogonal subspaces of T p M such that dim( j ) = n j , j = 1, . . . , k, is called a δ-invariant or Chen invariant.

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We denote [36]  n 2 (n + k − 1 − kj=1 n j ) d(n 1 , . . . , n k ) = ,  2(n + k − kj=1 )

b(n 1 , . . . , n k ) =

n(n − 1) −

k j=1

n j (n j − 1)

2

(46)

.

(47)

For an orthonormal basis {u 1 , . . . , u r } of  ⊂ T p M, dim() = r , we put () =



g 2 (Pu i , u j ).

1≤i< j≤r

3.1 Chen-Ricci Inequality Chen [15] proved an optimal inequality for submanifolds in Riemannian space forms in terms of the Ricci curvature and the squared mean curvature, well known as the Chen-Ricci inequality. Afterward, the Chen-Ricci inequality was improved in [23, 37] for Lagrangian submanifolds of complex space forms. First, we obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a complex space form M(4c), in terms of the main extrinsic invariant, namely the squared mean curvature. Theorem 3.1 ([31]) Let M be an n-dimensional θ -slant submanifold in an mdimensional complex space form M(4c) of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies  S≤

 n2 2 2 H  + (n − 1)c + 3c cos θ g. 4

(48)

The equality case of (48) holds identically if and only if either M is a totally geodesic submanifold or n = 2 and M is a totally umbilical submanifold. In particular, for totally real and complex submanifolds, respectively, we state the following: Corollary 3.2 ([31]) Let M be an n-dimensional totally real submanifold in an mdimensional complex space form M(4c) of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies  S≤

 n2 H 2 + (n − 1)c g. 4

(49)

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

301

The equality case of (49) holds identically if and only if either M is a totally geodesic submanifold or n = 2 and M is a totally umbilical submanifold. For a classification of totally umbilical submanifolds in non-flat complex space forms, we have the following: Corollary 3.3 ([31]) Let M be an n-dimensional complex submanifold in an mdimensional complex space form M(4c) of constant holomorphic sectional curvature 4c. Then the Ricci tensor S satisfies S ≤ 2(n + 1)cg.

(50)

The equality case of (50) holds identically if and only if M is a totally geodesic submanifold. In 2005, Aimin Song and Ximin Liu [45] obtained an inequality about Ricci curvature and squared mean curvature of slant submanifolds in generalized complex space forms and also an inequality about the squared mean curvature and the normalized scalar curvature of slant submanifolds in the same ambient space forms. The basic inequality for θ -slant submanifolds in generalized complex space forms is given by the following: Theorem 3.4 ([45]) Let M be an n-dimensional θ -slant submanifold in an mdimensional generalized complex space form M( f 1 , f 2 ). Then (1) For each unit vector X ∈ T p M, we have Ric(X ) ≤ (n − 1) f 1 + 3 f 2 cos2 θ +

n2 H 2 . 4

(51)

(2) If H ( p) = 0, then a unit tangent vector X at p satisfies the equality case of (51) if and only if X ∈ K er h p . (3) The equality case of (51) holds identically for all unit tangent vectors at p if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point. Proof From Lemma 3.8, we derive n 2 H 2 = 2τ + +2

2m 1  [(h a + · · · + h ann )2 + (h a11 − · · · − h ann )2 ] 2 a=n+1 11

2m  

(h iaj )2 − 2

a=n+1 i< j

2m 



h iia h aj j − [n(n − 1) f 1

a=n+1 2≤i< j≤n

+3n f 2 cos θ ]. 2

(52)

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And from the equation of Gauss, we get 

Ki j =

2m 



[h iia h aj j − (h iaj )2 ] +

a=n+1 2≤i< j≤n

2≤i< j≤n

(n − 1)(n − 2) f1 2

3 + n f 2 cos2 θ − 3 f 2 cos2 θ. 2

(53)

Substituting (53) in (52), we have n 2m    1 2 2 n H  ≥ 2τ + n H  + 2 (h a1 j )2 − 2 Ki j 2 a=n+1 j=2 2≤i< j≤n 2

2

+[(n − 1)(n − 2) − n(n − 1)] f 1 − 6 f 2 cos2 θ.

(54)

From inequality (54), we could get (51). (2)

Assume H ( p) = 0, equality holds in (51) if and only if 0 = h a12 = · · · = h a1n , h a11 = h a22 + · · · + h ann ,

for all a ∈ {n + 1, . . . , 2m}. Then h a1 j = 0, for all j ∈ {1, . . . , n}, a ∈ {n + 1, . . . , 2m}, that is, X ∈ K er h p . (3)

The equality case of (51) holds for all unit tangent vectors at p if and only

if h iaj = 0, i  = j, a ∈ {n + 1, . . . , 2m}, 2h iaj = h a11 + · · · + h ann , i ∈ {1, . . . , n}, a ∈ {n + 1, . . . , 2m}. We distinguish two cases. (i) n  = 2, then p is a totally geodesic point. (ii) n = 2, it follows that p is a totally umbilical point. The converse is trivial.



Corollary 3.5 ([45]) Let M be a n-dimensional totally real submanifold in an mdimensional generalized complex space form M( f 1 , f 2 ). Then, we have (1) For each unit vector X ∈ T p M, we have

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

Ric(X ) ≤ (n − 1) f 1 +

n2 H 2 . 4

303

(55)

(2) If H ( p) = 0, then a unit tangent vector X at p satisfies the equality case of (55) if and only if X ∈ K er h p . (3) The equality case of (55) holds identically for all unit tangent vectors at p if and only if either p is a totally geodesic point or n = 2 and p is a totally umbilical point.

3.2 B. Y. Chen Inequality In this section, we derive an inequality for the Chen invariant R and prove that any Kählerian slant submanifold M which satisfies the equality case is minimal. We denote by R the maximum Ricci curvature function on M, defined by [31] R( p) = max{S(u, u) ∈ T p1 M, p ∈ M}, where T p1 M = {u ∈ T p M|g(u, u) = 1}. If n = 3, R is the Chen first invariant δ M defined in [10]. For n > 3, R is the Chen invariant δ(n − 1) given in [14]. Theorem 3.6 ([31]) Let M be an n-dimensional Kählerian slant submanifold in an n-dimensional complex space form M(4c) of constant holomorphic sectional curvature 4c. Then, we have R≤

n2 H 2 + (n − 1)c + 3c cos2 θ. 4

(56)

If M satisfies the equality case of (56) identically, then M is a minimal submanifold. Corollary 3.7 ([31]) Let M be an n-dimensional Kählerian slant submanifold of an n-dimensional complex space form M(4c). If dim(K er h p ) is positive constant, then M satisfies the equality case of (56) identically and is foliated by totally geodesic submanifolds. Proof By the above proof, it follows that M satisfies the equality case of (56) at a point p ∈ M if and only if dim(K er h p ) ≥ 1. Assume that dim(K er h p ) is positive constant. It is known that K er h is involutive and its leaves are totally geodesic. This achieves the proof. 

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Next, Chen established the following inequality for submanifolds in real space forms [10]:

δM ≤

  n−2 n2 H 2 + (n + 1)c 2 (n − 1)

(57)

and investigated the equality case of the above inequality. He also derived this basic inequality for submanifold M in CPm (4c) and CHm (4c)) of constant holomorphic sectional curvature 4c as follows [12]: n 2 (n − 2) 1 H 2 + (n 2 + 2n − 2)c, 2(n − 1) 2   n−2 n2 2 ≤ H  + (n + 1)c , 2 (n − 1)

δM ≤ δM respectively.

In 2003, Kim et al. [27] extended the B.-Y. Chen inequality for θ-slant submanifolds in a generalized complex space form. One denotes the Riemannian invariant by δ M ( p) = τ ( p) − inf{K (π )|π ∈ T p M 2-plane section invariant by J }. Lemma 3.8 ([27, 45]) Let M be an n-dimensional submanifold of an m-dimensional generalized complex space form M( f 1 , f 2 ). Then the scalar curvature and the squared mean curvature satisfy 2τ = n(n − 1) f 1 + 3 f 2 P2 + n 2 H 2 − h2 .

(58)

Theorem 3.9 ([27]) Let M be an n-dimensional (n > 2) θ-slant submanifold isometrically immersed in a 2m-dimensional generalized complex space form M( f 1 , f 2 ). At every point p ∈ M and each plane section π ⊂ T p M, we have δM

  n−2 n2 2 2 ≤ H  + (n + 1) f 1 + 3 f 2 cos θ . 2 (n − 1)

(59)

Equality in (59) holds at p ∈ M if and only if there exist an orthonormal basis {e1 , . . . , en } of T p M and an orthonormal basis {en+1 , . . . , e2m } of T p⊥ M such that the shape operators of M in M( f 1 , f 2 ) at p have the forms

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

⎛

β1 0 0 ⎜ 0 β 0 2 ⎜ ⎜ ⎜ 0 0 γ An+1 = ⎜ ⎜ .. .. .. ⎝ . . .

... ... ... ...

305

⎞

0 0⎟ ⎟ ⎟ 0⎟ , .. ⎟ ⎟ .⎠

(60)

0 0 0 ... γ

and ⎛ a h 11 h a12 0 . . . ⎜ h a −h a 0 . . . ⎜ 12 11 ⎜ 0 0 0 ... Aa = ⎜ ⎜ .. .. . . ⎜ .. ⎝ . . . .

0

0

⎞

0 0⎟ ⎟ ⎟ 0⎟ , .. ⎟ ⎟ .⎠

(61)

0 ... 0

where γ = β1 + β2 , and Aa = Aea , a = n + 1, . . . , 2m.

(62)

Theorem 3.10 ([32]) For any 2m-dimensional generalized complex space form M(c, α) and a θ -slant submanifold M, dim(M) = n, n > 2, we have δM

  n−2 c 3α n2 2 2 2 ≤ H  + (n + 1 + 3 cos θ ) + (n + 1 − cos θ ) . 2 (n − 1) 4 4 (63)

The equality case of inequality (63) is identical with the equality case of inequality (59) from Theorem 3.9. Using a similar method, one may prove the Chen inequality for totally real submanifolds in generalized complex space forms. Proposition 3.11 ([32]) For any 2m-dimensional generalized complex space form M(c, α) and a totally real submanifold M, dim(M) = n, n > 2, we have δM

  n−2 c + 3α n2 2 ≤ H  + (n + 1) . 2 (n − 1) 4

(64)

The equality case of inequality (64) is identical with the equality case of inequality (59) from Theorem 3.9. Remark 3.12 For α = 0, we obtain the Chen first inequality for slant submanifolds in complex space forms (see [33]). We consider a plane section π-invariant by P and denote dim(D1 ) = 2d1 and dim(D2 ) = 2d2 . So, we have the following:

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Theorem 3.13 ([40]) Let M be an n-dimensional proper bi-slant submanifold of a 2m-dimensional generalized complex space form M(c, α). Then (1) For any plane section π-invariant by P and tangent to D1 , τ − K (π ) ≤

    c + 3α n2 n−2 H 2 + (n + 1) 2 (n − 1) 4   c−α + 3(d1 − 1) cos2 θ1 + 3d2 cos2 θ . 4

(65)

(2) For any plane section π-invariant by P and tangent to D2 , τ − K (π ) ≤

    c + 3α n2 n−2 H 2 + (n + 1) 2 (n − 1) 4   c−α + 3d1 cos2 θ1 + 3(d2 − 1) cos2 θ . 4

(66)

The equality case of inequalities (65) and (66) is identical with the equality case of inequality (59) from Theorem 3.9. B.-Y. Chen introduced in [11] proved that for every n-dimensional submanifold M in a complex space form M(4c), the invariant δ(n 1 , . . . , n k ) and the squared mean curvature H 2 satisfy the following basic inequality. 3 δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )c + P2 2 k  −3c m j ( j ),

(67)

j=1

where  j ⊂ T p M is a subspace of T p M, dim( j ) = n j , for all j = 1, . . . , k. In order to generalize B.Y. Chen inequality (see [36]) in terms of Chen invariants, we will use the following Lemma (see [14]). Lemma 3.14 ([36]) Let M(c) be a complex space form of dimension 2m and M an ndimensional submanifold of M(c). Let n 1 , . . . , n k be integers ≥ 2 satisfying n 1 < n, n 2 + · · · + n k ≤ n. For p ∈ M and 1 , . . . , k mutually orthogonal subspaces of T p M of dimensions n 1 , . . . , n k , we have τ ( p) −

  k  c τ ( j ) < 2b(n 1 , . . . , n k ) + 3P2 − 6 ( j ) 8 j=1 j=1

k 

+d(n 1 , . . . , n k )H 2 .

(68)

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Theorem 3.15 ([36]) Given a 2m-dimensional complex space form M(c) and an n-dimensional θ-slant submanifold M, n ≥ 3, we have  c c n j ) cos2 θ. δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H  + b(n 1 , . . . , n k ) + 3(n − 4 8 j=1 k

2

(69) Corollary 3.16 ([36]) Let M(c) a 2m-dimensional complex space form and M an n-dimensional totally real submanifold, n ≥ 3, we have c δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k ) . 4

(70)

Similarly, we can derive the following results on Chen invariants as in Lemma 3.14 and Theorem 3.15. Lemma 3.17 ([27]) Let M( f 1 , f 2 ) be a generalized complex space form of dimension 2m and M be an n-dimensional submanifold of M(c). Let n 1 , . . . , n k be integers ≥ 2 satisfying n 1 < n, n 2 + · · · + n k ≤ n. For p ∈ M, we have τ ( p) −

k  j=1

τ ( j ) < b(n 1 , . . . , n k ) f 1 + d(n 1 , . . . , n k )H 2   k  3 2 + P − 2 ( j ) f 2 . 2 j=1

(71)

Theorem 3.18 ([27]) Let M( f 1 , f 2 ) be a 2m-dimensional generalized complex space form and M be an n-dimensional θ -slant submanifold, n ≥ 3. Then we have δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k ) f 1   k  nj 3 + n−2 [ ] f 2 cos2 θ. 2 2 j=1

(72)

In order to prove the next result for θ -slant submanifold of generalized complex space form M(c, α), we need the following lemma. Lemma 3.19 ([32]) Let M(c, α) be a 2m-dimensional generalized complex space form and M be an n-dimensional θ -slant submanifold, n > 2, of M(c, α). Let {n 1 , . . . , n k be integers ≥ 2 satisfying n 1 < n, n 1 + · · · + n k ≤ n. For p ∈ M, let  j ∈ T p M be subspaces of T p M, invariant by P, such that dim( j ) = n j , for all j = 1, . . . , k. Then, we have

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τ ( j ) ≤ d(n 1 , . . . , n k )H 2 + [n(n − 1) −

j=1

k 

n j (n j − 1)]

j=1

−[3P2 − 6 −[P2 + 2

k 

c 8

 c c ( j )] + [n(n − 1) − n j (n j − 1)] 8 8 j=1 j=1

k 

k

( j )]

j=1

3α . 8

Theorem 3.20 ([32]) Let M(c, α) be a 2m-dimensional generalized complex space form and M be an n-dimensional θ -slant submanifold, n > 2, of M(c, α). Let {n 1 , . . . , n k be integers ≥ 2 satisfying n 1 < n, n 1 + · · · + n k ≤ n. Then, we have  c + 3α c + 3(n − n j ) cos2 θ 4 8 k

δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )

j=1

−3(n +

k  j=1

n j)

α cos2 θ. 8

(73)

Proof Let p ∈ M and {e1 , . . . , en } be an orthonormal basis of T p M. Since we use subspaces invariant by P, we may choose e2 = (sec θ )Pe1 , . . . , e2k = (sec θ )Pe2k−1 . It can be easily verified that P2 = n cos2 θ . Now, let 1 , . . . , k be k mutually orthogonal subspaces of T p M, dim( j ) = n j , defined by 1 = span{e1 , . . . , en 1 }, 2 = span{en 1 +1 , . . . , en 1 +n 2 }, ··· k = span{en 1 +···+n k−1 +1 , . . . , en 1 +···+n k−1 +n k }. In the same way, it follows that ( j ) = 3.19, we obtain the inequality (73).

nj 2

cos2 θ, j = 1, . . . , k. From Lemma 

Similarly, one can prove the following. Proposition 3.21 ([32]) For any 2m-dimensional generalized complex space form M(c, α) and a totally real submanifold M, dim(M) = n, n > 2, we have δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )

c + 3α . 4

Physical Interpretation of Ideal Immersions: An isometric immersion ψ : M → M is an ideal immersion means that M receives the least possible amount of tension

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 given by

δ(n 1 , . . . , n k ) − b(n 1 , . . . , n k ) d(n 1 , . . . , n k )

309



from the surrounding space at each point p in M. This is due to δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )c and the well-known fact that the mean curvature vector field is exactly the tension field for an isometric immersion of a Riemannian manifold in another Riemannian manifold. Thus, the squared mean curvature at each point in the submanifold simply measures the amount of tension the submanifold receives from the surrounding space at that point. For more similar inequalities, we can follow [32]. Next, we have the following. Lemma 3.22 ([26]) Let ψ : M → M(4c) be an n-dimensional Kählerian slant submanifold into an n-dimensional complex space form M(4c). If the equality in (67) holds at a point p ∈ M, then there exists an orthonormal basis {e1 , . . . , en } of T p M such that h(i , i ) ⊂ F(i ), h(i ,  j ) = {0}, h(X, en 1 +···+n k +1 ) = · · · = h(X, en ) = 0, tracei (h) = 0, for 1 ≤ i  = j ≤ k and X ∈ (T M). Proof This lemma can be proved by using the almost same methods as in [16].  Theorem 3.23 ([26]) There do not exist n-dimensional Kählerian slant submanifolds with Chen’s equality holding in an n-dimensional non-trivial complex space form satisfying either one of the following two conditions. (1) n 1 + · · · + n k < n, or (2) n 1 + · · · + n k = n and at least one of n i is 2. Proof By Lemma 3.22 and Theorem 2.6, (1) holds immediately. To prove (2), follow [26].  In the following, we shall quote a theorem obtained by B.-Y. Chen and list a result similar to that in [16], which we will use later. Lemma 3.24 ([26]) Given an n-dimensional submanifold M in a complex space form M(4c), then we have the inequality (68). The equality holds at a point p ∈ M if and only if there exists an orthonormal basis {e1 , . . . , e2m } at p such that (1)  j = span{en 1 +···+n j−1 +1 , . . . , en 1 +···+n j }.

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(2) the shape operators of M in M(4c) at p take the following forms ⎛ ⎜ ⎜ ⎜ Aa = ⎜ ⎜ ⎜ ⎝

⎞

Aa1 Aak

...

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

(74)

μa I

where each Aaj is a symmetric n j × n j submatrix satisfying trace(Aa1 ) = · · · = trace(Aak ) = μa .

(75)

Finally, for general slant submanifolds in a complex projective space, we have the following application. Theorem 3.25 ([26]) Let ψ : M → CPm (4c) (c > 0) be a θ-slant immersion. Then for each (n 1 , . . . , n k ) ∈ S(n), we have δ(n 1 , . . . , n k ) ≤ d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )c 3nc + cos2 θ. 2

(76)

Moreover, if the structure F is parallel, then equality in (76) holds if and only if M is immersed as one of the following submanifolds: (1) a holomorphic, totally geodesic submanifold of CPm (4c), (2) a ruled minimal totally real submanifold in some totally geodesic subspace CPn (4c) of CPm (4c). Proof Let ψ : M → CPm (4c) (c > 0) be a θ-slant immersion. Then, we have inequality (68) for any (n 1 , . . . , n k ) ∈ S(n). Since P2 = n cos2 θ and (i ) ≥ 0, we have (76). If the equality in (76) holds identically, then (i ) = 0 for all i = 1, . . . , k. Thus, by applying Lemma 3.24, we know that for each p ∈ M, there exists an orthonormal basis {e1 , . . . , en } of T p M and orthonormal basis {en+1 , . . . , e2m } of T p⊥ M such that Peβ1 doesn’t belong to i , where eβ1 ∈ i . Moreover, the shape operators of M take the forms of (74) and (75). Now, we assume the structure F is parallel. So, for any eβ1 , eβ2 ∈ i , we have f h(eβ1 , eβ2 ) = h(eβ1 , Peβ2 ) = 0.

(77)

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If h(eβ1 , eβ2 ) = 0, for any β1 , β2 ∈ {1, . . . , n}, then M is totally geodesic. Thus, by the equation of Codazzi, we know that θ = 0 or π2 . When θ = 0, M is holomorphic and also totally geodesic. Conversely, if M is holomorphic as well as totally geodesic, equality in (76) holds obviously. In this case, F = 0. If I m h p  = 0, then there is a non-zero vector N ∈ I m h p such that f N = 0. So, PtN = 0, which implies P = 0, that is, M is totally real. Now, we can choose a frame {e1 , . . . , en } of T p M and {en+1 , . . . , e2m } of T p⊥ M such that en+i = J ei , i = 1, . . . , n, J eγ ∈ {e2n+1 , . . . , e2m }, γ = 2n + 1, . . . , 2m. Thus, by last two relations, we have for any 2n + 1 ≤ γ ≤ 2m 0 = f h(ei , e j ) =

2m 

γ

h i j f eγ ,

γ =2n+1 γ

which implies h i j = 0 for any i, j ∈ {1, . . . , n}. This proves that I m h p ⊂ span{en+1 , . . . , e2n }. That is, M is contained in some totally geodesic subspace CPn (4c) of CPm (4c). Thus, by Lemma 3.3 of [16], the equality in (76) holds if and only if M is a ruled minimal submanifold in some totally geodesic subspace CPn (4c)  of CPm (4c). We assume that an n-dimensional Kählerian slant submanifold M in an ndimensional complex space form M(4c) is ideal. Then it satisfies [35] δ(n 1 , . . . , n k ) = d(n 1 , . . . , n k )H 2 + b(n 1 , . . . , n k )c   k  3c n j cos2 θ + n− 2 j=1

(78)

identically for some k-tuple (n 1 , . . . , n k ) ∈ S(n). We denote by Di the distribution generated by i , for i = 1, . . . , k. Here, we prove that ideal Kählerian slant submanifolds in a complex space form are minimal. Thus, we have the following: Theorem 3.26 ([35]) Let M be an n-dimensional Kählerian slant submanifold of an n-dimensional complex space form M(4c). If M is an ideal submanifold, then it is minimal. Proof Let M be a Kählerian slant submanifold of a complex space form M(4c), and let p ∈ M. We distinguish two cases. (i) g(h(u, v), Fw) = 0, for all u, v, w ∈ T p M. Obviously, it follows that H ( p) = 0.

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(ii) g(h(u, u), Fu) = 0,

for some u ∈ T p M. For this case, follow [35]. 

To prove the non-existence of n-dimensional ideal Kählerian slant submanifolds in an n-dimensional complex hyperbolic space with full first normal bundle, first we state the following. Proposition 3.27 ([35]) Every minimal slant submanifold of a hyperbolic complex space form is irreducible. Proof Assume that M is an n-dimensional reducible slant submanifold of an mdimensional complex space form M(4c), with c < 0. Then, locally, M is the Riemannian product of some Riemannian manifolds, say M = M1 × Ms , s ≥ 2. If dim(M1 ) = r , then we can choose an orthonormal basis {e1 , . . . , en } such that e1 , . . . , er are tangent to M1 and er+1 , . . . , en are tangent to M1 × Ms . Since M is minimal, the Gauss equation yields 0=

r n  

R(ei , e j , ei , e j )

i=1 j=r+1

= r (n − r )(1 + 3 cos2 θ )c − 

r 

h(ei , ei )2

i=1

−

n r  

h(ei , e j )2 ,

i=1 j=r+1

which is impossible.



The following lemma implies the integrability and the minimality of the distributions D1 , . . . , Dk . Lemma 3.28 ([35]) Let M be an n-dimensional ideal Kählerian slant submanifold of an n-dimensional complex space form M(4c) satisfying the equality (78) identically. If the first normal bundle of M is full, then n 1 + · · · + n k = n, h(Di , Di ) = F(Di ), h(Di , D j ) = {0}, 1 ≤ i = j ≤ k, and ∇Y j X i ∈ Di ⊕ D j , 1 ≤ i = j ≤ k, for vector fields X i in Di and Y j in D j , respectively. Moreover, D1 , . . . , Dk are completely integrable distributions and the leaves of D1 , . . . , Dk are totally geodesic submanifolds in M and minimal submanifolds in M(4c), respectively. Proof The proof follows from [16].



Using the above results, we will obtain a non-existence theorem for certain ideal slant submanifolds. Theorem 3.29 ([35]) There do not exist n-dimensional ideal Kählerian slant submanifolds in an n-dimensional complex hyperbolic space CHn whose first normal bundle is full.

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313

Proof Let M be n-dimensional ideal Kählerian slant submanifolds in an ndimensional complex hyperbolic space CHn . Then it satisfies the equality (78) for some k-tuple (n 1 , . . . , n k ) ∈ S(n). If the first normal bundle is full, then n 1 + · · · + n k = n and k ≥ 2. Hence, the tangent bundle T M of M is the direct sum D1 ⊕ · · · ⊕ Dk . According to Lemma 3.28, each Di is an integrable distribution with totally geodesic leaves. Moreover, by the form (74) of the shape operators of an ideal submanifold, any sum D j1 ⊕ · · · ⊕ D js , s ∈ {2, . . . , k} is also an integrable distribution with totally geodesic leaves. Therefore, de Rham’s decomposition theorem implies that M is locally the Riemannian product of k Riemannian manifolds N1 , . . . , Nk of dimensions n 1 , . . . , n k , respectively, where Ni is an integral submanifold of Di . Thus, M is a reducible Riemannian manifold. By applying Theorem 3.26, we know that the submanifold M is minimal. Hence, by using Proposition 3.27, we obtain the desired result.  Also, one can prove the following. Theorem 3.30 ([35]) There do not exist n-dimensional ideal Kählerian slant submanifolds in an n-dimensional complex projective space CPn . On the other hand, there do exist n-dimensional ideal Kählerian slant submanifolds in the complex Euclidean space Cn with full first normal bundle. In fact, we have the following. Theorem 3.31 ([35]) Let M be an n-dimensional Kählerian slant submanifold in Cn with full first normal bundle. Then M is ideal if and only if, locally, M is the Riemannian product of some minimal Kählerian slant submanifolds M j , j = 1, . . . , k, with full first normal bundle. We state a theorem of characterization of ideal Kählerian slant submanifolds in the complex Euclidean space. Theorem 3.32 ([35]) Let M be an n-dimensional Kählerian slant submanifold of the complex Euclidean space Cn such that Im h p  = T p⊥ M, at each point p ∈ M. Then M is ideal if and only if M is a ruled minimal submanifold.

3.3 Other Geometric Inequalities Theorem 3.33 ([45]) Let M be an n-dimensional θ-slant submanifold in an mdimensional generalized complex space form M( f 1 , f 2 ). Then we have H 2 ≥ ρ − f 1 −

3 f2 cos2 θ. n−1

(79)

The equality holds at a point p ∈ M if and only if p is a totally umbilical point.

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Proof Choose an orthonormal basis {e1 , . . . , en , en+1 , . . . , e2m } at p such that en+1 is parallel to the mean curvature vector and {e1 , . . . , en } diagonalize the shape operator An+1 . Then we have ⎛

b1 0 0 . . . ⎜ ⎜ 0 b2 0 . . . An+1 = ⎜ ⎜ .. .. .. . . ⎝ . . . .

⎞

0 0⎟ ⎟ .. ⎟ ⎟. . ⎠ 0 0 0 . . . bn

(80)

From the equation of Gauss, we have n 2 H 2 = 2τ − n(n − 1) f 1 − 3n f 2 cos2 θ +

n 

ai2

i=1

+

2m 

n 

(h iaj )2 .

(81)

a=n+2 i, j=1

On the other hand, since n

n 

bi2 ≥

i=1

 2 n bi ,

(82)

i=1

we get n 

bi2 ≥ nH 2 .

i=1

Combining this with (81), we obtain n(n − 1)H 2 ≥ 2τ − n(n − 1) f 1 − 3n f 2 cos2 θ n 2m   + (h iaj )2 ,

(83)

a=n+2 i, j=1

which implies inequality (79). If the equality sign of (79) holds at a point p ∈ M, then from (82) and (83), we get Aa = 0, a = n + 2, . . . , 2m and b1 = · · · = bn . Therefore, p is a totally umbilical point. The converse is trivial.  Corollary 3.34 ([45]) Let M be a n-dimensional totally real submanifold in an m-dimensional generalized complex space form M( f 1 , f 2 ). Then we have H 2 ≥ ρ − f 1 .

(84)

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315

The equality holds at a point p ∈ M if and only if p is a totally umbilical point.

4 Slant Submanifolds in Neutral Kähler Manifolds m

Let M i be a complex m-dimensional indefinite Kähler manifold with complex index m i. Thus, M i is endowed with an almost complex structure J and with an indefinite m Riemannian metric g, which is J -Hermitian. The complex index of M i is defined as the complex dimension of the largest complex negative definite subspace of the tangent space. When m = 2n and the complex index is n, the indefinite Kähler 2n manifold M n is called a neutral Kähler manifold. A neutral Kähler surface is nothing but a Lorentzian Kähler surface. The simplest examples of neutral Kähler manifolds are the neutral complex space forms. Denote by the inner product induced from the neutral metrics on neutral manifolds. A tangent vector v of a neutral manifold 2n M n is called (1) space-like if v = 0 or < v, v >> 0, (2) time-like if < v, v >< 0, (3) null or light-like if v  = 0 and < v, v >= 0. 2n

A distribution D on a neutral manifold M n is called space-like (respectively, timelike) if each non-zero vector v ∈ D is space-like (respectively, time-like). 2n

Definition 4.1 ([2]) An isometric immersion ψ : Mn2n → M n of a neutral 2nmanifold into a neutral Kähler manifold of complex dimension 2n is called θ -slant if there exist a real number θ and an orthogonal decomposition T Mn2n = Dsn ⊕ Dtn of the tangent bundle T Mn2n such that (1) Dsn is a space-like distribution and Dtn a time-like distribution; (2) P(Dsn ) = Dtn and P(Dtn ) = Dsn ; (3) P 2 = (sinh2 θ )I . 2n

For a Lagrangian submanifold Mn2n in a neutral Kähler manifold M n , the almost 2n complex structure J on M n interchanges the tangent bundle of Mn2n with its normal bundle of Mn2n . 2n

Assume that ψ : Mn2n → M n is a θ -slant immersion with distributions Dsn and Dtn given above. Let {e1 , . . . , en } be an orthonormal frame of the space-like distribution Dsn . Then we have [2] < Pei , Pe j >= − < ei , P 2 e j >= −δi j sinh2 θ.

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Hence, if we set Pei = sinh θ ei∗ , i = 1, . . . , n,

(85)

then {e1∗ , . . . , en∗ } form an orthonormal frame of the time-like distribution Dtn . Also, it follows that Pei∗ = sinh θ ei , i = 1, . . . , n.

(86)

Next, for i = 1, . . . , n, we put Fei = cosh θ ξi . Then we have J ei = sinh θ ei∗ + cosh θ ξi , i = 1, . . . , n.

(87)

From < J ei , J e j >= δi j and (87), we know that ξ1 , . . . , ξn are orthonormal spacelike normal vector fields of Mn2n . Similarly, if we take Fei∗ = cosh θ ξi∗ , i = 1, . . . , n. Then we obtain J ei∗ = sinh θ ei + cosh θ ξi∗ , i = 1, . . . , n,

(88)

where ξ1∗ , . . . , ξn∗ are orthonormal time-like normal vectors. Moreover, it is easy to verify that {ξ1 , . . . , ξn , ξ1∗ , . . . , ξn∗ } form an orthonormal frame of the normal bundle of the slant immersion ψ. From (2), (87), and (88), we also have J ξi = − cosh θei − sinh θ ξi∗ ,

(89)

J ξi∗ = − cosh θei∗ − sinh θ ξi

(90)

and

for i = 1, . . . , n. To characterize the minimal slant surfaces in a neutral Kähler surface among purely real surfaces in terms of ∇ F, the following results are required. 2n

Proposition 4.2 ([2]) Let ψ : Mn2n → M n be a slant immersion of a neutral manj∗ j ifold into a neutral Kähler manifold. Then ∇ P = 0 if and only if ωi∗ = ωi , i, j = 1, . . . , n with respect to an adapted slant frame of ψ. Proof Let us choose an adapted slant frame {e1 , . . . , en , e1∗ , . . . , en∗ , ξ1 , . . . , ξn , ξ1∗ , . . . , ξn∗ } of ψ. We set

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

∇ X ei =

n 

j

ωi (X )e j +

n 

j=1

j∗

ωi (X )e j∗ ,

317

(91)

j=1

and ∇ X ei∗ =

n 

j

ωi∗ (X )e j +

j=1

n 

j∗

ωi∗ (X )e j∗ ,

(92)

j=1

for i = 1, . . . , n, X ∈ (T Mn2n ). From < ei , e j∗ >= 0 and (91) and (92), we obtain j∗

j

ωi = ωi∗ ,

i, j = 1, . . . , n.

(93)

It follows from (91), (92), and (93) that (∇ X P)ei = sinh θ

n  j∗ j {ωi∗ (X ) − ωi (X )}e j∗ , j=1

n  j∗ j (∇ X P)ei∗ = sinh θ {−ωi∗ (X ) + ωi (X )}e j , j=1

for i = 1, . . . , n, which imply our assertion.



2

Corollary 4.3 ([2]) Let ψ : M12 → M 1 be a purely real surface in a neutral Kähler 2 2 surface M 1 . Then M12 is a slant surface in M 1 if and only ∇ P = 0 holds identically. 2

1∗ = ω11 = 0 with respect Proof Under the hypothesis, if M12 is slant in M 1 , we have ω1∗ to an adapted slant frame {e1 , e1∗ , ξ1 , ξ1∗ }. Hence, by applying Proposition 4.2, we know that ∇ P = 0 holds identically. Conversely, assume that M12 is a purely real 2 surface in M 1 satisfying ∇ P = 0. Let {e1 , e1∗ } be an orthonormal frame satisfying

< e1 , e1 >= 1, < e1 , e1∗ >= 0, < e1∗ , e1∗ >= −1.

(94)

Then there exists a function β1 such that Pe1 = sinh β1 e1∗ and P 2 = (sinh2 β1 )I . Hence, we have 0 = (∇ X P)e1 = ∇ X (sinh β1 e1∗ ) − P(ω11∗ (X )e1∗ ) = (Xβ1 ) cosh β1 e1∗ + sinh β1 ∇ X e1∗ − ω11∗ (X )Pe1∗ .

(95)

Since ∇ X e1∗ and Pe1∗ are parallel to e1 , (95) implies that β1 is constant. Therefore, the surface is slant.  2n

Proposition 4.4 ([2]) Let ψ : Mn2n → M n be a purely real immersion of a neutral manifold into a neutral Kähler manifold. Then ∇ P = 0 holds identically if and only if the shape operator satisfies

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A FY Z = A F Z Y,

(96)

for any Y, Z ∈ (T Mn2n ). 2n

Proposition 4.5 ([2]) Let ψ : Mn2n → M n be a purely real immersion of a neutral manifold into a neutral Kähler manifold. Then ∇ F = 0 holds if and only if the shape operator A satisfies f h(X, Y ) = h(X, PY ),

(97)

for any X, Y ∈ (T Mn2n ), or equivalently, A f ξ Y = −Aξ (PY ),

(98)

for Y ∈ (T Mn2n ) and ξ ∈ (T ⊥ Mn2n ). Proof Our assertions follow directly from Proposition 4.4.



2n

Corollary 4.6 ([2]) Let ψ : Mn2n → M n be a slant immersion of a neutral manifold into a neutral Kähler manifold. If ∇ F = 0 holds, then h(ei , ei ) = h(ei∗ , ei∗ ), i = 1, . . . , n,

(99)

with respect to the adapted slant frame {e1 , . . . , en , e1∗ , . . . , en∗ , ξ1 , . . . , ξn , ξ1∗ , . . . , ξn∗ }. In particular, a slant submanifold with ∇ F = 0 in a neutral Kähler manifold is a minimal submanifold. Proof Under the hypothesis, it follows from (85), (86), and (97) that h(ei∗ , ei∗ ) =  csch θ f h(ei, ei∗ ) = h(ei , ei ), which implies our assertion. Theorem 4.7 ([2]) Let ψ : M12 → C21 be a purely real surface in a neutral Kähler surface. Then ∇ F = 0 holds if and only if M12 is a minimal slant surface. Proof Under the hypothesis, if ∇ F = 0 holds, then the shape operator satisfies (98) proved in Proposition 4.5. Let {e1 , e1∗ } be an orthonormal frame on M12 satisfying (94). Then there is a function β1 and unit space-like and unit time-like normal vector fields ξ1 and ξ1∗ , respectively, satisfying J e1 = sinh β1 e1∗ + cosh β1 ξ1 ,

(100)

J e1∗ = sinh β1 e1 + cosh β1 ξ1∗ ,

(101)

Geometry of Pointwise Slant Immersions in Almost Hermitian Manifolds

319

and < ξ1 , ξ1 >= 1, < ξ1 , ξ1∗ >= 0, < ξ1∗ , ξ1∗ >= −1.

(102)

Hence, we have J ξ1 = − cosh β1 e1 − sinh β1 ξ1 , J ξ1∗ = − cosh β1 e1∗ − sinh β1 ξ1 . Thus, we obtain A Fe1 e1∗ = coth β1 Aξ1 (Pe1 ) = cosh β1 Aξ1∗ e1 = A Fe1∗ e1 . Therefore, according to Proposition 4.4 and Corollary 4.3, M12 is a slant surface. Consequently, M12 is a minimal slant surface according to Corollary 4.6. Conversely, if ψ is a minimal slant surface, with respect to an adapted slant frame {e1 , e1∗ , ξ1 , ξ1∗ }, we have Aξ1 e1∗ = Aξ1∗ e1 .

(103)

Since M12 is minimal, we also have h(e1∗ , e1∗ ) = h(e1 , e1 ).

(104)

So, it follows from (103) and (104) that the second fundamental form satisfies h(e1 , e1 ) = h(e1∗ , e1∗ ) = β2 ξ1 + γ ξ1∗ ,

(105)

h(e1 , e1∗ ) = −γ ξ1 − β2 ξ1∗

(106)

for some functions β2 , γ . Thus, after applying (85), (86), (89), (105), and (106), we obtain (97). Consequently, the slant surface satisfies ∇ F = 0. 

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5 Pointwise Slant Submanifolds in Almost Hermitian and Kähler Manifolds The main purpose of this section is to study pointwise slant submanifolds in almost Hermitian manifolds. The notion of pointwise slant submanifolds extends the notion of slant submanifolds in a very natural way. Several fundamental results on pointwise slant submanifolds are shown in this section. Lemma 5.1 ([18]) An immersion ψ : M −→ M of a manifold M into an almost Hermitian manifold (M, J, g) is a pointwise slant immersion if and only if P 2 = −(cos2 θ )I for some real-valued function θ defined on the tangent bundle T M of M. The following result is an immediate consequence of Lemma 5.1. Corollary 5.2 ([18]) Let ψ : M −→ M be a pointwise slant immersion of an mmanifold into an almost Hermitian manifold (M, J, g). If ψ is not a totally real immersion, then M is even-dimensional. The following proposition provides another simple characterization of pointwise slant immersions. Proposition 5.3 ([18]) Let ψ : M −→ M be an immersion of a manifold M into an almost Hermitian manifold (M, J, g). Then ψ is a pointwise slant immersion if and only if P : T M −→ T M preserves orthogonality, that is, P carries each pair of orthogonal vectors into orthogonal vectors. For any X ∈ (T M), we have g(P X, P X ) = (cos2 θ )g(X, X ). If we put g = e g for a function f on M, then 2f

g(P X, P X ) = e2 f g(P X, P X ) = (cos2 θ )e2 f g(X, X ) = (cos2 θ )g(X, X ). From this, we conclude that ψ : M −→ (M, J, e2 f g) is also a pointwise slant immersion with the same slant function as ψ : M −→ (M, J, g). Hence, we have the following result. Proposition 5.4 ([18]) Let ψ : M −→ (M, J, g) be a pointwise slant immersion of a manifold M into an almost Hermitian manifold M. Then, for any given function f on M, the immersion ψ : M −→ (M, J, e2 f g) is pointwise slant with the same slant function as the immersion ψ : M −→ (M, J, g). Remark 5.5 Proposition 5.4 shows the important facts that the notions of pointwise slant submanifolds and slant functions are conformal invariant. The following conformal property of slant immersions is an immediate consequence of Proposition 5.4.

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Corollary 5.6 ([18]) Let ψ : M −→ (M, J, g) be a slant immersion of a manifold M into an almost Hermitian manifold M. Then, for any given function f on M, the immersion ψ : M −→ (M, J, e2 f g) is also a slant with the same slant angle. Another interesting application of Proposition 5.4 is the following. Corollary 5.7 ([18]) For each integer m ≥ 1, there exist infinitely many 2mdimensional totally umbilical proper slant submanifolds in locally conformal Kähler 2m-manifolds. We have the following result in terms of the operators A, F, and P. Proposition 5.8 ([18]) A pointwise slant submanifold M in a Kähler manifold is slant if and only if the shape operator of M satisfies A F X P X = A F P X X, for any X ∈ (T M). Some consequences of Proposition 5.8 are the following. Corollary 5.9 ([18]) Every totally geodesic pointwise slant submanifold of any Kähler manifold is slant. Proof It follows from Proposition 5.8 and the fact that the shape operator A vanishes identically for totally geodesic submanifolds.  Corollary 5.10 ([18]) Let M be a 2n-dimensional totally umbilical pointwise proper slant submanifold of a Kähler 2n-manifold M. If M is non-totally geodesic in M, then M is always non-slant in M. The special case of Corollary 5.9 is given below. Proposition 5.11 ([18]) Every totally geodesic surface in any Kähler surface is slant. The following result follows from Corollary 5.10 and the fact that every complex surface in a Kähler manifold is minimal. Proposition 5.12 ([18]) Every non-totally geodesic, totally umbilical surface in any Kähler surface is a non-slant, pointwise slant surface, unless it is Lagrangian.

5.1 Cohomology of Pointwise Slant Submanifolds Cohomology groups encode a lot of interesting information about a topological manifold M. Indeed, such groups give information as measuring the number of

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holes in M which measures the obstructions to a circle being able to shrink to a point. Recall that a 2n-dimensional manifold M is called a symplectic manifold if it has a non-degenerate closed 2-form φ, that is, φ is a 2-form satisfying dφ = 0 and φ n  = 0 at each point on M. For a given pointwise proper slant 2n-submanifold M of a Kähler manifold (M, J, g), we put [18] (X, Y ) = g(X, PY ),

(107)

for any X, Y ∈ (T M). Then,  is a non-degenerate 2-form on M. To prove the main result of this section, we need the following lemma. Lemma 5.13 ([18]) Let M be a submanifold of a Kähler manifold M. Then for any X, Y ∈ (T M), we have (∇ X P)Y = th(X, Y ) + A FY X . By definition of the exterior differentiation, we have 3d(X, Y, Z ) = X (Y, Z ) + Y (Z , X ) + Z (X, Y ) −([X, Y ], Z ) − ([Y, Z ], X ) − ([Z , X ], Y ). Thus, by applying the definition of , we obtain 3d(X, Y, Z ) = g(∇ X Y, P Z ) + g(Y, ∇ X P Z ) + g(∇Y Z , P X ) +g(Z , ∇Y P X ) + g(∇ Z X, PY ) + g(X, ∇ Z PY ) −g([X, Y ], P Z ) − g([Z , X ], PY ) − g([Y, Z ], P X ) = g(Y, ∇ X P Z ) + g(Z , ∇Y P X ) + g(X, ∇ Z PY ) +g(∇ X Z , PY ) + g(∇Y X, P Z ) + g(∇ Z Y, P X ). Therefore, by using (∇ X P)Y = ∇ X PY − P∇ X Y , we find 3d(X, Y, Z ) = < X, (∇ Z P)Y > + < Y, (∇ X P)Z > + < Z , (∇Y P)X >,

(108)

for any X, Y ∈ (T M). Therefore, after applying Lemma 5.13 and formula (108), we get 3d(X, Y, Z ) = g(X, th(Y, Z )) + g(X, A FY Z ) + g(Y, th(Z , X )) +g(Y, A F Z X ) + g(Z , th(X, Y )) + g(Z , A F X Y ). Consequently, by applying formula (109), we state the following result.

(109)

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Theorem 5.14 Let M be a proper pointwise slant submanifold of a Kähler manifold M. Then  is closed. Consequently,  defines a canonical cohomology class of M: [] ∈ H2 (M; R). Corollary 5.15 ([18]) Every 2n-dimensional proper pointwise slant submanifold M of a Kähler manifold is a symplectic manifold. Proof Under the hypothesis, it follows from Theorem 5.14 that d = 0. On the other hand, it follows from the formula (107) that n is a positive multiple of the volume element of M. Thus, n  = 0. Consequently,  defines a symplectic structure on M. 

5.2 Some Examples In this section, we provide some examples of pointwise slant submanifolds in almost Hermitian manifolds. Example 8 ([18]) Every 2-dimensional submanifold in an almost Hermitian manifold is pointwise slant. Example 9 ([18]) Every slant (resp. proper slant) submanifold in an almost Hermitian manifold is pointwise slant (resp. pointwise proper slant). Example 10 ([18]) Let E4n = (R4n , g0 ) be the Euclidean 4n-space endowed with the standard Euclidean metric g0 and let {J0 , J1 } be a pair of almost complex structures on E4n satisfying J0 J1 = −J1 J0 . Assume that J0 , J1 are orthogonal almost complex structures, that is, they are compatible with g0 . Thus, g0 (Ji X, Ji Y ) = g0 (X, Y ), i = 0, 1, for X, Y ∈ (T E4n ). Let us denote (R4n , J0 , g0 ) by C2n 0 . For any realvalued function f : E4n −→ R, we define an almost complex structure J f on E4n 4n by J f = (cos f )J0 + (sin f )J1 . Then C2n f = (R , J f , g0 ) is an almost Hermitian manifold. For any given pointwise slant submanifold (resp. slant submanifold) M in (E4n , J0 , g0 ), M is also a pointwise slant submanifold (resp. slant submanifold) of 2n C2n f . In particular, if M is a complex submanifold of C0 , then M is a pointwise slant minimal submanifold in C2n f whose slant function θ is the restriction of f on M, that is, θ = f | M . Acknowledgements The authors thank the referee for many valuable suggestions to improve the presentation of this article.

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References 1. Abe, K.: Applications of Riccati type differential equation to Riemannian manifolds with totally geodesic distributions. J. Tohoku Math. 25, 425–444 (1973) 2. Arslan, K., Carriazo, A., Chen, B.-Y., Murathan, C.: On slant submanifolds of neutral Kähler manifolds. Taiwanese J. Math. 14(2), 561–584 (2010) 3. Bejancu, A.: Geometry of CR-Submanifolds. D. Reidel Publishing Company, Dordrecht (1986) 4. Bejancu, A., Kon, M., Yano, K.: CR-submanifolds of a complex space form. J. Differ. Geo. 16, 137–145 (1981) 5. Carriazo, A.: Bi-slant immersions. In: Proceedings of the ICRAMS 2000, pp. 88–97. Kharagpur, India (2000) 6. Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedi. 78, 183–199 (1999) 7. Chen, B.-Y.: Cohomology of CR-submanifolds. Ann. Fac. Sci. Toulouse Math. 3(2), 167–172 (1981) 8. Chen, B.-Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Belgium (1990) 9. Chen, B.-Y.: Slant immersions. Bul. Math. Soc. Sci. Math. 41, 135–147 (1990) 10. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60, 568–578 (1993) 11. Chen, B.-Y.: A Riemannian invariants and its applications to submanifolds theory. Result Math. 27, 17–26 (1995) 12. Chen, B.-Y.: A general inequality for submanifolds in complex space forms and its applications. Arch. Math. 67, 519–528 (1996) 13. Chen, B.-Y.: Special slant surfaces and a basic inequality. Results Math. 33, 65–78 (1998) 14. Chen, B.-Y.: Some new obstructions to minimal and Lagrangian isometric immersions. Japan. J. Math. 26(1), 105–127 (2000) 15. Chen, B.-Y.: On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms. Arch. Math. 74, 154–160 (2000) 16. Chen, B.-Y.: Ideal Lagrangian immersions in complex space forms. Math. Proc. Camb. Phil. Soc. 128, 511–533 (2000) 17. Chen, B.-Y., Mihai, A., Mihai, I.: A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Results Math. 74, 165 (2019) 18. Chen, B.-Y., Garay, O.J.: Pointwise slant submanifolds in almost Hermitian manifolds. Turk. J. Math. 36, 630–640 (2012) 19. Chen, B.-Y., Ogiue, K.: On totally real submanifolds. Trans. Am. Math. Soc. 193, 257–266 (1974) 20. Chen, B.-Y., Tazawa, Y.: Slant surfaces with codimension two. Ann. Sci. Toul. Math. 3, 29–43 (1990) 21. Chen, B.-Y., Tazawa, Y.: Slant submanifolds in complex Eulidean spaces. J. Tokyo Math. 14, 101–120 (1991) 22. Chen, B.-Y., Tazawa, Y.: Slant submanifolds of complex projective and complex hyperbolic spaces. Glasgow Math. J. 42, 439–454 (2000) 23. Deng, S.: An improved Chen-Ricci inequality. Int. Electron. J. Geom. 2, 39–45 (2009) 24. Etayo, F.: On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. Debrecen 53, 217–223 (1998) 25. Gray, A.: Nearly Kählerian manifolds. J. Differ. Geom. 4, 283–309 (1970) 26. Guanghan, L., Chuanxi, W.: Slant immersions of complex space forms and Chen’s inequality. Acta Math. Sci. 25B(2), 223–232 (2005) 27. Kim, J.-S., Song, Y.-M., Tripathi, M.M.: B.-Y. Chen inequalities for submanifolds in generalized complex space forms. Bull. Korean Math. Soc. 40(3), 411–423 (2003) 28. Lotta, A.: Slant submanifolds in contact geometry. Bul. Math. Soc. Sci. Math. 39, 183–198 (1996) 29. Ludden, G.D., Okumura, M., Yano, K.: A totally real surface in CP2 that is not totally geodesic. Proc. Am. Math. Soc. 53 (1975)

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30. Madsen, I., Tornehave, J.: From Calculus to Cohomology, de Rham Cohomology and Characteristic Classes. Cambridge University Press, Cambridge (1997) 31. Matsumoto, K., Mihai, I., Tazawa, Y.: Ricci tensor of slant submanifolds in complex space forms. Kodai Math. J. 26, 85–94 (2003) 32. Mihai, A.: B.-Y. Chen inequalities for slant submanifolds in generalized complex space forms. Radovi Matematicki 12, 215–231 (2004) 33. Mihai, A., Mihai, I.: Some Basic Inequalities on Slant Submanifolds in Space Forms. In: Chen, B.-Y., Hasan Shahid, M., Al-Solamy, F. (eds.), Contact Geometry of Slant Submanifolds. Springer, Berlin (2022) 34. Mihai, A., R˘adulescu, I.N.: An improved Chen-Ricci inequality for Kählerian slant submanifolds in complex space forms. Taiwanese J. Math. 16(2), 761–770 (2012) 35. Mihai, I.: Ideal Kählerian slant submanifolds in complex space forms. Rocky Mountain J. Math. 35(3) (2005) 36. Oiaga, A., Mihai, I.: B. Y. Chen inequalities for slant submanifolds in complex space forms. Demonstr. Math. XXXII(4) (1999) 37. Oprea, T.: On a geometric inequality. arXiv:math.DG/0511088 38. Papaghiuc, N.: Semi-slant submanifolds of a Kählerian manifold. An. Stiint. Al. I. Cuza Univ. Iasi 40, 55–61 (1994) 39. Shahid, M.H.: CR-submanifolds of Kählerian product manifold. Indian J. Pure Appl. Math. 23(12), 873–879 (1992) 40. Shukla, S.S., Rao, P.K.: B.-Y. Chen inequalities for bi-slant submanifolds in generalized complex space forms. J. Nonlinear Sci. Appl. 3(4), 282–293 (2010) 41. Siddiqui, A.N., Ozgur, C.: An integral formula for bi-slant submanifolds in complex space forms. Hacettepe J. Math. Stat. 49(6), 2028–2036 (2020) 42. Siddiqui, A.N., Shahid, M.H., Jamali, M.: Simons type formula for Kählerian slant submanifolds in complex space forms. Kyungpook Math. J. 58, 149–165 (2018) 43. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968) 44. Senlin, X., Yilong, N.: Submanifoldsof product Riemannian manifold. Acta Math. Sci. Ser. B Engl. Ed. 20(2), 213–218 (2000) 45. Song, A., Liu, X.: Some inequalities of slant submanifolds in generalized complex space forms. Tamkang J. Math. 36(3), 223–229 (2005) 46. Tripathi, M.M.: On CR-submanifolds of nearly and closely cosympletic manifolds. Ganita 51(1), 45–56 (2000) 47. Yano, K., Kon, M.: Structures on Manifolds. Worlds Scientific, Singapore (1984)

Lorentzian Slant Submanifolds in Indefinite Kähler Manifolds Yu Fu and Dan Yang

2000 Mathematics Subject Classification. 53D12 · 53C40 · 53C42

1 Introduction The theory of submanifolds in Kähler manifolds is one of the most interesting topics in differential geometry. As a natural generalization of both complex and totally real submanifolds, B. Y. Chen in 1990 introduced a well-known notion—slant submanifolds in Kähler manifolds. Since then, the field of studying slant submanifolds has been developed as an important research direction by geometers from all over the world (c.f. [3, 4]). Some interesting geometric properties and structures of slant submanifolds were investigated, which enrich greatly the geometry of submanifolds in Kähler manifolds. We denote M˜ in (4c) by a simply connected indefinite complex space form with complex dimension n and complex index i, where the complex index is the complex dimension of the largest complex negative definite subspace of the tangent space. When n = 2m and the complex index is m, the indefinite Kähler manifold M˜ m2m is called a neutral Kähler manifold. If i = 1, M˜ 1n (4c) is Lorentzian. It is well known that a neutral Kähler surface is nothing but a Lorentzian Kähler surface. Thus, M˜ in (4c) is endowed with an almost complex structure J and with an indefinite Riemannian metric g, ˜ which is J -Hermitian, i.e., for all p ∈ M˜ in and n X, Y ∈ T p M˜ i , we have

Y. Fu (B) School of Mathematics, Dongbei University of Finance and Economics, Dalian 116025, People’s Republic of China e-mail: [email protected] D. Yang School of Mathematics, Liaoning University, Shenyang 110044, People’s Republic of China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_11

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g(J ˜ X, J Y ) = g(X, ˜ J ), ∇˜ J = 0, where ∇˜ is the Levi-Civita connection of g. ˜ The simplest example of neutral Kähler manifolds is the neutral complex space 2m denote the complex 2m-space with form C2m m , which is defined as follows: Let C complex coordinates z 1 , . . . , z m , z m+1 , . . . , z 2m . Then C2m endowed with gm,2m , i.e., the real part of the Hermitian form bm,2m (z, ω) = −

m  k=1

z¯k ωk +

2m 

z¯j ω j , z, ω ∈ C2m ,

(1)

j=m+1

defines a flat indefinite complex space form of complex index m. We denote the pair (C2m , gm,2m ) by C2m m briefly, which is the flat Lorentzian complex 2m-space. In particular, C21 is the flat complex Lorentzian plane. It is easy to see that the curvature tensor R˜ of the ambient space M˜ in (4c) is ˜ R(X, Y )Z = c{Y, Z X − X, Z Y + J Y, Z J X −J X, Z J Y + 2X, J Y J Z }.

(2)

Let us consider the differentiable manifold: 4m+1 S2m (c) = {z ∈ C2m+1 ; bm,2m+1 (z, z) = c−1 > 0}. m

This is an indefinite real space form with constant curvature c. The Hopf-fibration 4m+1 π : S2m (c) → C Pm2m (4c) : z  → z · C∗

is a submersion and there is a unique pseudo-Riemannian metric with complex index m on C Pm2m (4c) such that π is a Riemannian submersion. In a similar way, if c < 0, consider 4m+1 −1 (c) = {z ∈ C2m+1 < 0}, H2m m+1 ; bm+1,2m+1 (z, z) = c

which is an indefinite real space form of constant sectional curvature c. The Hopffibration 4m+1 π : H2m (c) → C Hm2m (4c) : z  → z · C∗

is a submersion and there is a unique pseudo-Riemannian metric with complex index m on C Hm2m (4c) such that π is a Riemannian submersion. We know that a complete simply connected neutral complex space form M˜ m2m (4c) 2m 2m is holomorphic isometric to C2m m , C Pm (4c), or C Hm (4c) for c = 0, c > 0 or c < 0, respectively.

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A real submanifold in a Kähler manifold with almost complex structure J is called slant if its Wirtinger angle is always constant (c.f. [3, 6]). It is obvious that complex submanifolds and totally real submanifolds are special classes of slant submanifolds. A slant submanifold is called proper if it is neither a complex submanifold nor a totally real submanifold. From the J -action point of views, slant submanifolds are the simplest and the most ˜ g, natural submanifolds of an indefinite Kähler manifolds ( M, ˜ J ). Slant submanifolds arise naturally and play extremely important roles in the study of submanifolds of Kähler manifolds. A remarkable result due to Kenmotsu and Zhou [23] says that every surface in a complex space form M˜ 2 (4c) is proper slant if it has constant curvature and nonzero parallel mean curvature vector. In the last two decades, the differential geometry of Lorentzian surfaces in any Lorentzian kähler surfaces were studied by B. Y. Chen and other geometers. Many interesting theories and classification results are obtained, see [7, 9, 13, 14]. In general, Lorentzian geometry is a useful field of mathematical research that builds the mathematical foundation of the general theory of relativity. In the case of Lorentzian surfaces in Lorentzian kähler surfaces, especially, Chen [9] discovered a surprising phenomenon that Ricci equation is a consequence of Gauss and Codazzi equations. This phenomenon means that the geometry of Lorentzian surfaces in a kähler surface carry much interesting geometric properties and very different from surfaces in a Riemannian 4-manifold. Generally, there exist many proper slant surfaces in complex projective plane C P 2 and complex hyperbolic plane C H 2 , for instance, see [3, 6]. However, interestingly there does not exist proper slant surface in a non-flat Lorentzian complex space form [17, 22], see Theorem 3.3 in Sect. 3. This is an interesting phenomenon for Lorentzian slant surfaces in the Lorentzian complex space forms, which shows the difference between Riemannian geometry and Lorentzian geometry. In Sect. 4, we survey some classification results for slant surfaces in Lorentzian complex space forms developed by some geometers in recent years. It includes mainly the classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, biharmonic slant surfaces and quasi-biharmonic slant surfaces in the Lorentzian complex space forms. It should be pointed out that some nonlinear hyperbolic equations play an important role in the classification problem of slant submanifolds in Lorentzian complex space forms. For example, the nonflat minimal slant surfaces in Lorentzian complex plane C21 could be constructed via the solution of a modified Liouville-type hyperbolic equation ([8], [1]). Also, minimal slant surfaces with non-constant Gauss curvature in Lorentzian complex projective plane C P12 (4) and complex hyperbolic plane C H12 (−4) are constructed via the solution of a nonlinear Klein-Gordon-type equation (see [10]).

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2 Preliminaries Let M be a submanifold of an indefinite Kähler manifold M˜ in equipped with an almost structure J and metric g. ˜ Let  ,  denote the inner product associated with g. ˜ ˜ respectively. We denote the Levi-Civita connections of M and M˜ in by ∇ and ∇, The formulas of Gauss and Weingarten are written, respectively, by (see [2, 4]) ∇˜ X Y = ∇ X Y + h(X, Y ), ∇˜ X ξ = −Aξ X + D X ξ for any vector fields X, Y tangent to M and ξ normal to M. Note that here D, h and A are the normal connection, the second fundamental form and the shape operator, respectively. Moreover, the following relation concerning h and A is essential: h(X, Y ), ξ = Aξ X, Y ,

(3)

where X, Y tangent to M and ξ normal to M. A vector v is called spacelike or timelike if v, v > 0 or v, v < 0, respectively. A nonzero vector v is called lightlike if it satisfies v, v = 0. For M in M˜ in , the mean curvature vector is given by H=

1 trace h. m

(4)

A submanifold M is called minimal or quasi-minimal (or marginally trapped) if its mean curvature vector is zero or lightlike at each point on M, respectively. Let LC denote the light cone in Cin , i = 1, 2, which is defined by LC = {z ∈ Cin : z, z = 0}. A curve z is called null if z , z  = 0 holds identically. We recall the Gauss, Codazzi and Ricci equations as follows: ˜ R(X, Y )Z , W  =  R(X, Y )Z , W  + h(Y, Z ), h(X, W ) − h(X, Z ), h(Y, W ),

(5)

˜ ¯ ¯ ( R(X, Y )Z )⊥ = (∇h)(X, Y, Z ) − (∇h)(Y, X, Z ),

(6)

¯ R D (X, Y )ξ, η =  R(X, Y )ξ, η + [ Aξ , Aη ]X, Y ,

(7)

¯ is defined by where X, Y, Z , W are tangent vectors on M. Here, ∇h ¯ (∇h)(X, Y, Z ) = D X h(Y, Z ) − h(∇ X Y, Z ) − h(Y, ∇ X Z ) ¯ = 0. and M is called parallel in M˜ in if ∇h

(8)

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3 Some Basic Results on Lorentzian Slant Surfaces In this section, we recall some basic results for Lorentzian slant surfaces in a Lorentzian Kähler surface ( M˜ 12 , g, J ) (see [10, 17, 22] for example). For any tangent vector X of M, we have the following decomposition form J X = P X + F X,

(9)

where P X and F X are the tangential and normal components of J X . We choose a pseudo-orthonormal local frame {e1 , e2 } on the Lorentzian surface M such that e1 , e1  = e2 , e2  = 0, e1 , e2  = −1.

(10)

Hence, it follows from (9), (10) and J X, J Y  = X, Y  that Pe1 = (sinh θ)e1 ,

Pe2 = −(sinh θ)e2

(11)

for some smooth function θ defined on M. It is well known that this function θ is the Wirtinger angle of M. In the case that the Wirtinger angle θ is constant on M, the Lorenzian surface M is called a slant surface and θ is called as the slant angle [3]. In this sense, the slant surface is then called θ-slant. Moreover, a θ-slant surface is called Lagrangian if θ = 0 and proper slant if the slant angle θ  = 0. Letting e3 = (sech θ)Fe1 , e4 = (sech θ)Fe2 , from (9), (10) and (11) we have J e1 = sinh θe1 + cosh θe3 , J e2 = − sinh θe2 + cosh θe4 , J e3 = − cosh θe1 − sinh θe3 , J e4 = − cosh θe2 + sinh θe4 , e3 , e3  = e4 , e4  = 0, e3 , e4  = −1.

(12) (13) (14)

Such a frame {e1 , e2 , e3 , e4 } is an adapted pseudo-orthonormal frame for M in M˜ 12 . Lemma 3.1 ([17]) Let M is a slant surface in a Lorentzian Kähler surface M˜ 12 . We have ∇ X e1 = ω(X )e1 , D X e3 = (X )e3 , for some 1-forms ω,  on M.

∇ X e2 = −ω(X )e2 , D X e4 = −(X )e4

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If we put h(ei , e j ) = h i3j e3 + h i4j e4 , then the following basic results are obtained in [17]. Lemma 3.2 ([17]) For a θ-slant surface in a Lorentzian Kähler surface M˜ 12 , with respect to an adapted pseudo-orthonormal frame we have ω j −  j = 2h 31 j tanh θ, A F X Y = A FY X, Ae3 e j = h 31 j e1 + h 41 j e2 , Ae4 e j = h 3j2 e1 + h 4j2 e2 for any X, Y ∈ T M and j = 1, 2, where ω j = ω(e j ) and  j = (e j ). As we know well, there are lots of proper slant surfaces in C P 2 and C H 2 , see [3, 6]. However, we will show that the Lorentzian case is quite different. In fact, it is interestingly proved in [22] (see also [7, 17] for special cases) that Theorem 3.3 ([22]) Any slant surface in a non-flat Lorentzian complex space form M˜ 12 (4c) has to be Lagrangian.

4 Classification Results of Lorentzian Slant Surfaces 4.1 Minimal Slant Surfaces in C21 The study of minimal surfaces has a very long history. Many interesting and important results on minimal surfaces in various ambient spaces had appeared in the past several centuries. From the view of differential geometry, one of the most interesting problems concerning the study of minimal surfaces is to construct concrete examples of minimal surfaces. Minimal flat Lagrangian surfaces in the Lorentzian complex plane C21 were classified by B. Y. Chen and L. Vrancken in [20]. Clearly, Lagrangian surfaces in C21 are Lorentzian surfaces automatically. Minimal flat slant surfaces in Lorentzian complex plane C21 , Lorentzian complex projective plane C P12 and Lorentzian complex hyperbolic plane C H12 were classified by B. Y. Chen in [7]. Theorem 4.1 ([7]) Every minimal flat θ-slant surface in C21 is either an open portion of a totally geodesic slant plane or congruent to the surface defined by  y L(x, y) = x + cosh2 θ + (i − sinh θ) f (y), x − y 2  y + cosh2 θ + (i − sinh θ) f (y) − i y sinh θ 2 for some function f (y).

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The following theorems completely classify minimal flat slant surfaces in the Lorentzian complex projective plane C P12 (4) and complex hyperbolic plane C H12 (−4), respectively. Theorem 4.2 ([7]) If L : M → C P12 (4) is a minimal flat slant surface in the Lorentzian complex projective plane C P12 (4), then L is Lagrangian. Moreover, the immersion is congruent to π ◦  L, where √  3  i 2 1 √ i (x−a 2 y)  2e 2a cosh (x + a 2 y) , e a (a y−x) , L(x, y) = √ 2a 3 √ √ i (x−a 2 y)  3  2e 2a sinh (x + a 2 y) , 2a a is a nonzero real number and π : S25 (1) → C P12 (4) is the Hopf-fibration. Theorem 4.3 ([7]) If L : M → C H12 (−4) is a minimal flat slant surface in the Lorentzian complex hyperbolic plane C H12 (−4), then L is Lagrangian. Moreover, the immersion is congruent to π ◦  L, where √ √  3  i 2 1 i 2 − (x+a y)  2e 2a cosh (x − a 2 y) , e a (a y+x) , L(x, y) = √ 2a 3 √ √ − i (x+a 2 y)  3  2e 2a sinh (x − a 2 y) , 2a a is a nonzero real number and π : H25 (−1) → C H12 (−4) is the Hopf-fibration. Later, Chen et al. [1] considered the remaining cases. The authors characterized the nonflat minimal slant surfaces in Lorentzian complex space forms. Theorem 4.4 ([1]) Let M be a nonflat minimal slant surface in the Lorentzian complex plane C21 . Then, up to rigid motions of C21 , M is locally given by  x L(x, y) = x0

f (x)d x i − √ f (x) 2 cosh θ

1 + i sinh θ − 2b cosh2 θ +

y y0

b(sinh θ − i) cosh θ

x0

dy √ , h (y)

y y0

x

dx b(sinh θ − i) + √ cosh θ f (x) x x0

f (x)d x i + √ f (x) 2 cosh θ

h(y)dy 1 + i sinh θ + √ h (y) 2b cosh2 θ

y √ y0

y y0

h(y)dy √ h (y)

x √ x0

dx f (x)

dy , h (y)

where θ is the Wirtinger angle, b is a nonzero real number, f (x) and h(y) are two differentiable functions with f (x) > 0 and h (y) > 0 on two open interval I1  x0 and I2  y0 , respectively.

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Remark 4.5 Note that the nonflat minimal slant surfaces defined in the above theorem were constructed via the solution of a modified Liouville hyperbolic type equation (cf. [1, 8]) (ln δ)x y = −δ 2 , whose exact solution is given by √ b f (x)k (y) δ= 2 , b k(y) − f (x) where b is a nonzero real number and f (x) and k(y) are differentiable functions satisfying f (x)k (y) > 0 on some open rectangle I1 × I2 . Moreover, nonflat minimal slant surfaces in the Lorentzian complex projective plane C P12 (4) and the complex hyperbolic plane C H12 (−4) were classified by Chen in [10], which provides a nice method to construct minimal Lagrangian immersions (containing no open subset of constant curvature) in the projective and hyperbolic Lorentzian complex planes via the solution of a nonlinear Klein-Gordon-type equation 1 (ln K )x y = ± − K 2 . K

4.2 Quasi-minimal Slant Surfaces A surface in a pseudo-Riemannian manifold is called quasi-minimal (or marginally trapped) if its mean curvature vector field is lightlike. In the theory of cosmic black holes, a marginally trapped surface in a space-time plays an extremely important role. From the viewpoint of differential geometry, some interesting classification results on quasi-minimal (or marginally trapped) surfaces have been obtained by some geometers (see [11, 12, 16]). In particular, Chen and Dillen [12] contributed a complete classification of quasi-minimal Lagrangian surfaces in the Lorentzian complex space forms. In 2009, Chen and Mihai [17] contributed a complete classification of quasiminimal surfaces in Lorentzian complex plane. In fact, there exist five large families of quasi-minimal proper slant surfaces in C21 . Theorem 4.6 ([17]) Up to rigid motions of C21 , every quasi-minimal proper slant surface in C21 is given by one of the following five families: (1) A flat θ-slant surface defined by L(x, y) = z(x)e(i−sinh θ)y , where z(x) is a null curve in the light cone LC satisfying z, i z  = 1.

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(2) A flat θ-slant surface defined by L(x, y) = z(x)y 2 (1− sinh θ ) , 1

i

where z(x) is a null curve in the light cone LC satisfying z, i z  = 2 sinh θ. (3) A flat θ-slant surface defined by L(x, y) =



2 sinh θ y

1 i 2 ( sinh θ −1)

y

yγ(y)dy + y

1 i 2 ( sinh θ −1)



1

y

γ(y)y 2 (3− sinh θ ) dy 1

i

1

y y i s 1 i 1 i − − yγ(y)dy + y 2 ( sinh θ −1) γ(y)y 2 (3− sinh θ ) dy , 2 2 sinh θ 1 1

i s + − , 2 2 sinh θ where γ(y) is an arbitrary function defined on an open interval I  1. (4) A flat θ-slant surface defined by 





 y + (sinh θ − i) (y) e(i+sinh θ)F(u) du dy 0 0 y  1   y (i+sinh θ)F(y) e dy (y)dy , +i + (1 + i sinh θ) 2 0 0 y  y  (i−sinh θ)F(y) (i+sinh θ)F(u) + (sinh θ − i) (y) e du dy xe 0 0 y  y  1 

+ (i − sinh θ) e(i+sinh θ)F(y) dy (y)dy , +i 2 0 0

L(x, y) = xe

(i−sinh θ)F(y)

y

y where F(y) := 0 μ(y)dy, (y) := ϕ(y)e−2F(y) sinh θ and μ(y), ϕ(y) are two functions defined on an open interval I  0. (5) A non-flat θ-slant surface defined by

y

L(x, y) =

φz − φx z + cosh2 θe− 3 y sinh θ z 4

(sinh θ + i)e−y(i+ 3 sinh θ) 1

0

dy + z(x)e y(i−sinh θ) ,

where z is a null curve in C21 satisfying z , z  = 0 and z , i z  = cosh2 θ, and φ is a solution of the second-order differential equation φx x − p(x)φ = cosh2 θe− 3 y sinh θ , 4

and p(x) is defined on an open interval I  0. Here φ is not the product of two functions of a single variable.

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4.3 Slant Surfaces with Parallel Mean Curvature Vector A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel in the normal bundle. Ofcourse, minimal submanifolds have parallel mean curvature vector automatically. Submanifolds with parallel mean curvature vector are important as they are critical points of some natural functionals. In the mid 1960s, S. S. Chern firstly introduced the notion of parallel mean curvature vector as the natural extension of constant mean curvature for hypersurfaces. In 1990, B. Y. Chen studied slant surfaces with nonzero parallel mean curvature vector and proved that a slant surface with parallel mean curvature vector in complex Euclidean plane C21 is either an open portion of the product surface of two plane circles, or an open portion of a circular cylinder which is contained in a hyperplane. Lorentzian slant surfaces in C21 with parallel mean curvature vector were studied by Arslan et al. in [1]. In order to clarify the properties of Lorentzian slant surfaces, here we provide two results on the geometry of Lorentzian slant surfaces with parallel mean curvature vector in C21 . Theorem 4.7 Every proper slant surface with nonzero parallel mean curvature vector in C21 is quasi-minimal. Proof Let M be a θ-slant surface with D H = 0 in C21 . Assume that M is not minimal. There is a pseudo-orthonormal local frame field {eˆ1 , eˆ2 } such that eˆ1 , eˆ1  = eˆ2 , eˆ2  = 0,

eˆ1 , eˆ2  = −1,

H = −h(eˆ1 , eˆ2 ).

(15) (16)

ˆ Since Assume that h(eˆ1 , eˆ2 ) = αF ˆ eˆ1 + βˆ F eˆ2 for some real-valued functions α, ˆ β. M is not minimal, without loss of generality, we assume αˆ is not vanishing. By putting e1 = αˆ eˆ1 , e2 = αˆ −1 eˆ2 , (15) and (16) become e1 , e1  = e2 , e2  = 0,

e1 , e2  = −1,

h(e1 , e2 ) = Fe1 + β Fe2 ,

(17) (18)

where β = βˆ αˆ −1 . Applying (10) and the total symmetry of h(X, Y ), F Z , we have h(e1 , e1 ) = β Fe1 + λFe2 , h(e1 , e2 ) = Fe1 + β Fe2 , h(e2 , e2 ) = γ Fe1 + Fe2

(19)

for some real-valued functions α, β, γ, λ. Then the mean curvature vector is given by H = −h(e1 , e2 ) = −Fe1 − β Fe2 .

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It follows from (19), Lemma 3.1, and Codazzi equation that ˜ 1 , e2 )e1 )⊥ = (∇¯ e1 h)(e1 , e2 ) − (∇¯ e2 h)(e1 , e1 ) ( R(e = e1 (α)Fe1 + e1 (β)Fe2 + α1 Fe1 − β1 Fe2 − e2 (β)Fe1 − e2 (λ)Fe2 − β2 Fe1 + λ2 Fe2 + 2ω2 (β Fe1 + λFe2 ), ⊥ ˜ 2 , e1 )e2 ) = (∇¯ e2 h)(e1 , e2 ) − (∇¯ e1 h)(e2 , e2 ) ( R(e = e2 (α)Fe1 + e2 (β)Fe2 + α2 Fe1 − β2 Fe2

(20)

(21)

− e1 (γ)Fe1 − e1 (α)Fe2 − γ1 Fe1 + α1 Fe2 − 2ω1 (γ Fe1 + αFe2 ). On the other hand, by applying (2) and (12)–(13) we have ˜ 1 , e2 )e1 )⊥ = 3c(sinh θ)Fe1 , ( R(e ˜ 2 , e1 )e2 )⊥ = 3c(sinh θ)Fe2 . ( R(e Thus, combining (20), (21) with (22) gives ⎧ −3c sinh θ + e1 (α) − e2 (β) + α1 − β2 + 2βω2 = 0, ⎪ ⎪ ⎪ ⎨e (β) − e (λ) − β + λ + 2λω = 0, 1 2 1 2 2 ⎪ (β) − e (α) − β + α1 − 2αω1 = 0, −3c sinh θ + e 2 1 2 ⎪ ⎪ ⎩ e2 (α) − e1 (γ) + α2 − γ1 − 2γω1 = 0.

(22)

(23)

It follows from Lemma 3.1 that the assumption D H = 0 reduces to (X )Fe1 − β(X )Fe2 + X (β)Fe2 = 0

(24)

for arbitrary tangent vector field X , which implies immediately that β is a constant and (X ) = 0. Then equations (23) become ⎧ βω2 = 0, ⎪ ⎪ ⎪ ⎨−e (λ) + 2λω = 0, 2 2 (25) ⎪ = 0, ω 1 ⎪ ⎪ ⎩ e1 (γ) + 2γω1 = 0. Furthermore, Lemma 3.2 yields ω1 = 2β sinh θ, ω2 = 2 sinh θ.

(26)

Combining (26) with (25) gives β sinh θ = 0. Therefore, the surface is quasi-minimal if θ  = 0.



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Theorem 4.8 Every slant surface with nonzero parallel mean curvature vector in C21 is flat. Proof If θ = 0, then M is Lagrangian. If  = 0, (26) implies clearly that M is flat. If β = 0, then from (26) we have ω1 = 0,

ω2 = 2 sinh θ,

which show that the Levi-Civita connection of g satisfies ∇e1 e1 = ∇e1 e2 = 0, ∇e2 e1 = 2 sinh θe1 , ∇e2 e2 = −2 sinh θe2 . A direct computation about curvature tensor yields R = 0. Thus we complete the proof of Theorem.  Slant surfaces with parallel mean curvature vector in the Lorentzian complex plane C21 were completely classified by Arslan et al. in [1]. Theorem 4.9 ([1]) Up to rigid motions of C21 , every slant surface with nonzero parallel mean curvature vector in C21 is given by one of the following nine families: (1) A Lagrangian surface defined by L(s, y) = z(s)eiay , where a is a real number and z(s) is a null curve in the light cone LC satisfying z, i z  = 1/a. (2) A Lagrangian surface defined by 

y  1 y icy eicy  u(y)dy − e u(y)dy , 2cx − i + 2 L(x, y) = 2c c 0 0 y y 

icy  1 e icy u(y)dy − e u(y)dy , 2cx + i + 2 2c c 0 0 where c is a nonzero real number and u(y) is a nonzero real-valued function defined on an open interval I  0. (3) A Lagrangian surface defined by  L(x, y) = where b is a positive real number. (4) A Lagrangian surface defined by

x + by e2i(x−by) , √ , √ 2b 2 2b

Lorentzian Slant Submanifolds in Indefinite Kähler Manifolds

 L(x, y) =

339

e2i(x+by) x − by , √ , √ 2 2b 2b

where b is a positive real number. (5) A Lagrangian surface defined by √  i(1+a −1 )(ax+by) i(a −1 −1)(ax−by)

a e e , , L(x, y) = √ a + 1 a−1 2b where a and b are positive real numbers with a  = 1. (6) A Lagrangian surface defined by √  i(a −1 −1)(ax+by) i(1+a −1 )(ax−by)

a e e , , L(x, y) = √ a−1 a+1 2b where a and b are positive real numbers with a  = 1. (7) A Lagrangian surface defined by L(x, y) = e(i−a)x+(i−a

−1

)by

 e2ax −

−1 −1 (a + i)4 e2b ay 2ax e2a by , e − , 8b(1 + a 2 )2 8b

where a is a positive real number and b is a nonzero real number. (8) A proper θ-slant surface defined by (2y sinh θ − a cosh θ) 2 − 2 sinh θ , sinh θ − i 1

L(x, y) = z(x)

i

where a is a real number and z(x) is a null curve in the light cone LC satisfying z , i z = cosh2 θ. (9) A proper θ-slant surface defined by  L(x, y) =

1 cosh2 θ

0

y

u(y)(2y sinh θ − a cosh θ) 2 − 2 sinh θ dy 3

i

 i 1 i + (2y sinh θ − a cosh θ) 2 − 2 sinh θ x + 2 y  1 u(y)(2y sinh θ − a cosh θ)dy , + cosh2 θ 0 y 1 3 i u(y)(2y sinh θ − a cosh θ) 2 − 2 sinh θ dy cosh2 θ 0  i 1 i + (2y sinh θ − a cosh θ) 2 − 2 sinh θ x − 2 y 

1 − u(y)(2y sinh θ − a cosh θ)dy , cosh2 θ 0

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where a is a real number and u(y) is a nonzero real-valued function defined on an open interval I  0. Conversely, locally every Lorentzian θ-slant surfaces with D H = 0 in C21 is congruent to the six families of surfaces defined above.

4.4 Pseudo-umbilical Slant Surfaces Let us recall the definition of a pseudo-umbilical surfaces. A surface is called pseudoumbilical if the shape operator with respect to its mean curvature vector is proportional to the identity map (c.f. [2]). The notion of pseudo-umbilical submanifolds is a natural generalization of minimal submanifolds. In 2002, Chen [6] completely classified pseudo-umbilical slant submanifolds in Riemannian complex space forms. In 2011, the first author and Hou [22] studied pseudo-umbilical slant surfaces in Lorentzian complex space forms and obtained Theorem 4.10 ([22]) Any pseudo-umbilical slant surface in a Lorentzian complex space form M˜ 12 (4c) has constant Gaussian curvature c. Moreover, after solving some special differential equations the authors [22] classified completely pseudo-umbilical slant surfaces in Lorentzian complex space forms. In fact, it was proved that there exist two families of pseudo-umbilical slant surfaces in C21 , three families in C P12 , and three families in C H12 . The following theorem completely classifies pseudo-umbilical slant surfaces in C21 . Theorem 4.11 ([22]) Any pseudo-umbilical slant surface in C21 is locally given by one of the following two families: (1) A flat Lagrangian surface defined by L(x, y) =

1 1 1 e(ib+b)x+(−1+i)y + e(ib−b)x+(1+i)y , e(ib+b)x+(−1+i)y 2b 2 2b  1 − e(ib−b)x+(1+i)y 2

for b ∈ R \ 0. (2) A flat θ-slant surface defined by

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       a2 a a2 a 2 2 L(x, y) = (1 + i)e sinh θ+ cosh θ+ 4 − 2 +i x+ −sinh θ− cosh θ+ 4 − 2 +i y       a2 a a2 a 2 2 + (−m + ni)e sinh θ− cosh θ+ 4 − 2 +i x+ −sinh θ+ cosh θ+ 4 − 2 +i y ,       2 2 sinh θ+ cosh2 θ+ a4 − a2 +i x+ −sinh θ− cosh2 θ+ a4 − a2 +i y (1 + i)e        a2 a a2 a 2 2 + (n + mi)e sinh θ− cosh θ+ 4 − 2 +i x+ −sinh θ+ cosh θ+ 4 − 2 +i y

with m = − a 2c+4 and n = −

(a 2 +4)

ac sinh θ √ 2

4 cosh θ+a 2

, where a ∈ R and c ∈ R \ 0.

The following two theorems state all the pseudo-umbilical slant surfaces in the Lorentzian complex projective plane C P12 (4) and the Lorentzian complex hyperbolic plane C H12 (−4). Theorem 4.12 ([22]) Let M be a Lorentzian pseudo-umbilical θ-slant surface in C P12 (4). Then M is Lagrangian, with Gaussian curvature 1, and the immersion is congruent to π ◦ L, where π : S25 (1) → C P12 (4) is the Hopf-fibration and L : M → S25 (1) ∈ C31 is locally one of the following three families of surfaces: (1) A Lagrangian surface defined by  1 2 a 2 | b | ( a +i)s 2 cosh( a + 4t) , e 2 L(s, t) = √ 2 2 | 1 − beas | a +4  1 2 a 2 | b | ( a +i)s abeas + a  2 2 sinh( a + 4t) , −2i − e 2 | 1 − beas | beas − 1 1



with a, b ∈ R \ 0. (2) A Lagrangian surface defined by     1 | c | eis | c | eis cosh( 1 − c2 t) L(s, t) =  , sinh( 1 − c2 t) , −i + c tan(cs) | cos(cs) | | cos(cs) | 1 − c2

with 0 1 and c ∈ R. Theorem 4.13 ([22]) Let M be a Lorentzian pseudo-umbilical θ-slant surface in C H12 (−4). Then M is Lagrangian, with Gaussian curvature -1, and the immersion is congruent to π ◦ L, where π : H35 (−1) → C H12 (−4) is the Hopf-fibration and L : M → H35 (−1) ∈ C32 is locally one of the following three families of surfaces:

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(1) A Lagrangian surface defined by  abeas + a a 2 | b | ( a +i)s 1 2 2i + a + 4t) , , 2 cosh( e 2 L(s, t) = √ as 2 be − 1 2 | 1 − beas | a +4  1 2 a 2 | b | ( a +i)s  2 sinh( a + 4t) e 2 2 | 1 − beas | 1



with a, b ∈ R \ 0. (2) A Lagrangian surface defined by    1 | c | eis | c | eis  i − c tan(cs), cosh( 1 − c2 t) , sinh( 1 − c2 t) L(s, t) = √ | cos(cs) | | cos(cs) | 1 − c2

with 0 1 and c ∈ R.

4.5 Biharmonic and Quasi-biharmonic Slant Surfaces During the past twenty years, the theory of biharmonic submanifolds has been developed greatly. There are a lot of classification results and non-existence results on biharmonic submanifolds, see for instance, [16, 24], or a general reference [25] for biharmonic submanifolds due to Y. L. Ou and B. Y. Chen in 2020. We recall some known results in [21] on the bitension field of quasi-minimal slant surfaces in the Lorentzian complex space forms. Note that biharmonic quasi-minimal slant surfaces and quasi-biharmonic quasi-minimal slant surfaces in the Lorentzian complex space forms have been classified, which extend Sasahara’s results from Lagrangian case to slant case in the Lorentzian complex space forms. Also biharmonic quasi-minimal surfaces in pseudo-Euclidean spaces have been studied in [15, 16]. For a smooth map φ : (M n , g) −→ ( M˜ m , , ), the tension field τ (φ) is a section of the vector bundle φ∗ T M˜ defined by τ (φ) = trace∇dφ =

n  ei , ei {(∇eφi dφ(ei ) − dφ(∇ei ei )}, i=1

˜ which is the pullback where ∇ φ is the induced connection by φ on the bundle φ∗ T M, ˜ of ∇. If φ is an isometric immersion, then τ (φ) and the mean curvature vector field

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H of M are related by τ (φ) = n H.

(27)

If τ (φ) = 0 at each point on M, then φ is called a harmonic map. The harmonic map between two Riemannian manifolds is a critical point of the energy functional E(φ) =

1 2

|dφ|2 vg M

for a smooth map φ : (M n , g) −→ ( M˜ m , , ). The bitension field is given by τ2 (φ) =

n  ˜ φ ei , ei {(∇eφi ∇eφi − ∇∇e ei )τ + R M (τ , dφ(ei ))dφ(ei )}, i

(28)

i=1 ˜ ˜ where R M is the curvature tensor of M. In the case that φ is an isometric immersion and M˜ is the complex space form M˜ in (4c), it follows from (27) and (28) that

τ2 (φ) = −nH + 5ncH,

(29)

n where  = − i=1 ei , ei (∇˜ ei ∇˜ ei − ∇˜ ∇ei ei ). A smooth map φ is called biharmonic if τ2 (φ) = 0 at each point on M. It is easy to see that a harmonic map is trivially biharmonic. A biharmonic map φ : (M n , g) −→ ( M˜ m , , ) between Riemannian manifolds is a critical point of the bienergy functional E 2 (φ) =

1 2

|τ (φ)|2 vg . M

T. Sasahara proposed the notion of quasi-biharmonic submanifolds in [28] as follows: Definition 4.14 A pseudo-Riemannian submanifold M isometrically immersed in a pseudo-Riemannian manifold is called quasi-biharmonic if τ2 is lightlike at each point on M. From [28], we know that the class of quasi-biharmonic submanifolds are quite different from the class of biharmonic submanifolds. For quasi-minimal Lagrangian surfaces immersed into the Lorentzian complex space forms, Sasahara [28] proved that the formula  D H = −K H holds always. It is shown in [21] that the formula  D H = −K H holds as well for slant surfaces in the Lorentzian complex space forms.

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Theorem 4.15 ([21]) For a quasi-minimal slant surface in C21 , the Gauss curvature K and  D H are related by  D H = −K H. Consider biharmonic and quasi-biharmonic quasi-minimal slant surfaces in Lorentzian complex plane C21 . The following results were proved: Theorem 4.16 ([21]) Every biharmonic quasi-minimal θ-slant surface in C21 is locally given by a flat slant surface defined as L(x, y) = c1 xe(i−sinh θ)y + w(y), where c1 is lightlike vector, and w(y) is a null curve in C21 satisfying c1 (i − sinh θ)e(i−sinh θ)y , w (y) = 0, c1 e(i−sinh θ)y , w (y) = −β(y) for some nonzero real-valued function β(y). Theorem 4.17 ([21]) Every quasi-biharmonic quasi-minimal θ-slant surface in C21 is locally given by a flat slant surface defined as L(x, y) = p(x)e(i−sinh θ)y , where p(x) is a null curve lying in the light cone LC satisfying  p(x), p(x) =  p (x), p (x) = 0,  p(x), i p (x) = 1. Remark 4.18 Combining the above theorems with Sasahara’s results in [27, 28], one completes the classifications of biharmonic quasi-minimal slant surfaces and quasi-biharmonic quasi-minimal slant surfaces in the Lorentzian complex forms, respectively. As the above results are under the assumption that the surfaces are quasi-minimal, a natural problem arises: Problem: To classify biharmonic slant surfaces and quasi-biharmonic slant surfaces in the Lorentzian complex forms. Acknowledgements The authors would like to take this opportunity to appreciate Professor BangYen Chen for his directions, encouragement, and help in the past ten years. The first author is supported by the NSFC (No.11601068), the General Project for Department of Liaoning Education (No.LN2019J05), Liaoning Provincial Science and Technology Department Project (No.2020-MS340), and Liaoning BaiQianWan Talents Program. The second author is supported by the NSFC (No.11801246) and the General Project for Department of Liaoning Education (No. LJC201901).

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References 1. Arslan, K., Carriazo, A., Chen, B.Y., Murathan, C.: On slant submanifolds of neutral Kähler manifolds. Taiwanese J. Math. 14(2), 561–584 (2010) 2. Chen, B.Y.: Geometry of Submanifolds. M. Dekker, New York (1973) 3. Chen, B.Y.: Geometry of Slant Submanifolds. Universiteit Leuven, Belgium, Katholieke (1990) 4. Chen, B.Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. Word Scientific, Hackensack (2011) 5. Chen, B.Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. World Scientific, Hackensack (2015) 6. Chen, B.Y.: Classification of slumbilical submanifolds in complex space forms. Osaka J. Math. 39, 23–47 (2002) 7. Chen, B.Y.: Minimal flat Lorentzian surfaces in Lorentzian complex space forms. Publ. Math. (Debrecen) 73, 233–248 (2008) 8. Chen, B.Y.: Lagrangian minimal surfaces in Lorentzian complex plane. Arch. Math. 91, 366– 371 (2008) 9. Chen, B.Y.: Dependence of the Gauss-Codazzi equations and the Ricci equation of Lorentz surfaces. Publ. Math. (Debrecen) 74, 341–349 (2009) 10. Chen, B.Y.: Nonlinear Klein-Gordon equations and Lorentzian minimal surfaces in Lorentzian complex space forms. Taiwanese J. Math. 13(1), 1–24 (2009) 11. Chen, B.Y.: Black holes, Marginally trapped surfaces and quasi-minimal surfaces. Tamkang J. Math. 40, 313–341 (2009) 12. Chen, B.Y., Dillen, F.: Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms. J. Math. Phys. 48, 013509, 23 (2007) 13. Chen, B.Y., Dillen, F., Van der Veken, J.: Complete classification of parallel Lorentzian surfaces in Lorentzian complex space forms. Intern. J. Math. 21, 665–686 (2010) 14. Chen, B.Y., Fastenakels, J.: Classification of flat Lagrangian Lorentzian surfaces in complex Lorentzian plane. Acta Math. Sinica (Eng. Ser.) 23, 2111–2144 (2007) 15. Chen, B.Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. Ser. A 45(2), 323–347 (1991) 16. Chen, B.Y., Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu J. Math. 52(1), 167–185 (1998) 17. Chen, B.Y., Mihai, I.: Classification of quasi-minimal slant surfaces in Lorentzian complex space forms. Acta Math. Hungar. 122(4), 307–328 (2009) 18. Chen, B.Y., Tazawa, Y.: Slant submanifold of complex Projective and complex hyperbolic spaces. Glasgow Math. J. 42, 439–454 (2000) 19. Chen, B.Y., Tazawa, Y.: Slant submanifolds in complex Euclidean spaces. Tokyo J. Math. 14, 101–120 (1991) 20. Chen, B.Y., Vrancken, L.: Lagrangian minimal isometric immersions of a Lorentzian real space form into a Lorentzian complex space form. Tohoku Math. J. 54, 121–143 (2002) 21. Fu, Y.: Biharmonic and quasi-biharmonic slant surfaces in Lorentzian complex space forms. Abstr. Appl. Anal. ID 412709, 7 (2013) 22. Fu, Y., Hou, Z.H.: Classification of pseudo-umbilical slant surfaces in Lorentzian complex space forms. Taiwanese J. Math. 15(5), 1919–1938 (2011) 23. Kenmotsu, K., Zhou, D.: Classification of the surfaces with parallel mean curvature vector in two dimensional complex space forms. Am. J. Math. 122(8), 295–317 (2000) 24. Oniciuc, C.: Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 48(2), 237–248 (2002) 25. Ou, Y.L., Chen, B.Y.: Biharmonic Submanifolds and Biharmonic Maps in Riemannian Geometry. World Scientific Publishing, Hackensack (2020) 26. Sasahara, T.: Quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors. Demonstr. Math. 38, 185–196 (2005) 27. Sasahara, T.: Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms. Glasgow Math. J. 49, 487–507 (2007)

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Slant Lightlike Submanifolds of Indefinite Kaehler Manifolds and Their Warped Product Manifolds Rashmi Sachdeva, Garima Gupta, Rachna Rani, Rakesh Kumar, and Akhilesh Yadav

2010 Mathematics Subject Classification: 53C15 · 53C40 · 53C50

1 Introduction As a generalization of both holomorphic and totally real immersions, slant immersions of a Riemannian manifold into an almost Hermitian manifold were introduced by Chen in [8, 9]. The study of slant immersions into complex Euclidean spaces C2 and C4 was presented by Chen and Tazawa in [7, 10], while slant immersions of Kaehler C-spaces into complex projective spaces were given by Maeda et al. in [29]. Slant immersions of a Riemannian manifold into an almost contact metric manifold were introduced by Lotta in [27]. Then the geometry of slant immersions and slant submanifolds attained momentum after the research on this subject matter by [2, 30, 38] and many other references therein. It is well known that for a semi-Riemannian manifold there is a natural existence of lightlike (null) subspaces, therefore, Duggal and Bejancu [14] introduced the geomR. Sachdeva · G. Gupta · R. Kumar (B) Department of Basic and Applied Sciences, Punjabi University, Patiala, India e-mail: [email protected] G. Gupta e-mail: [email protected] R. Rani Department of Mathematics, University College, Ghanaur, Patiala, India e-mail: [email protected] A. Yadav Department of Mathematics, Banaras Hindu University, Varanasi, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 B.-Y. Chen et al. (eds.), Complex Geometry of Slant Submanifolds, https://doi.org/10.1007/978-981-16-0021-0_12

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etry of lightlike submanifolds of semi-Riemannian manifolds to fill an important missing part in the general theory of submanifolds. Since the geometry of lightlike submanifolds has extensive uses in general theory of relativity and has potential applications in the study of asymptotically flat spacetimes, radiation and electromagnetic fields and event horizons of the Kerr and Kruskal black holes [14, 17, 23], hence, the notion of lightlike submanifolds becomes the demand of the present scenario. On the other hand, most of the authors studied the geometry of slant submanifolds in complex or contact geometry endowed with a definite metric. Since most of the branches of physics and mathematics use the geometry of manifolds and their submanifolds endowed with indefinite metric (metric with non-zero index), therefore, the notion of slant submanifolds endowed with definite metric may not be useful there. Thus, the notion of slant lightlike submanifolds of indefinite Hermitian manifolds was introduced by Sahin in [37] and further studied in [32, 33]. In this chapter, we study the geometry of slant lightlike submanifolds of indefinite Kaehler manifolds and discuss the necessary and sufficient conditions for the existence of such submanifolds. We study totally umbilical slant lightlike submanifolds of indefinite Kaehler manifolds and show that there do not exist totally umbilical proper slant lightlike submanifolds in indefinite Kaehler manifolds other than totally geodesic proper slant lightlike submanifolds. Further, we study hemi-slant lightlike submanifolds and define the axiom of indefinite hemi-slant 3-planes and 3-spheres for an indefinite almost Hermitian manifold with lightlike submanifolds. We obtain some characterization theorems for the non-existence of warped product slant lightlike submanifolds of indefinite Kaehler manifolds. We study pointwise slant lightlike submanifolds of indefinite Hermitian manifolds and obtain some conditions for a pointwise slant lightlike submanifold to be a slant lightlike submanifold of an indefinite Kaehler manifold. We provide some non-trivial examples of pointwise slant lightlike submanifolds and also study totally umbilical pointwise slant lightlike submanifolds.

2 Lightlike Submanifolds Let (M, g) be an m−dimensional submanifold of a semi-Riemannian manifold ¯ g) ( M, ¯ of real dimension (m + n), where the metric g¯ is of non-zero index q. If g¯ is degenerate on the tangent bundle T M of M, then the tangent space T p M and normal space T p M ⊥ are also degenerate and not complementary to each other. So, there exists a subspace Rad(T p M) = T p M ∩ T p M ⊥ , called a radical subspace. If Rad(T M) : p ∈ M −→ Rad(T p M) becomes a smooth distribution on M of rank r (> 0), then M is said to be an r −lightlike submanifold and the distribution Rad(T M) is said to be the radical distribution on M. The non-degenerate complementary subbundle of Rad(T M) in T M (respectively, in T M ⊥ ) is known as the screen distribution S(T M) (respectively, the screen transversal bundle S(T M ⊥ )) and then T M = Rad(T M)⊥S(T M) and T M ⊥ = Rad(T M)⊥S(T M ⊥ ). A vector bundle complementary to T M, but not orthogonal to it, is known as the transversal vector bundle tr (T M) and is given by tr (T M) =

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ltr (T M)⊥S(T M ⊥ ), where the bundle ltr (T M) is known as the lightlike transversal vector bundle which is complementary to Rad(T M) in S(T M ⊥ )⊥ . Hence, the tangent bundle T M¯ splits as T M¯ | M = T M ⊕ tr (T M) = S(T M)⊥(Rad(T M) ⊕ ltr (T M))⊥S(T M ⊥ ), (for details, see [14]). If U is a local coordinate neighborhood of M, then the local quasi-orthonormal field of frames on M¯ (along M) is {X r +1 , ..., X m , ξ1 , ..., ξr , Wr+1 , ..., Wn , N1 , ..., Nr }, where {Wα }nα=r +1 and m are orthonormal bases of (S(T M ⊥ )|U ) and (S(T M)|U ), respectively, {X a }a=r+1 r {Ni }i=1 and {ξi }ri=1 are the lightlike bases of (ltr (T M)|U ) and (Rad(T M)|U ), respectively and satisfy ¯ i , X a ) = g(N ¯ i , Wα ) = 0, g(N ¯ i , ξ j ) = δ ij . g(N ¯ i , N j ) = g(N

(1)

The following four cases of a lightlike submanifold occur: Case 1: Case 2: Case 3: Case 4:

r -lightlike submanifold if r < min{m, n}; Co-isotropic submanifold if r = n < m; Isotropic submanifold if r = m < n; Totally lightlike submanifold if r = m = n.

¯ Then the Gauss-Weingarten Assume that the Levi-Civita connection on M¯ is ∇. formulae, for X, Y ∈ (T M) and V ∈ (tr (T M)), are given by ∇¯ X Y = ∇ X Y + h(X, Y ), ∇¯ X V = −A V X + ∇ X⊥ V, where {h(X, Y ), ∇ X⊥ V } and {∇ X Y, A V X } belong to (tr (T M)) and (T M), respectively. It should be noted that the induced connection ∇ is not a metric connection on M. Here, h is a symmetric bilinear form on T M which is called the second fundamental form, and A V is linear operator on M which is called the shape operator. Let L and S be the projection morphisms of tr (T M) on ltr (T M) and S(T M ⊥ ), respectively, then we get ∇¯ X Y = ∇ X Y + h l (X, Y ) + h s (X, Y ),

(2)

∇¯ X V = −A V X + DlX V + D sX V, where L(h(X, Y )) = h l (X, Y ) ∈ (ltr (T M)), S(h(X, Y )) = h s (X, Y ) ∈  l s ⊥ ⊥ ⊥ (S(T M )), L(∇ X V ) = D X V and S(∇ X V ) = D X V . Particularly, for N ∈ (ltr (T M)) and W ∈ (S(T M ⊥ )), we have ∇¯ X N = −A N X + ∇ Xl N + D s (X, N ),

(3)

∇¯ X W = −A W X + ∇ Xs W + Dl (X, W ),

(4)

and using (2) and (4), we have g(A W X, Y ) = g(h ¯ s (X, Y ), W ) + g(Y, ¯ Dl (X, W )).

(5)

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Now, assume P¯ is the projection morphism of the tangent bundle of M on its screen distribution; then, using decomposition T M = Rad(T M)⊥S(T M), we can write ¯ ), ∇ X ξ = −A∗ξ X + ∇ X∗t ξ, ¯ = ∇ X∗ PY ¯ + h ∗ (X, PY ∇ X PY ¯ ), ∇ X∗t ξ } ∈ (Rad(T M)) and {∇ X∗ PY ¯ , A∗ξ X } ∈ (S(T M)). Furwhere {h ∗ (X, PY l ¯ ), ξ ) = g(A∗ξ X, PY ¯ ). ther, using (3), (4) and the last equation, we attain g(h ¯ (X, PY Denote by R¯ and R the curvature tensors of ∇¯ and ∇, respectively, then by straightforward calculations ([14]), we have ¯ R(X, Y )Z = R(X, Y )Z + Ahl (X,Z ) Y − Ahl (Y,Z ) X + Ah s (X,Z ) Y − Ah s (Y,Z ) X + (∇ X h l )(Y, Z ) − (∇Y h l )(X, Z ) + Dl (X, h s (Y, Z )) − Dl (Y, h s (X, Z )) + (∇ X h s )(Y, Z ) − (∇Y h s )(X, Z ) + D s (X, h l (Y, Z )) − D s (Y, h l (X, Z )),

(6)

where (∇ X h s )(Y, Z ) = ∇ Xs h s (Y, Z ) − h s (∇ X Y, Z ) − h s (Y, ∇ X Z ),

(∇ X h l )(Y, Z ) = ∇ Xl h l (Y, Z ) − h l (∇ X Y, Z ) − h l (Y, ∇ X Z ).

(7)

3 Slant Lightlike Submanifolds In this section, we present the study of slant lightlike submanifolds of an indefinite Hermitian manifold. To define the notion of slant submanifolds, one needs to consider an angle between two vector fields. A lightlike submanifold has two distributions, namely the radical distribution and the screen distribution. The radical distribution is totally lightlike, and therefore it is not possible to define the angle between two of its vector fields. On the other hand, the screen distribution is non-degenerate. Using these facts, the notion of a slant lightlike submanifold of an indefinite Hermitian manifold was introduced by Sahin in [37]. ¯ g, A 2n−dimensional semi-Riemannian manifold ( M, ¯ J¯) of constant index q, 0 < q < 2n, is called an indefinite almost Hermitian manifold if there exists a tensor field ¯ J¯ X, J¯Y ) = g(X, ¯ Y ), for any vector J¯ of type (1, 1) on M¯ such that J¯2 = −I and g( ¯ where I denotes the identity transformation fields X, Y of tangent bundle T M¯ of M, ¯ Let ( M, ¯ J¯, g) ¯ be an indefinite almost Hermitian manifold and ∇¯ be the Leviof T p M. Civita connection on M¯ with respect to g. ¯ Then M¯ is called an indefinite Kaehler ¯ that is, (∇¯ X J¯)Y = 0, for any manifold (see [3]) if J¯ is parallel with respect to ∇, ¯ X, Y ∈ (T M).

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We present the following important lemmas from [37] which are used to define the notion of slant lightlike submanifolds of indefinite Hermitian manifolds. Lemma 3.1 Let M be an r −lightlike submanifold of an indefinite Hermitian manifold M¯ of index 2q. Suppose that J¯ Rad(T M) is a distribution on M such that ¯ (T M) is a subbundle of the screen disRad(T M) ∩ J¯ Rad(T M) = {0}. Then Jltr ¯ ¯ tribution S(T M) and J Rad(T M) ∩ Jltr (T M) = {0}. Lemma 3.2 Under the hypothesis of Lemma 3.1, if r = q then any complementary ¯ (T M) in S(T M) is Riemannian. distribution to J¯(Rad(T M)) ⊕ Jltr Thus, using Lemma 3.2, the notion of a slant lightlike submanifold of a Hermitian manifold is given as below. Definition 3.3 ([37]) Let M be a q−lightlike submanifold of an indefinite Hermitian manifold M¯ of index 2q. Then M is said to be a slant lightlike submanifold of M¯ if the following conditions are satisfied: (A) Rad(T M) is a distribution on M such that J¯ Rad(T M) ∩ Rad(T M) = {0}. (B) For each non-zero vector field X tangent to D at p ∈ U ⊂ M, the angle θ (X ) between J¯ X and the vector space D p is constant, that is, it is independent of the choice of p ∈ U ⊂ M and X ∈ D p , where D is a complementary distribution ¯ (T M) in the screen distribution S(T M). to J¯ Rad(T M) ⊕ Jltr This constant angle θ (X ) is called the slant angle of the distribution D. A slant lightlike submanifold is said to be proper if D = {0} and θ = 0, π2 . It is known that a submanifold M is said to be an invariant (respectively, anti-invariant) submanifold if J¯ T p M ⊂ T p M, (respectively, J¯ T p M ⊂ T p M ⊥ ), for any p ∈ M. Therefore, from the above definition, it is clear that M is an invariant (respectively, anti-invariant) submanifold if θ (X ) = 0 (respectively, θ (X ) = π2 ). From the definition of a slant lightlike submanifold, it is clear that the tangent bundle T M of M is decomposed as follows: ¯ (T M))⊥D. T M = RadT M⊥S(T M) = RadT M⊥( J¯ RadT M ⊕ Jltr

(8)

¯ be a semi-Riemannian manifold, where R82 is Example 1 ([37]) Let M¯ = (R82 , g) a semi-Euclidean space of signature (−, −, +, +, +, +, +, +) with respect to the canonical basis {∂ x 1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ x7 , ∂ x8 }. Let M be a submanifold of R82 given by X (u, v, θ, t, s) = (u, v, sin θ, cos θ, −θ sin t, −θ cos t, u, s). Then, the tangent bundle T M is spanned by Z 1 = ∂ x1 + ∂ x7 , Z 2 = ∂ x2 Z 3 = cos θ ∂ x3 − sin θ ∂ x4 − sin t∂ x5 − cos t∂ x6 Z 4 = −θ cos t∂ x5 + θ sin t∂ x6 ,

Z 5 = ∂ x8 .

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Clearly, M is a 1-lightlike submanifold of R82 with Rad(T M) = span{Z 1 }. Here, J¯ Rad(T M) = span{Z 2 + Z 5 }, therefore, it is a distribution on M. Choose D = span{Z 3 , Z 4 }, which is Riemannian. Then M has a slant distribution with slant angle π4 , and the screen transversal bundle S(T M ⊥ ) is spanned by W1 = − csc θ ∂ x4 + sin t∂ x5 + cos t∂ x6 , W2 = (2 sec θ − cos θ )∂ x3 + sin θ ∂ x4 + sin t∂ x5 + cos t∂ x6 , which is also Riemannian and the lightlike transversal bundle ltr (T M) is spanned by 1 N = (−∂ x1 + ∂ x7 ). 2 Then it is easy to verify that J¯ N = −Z 2 + Z 5 ∈ (S(T M)) and g( J¯ N , J¯ Z 1 ) = 1. Thus, M is a proper slant lightlike submanifold of R82 . Definition 3.4 ([17]) Let M be a lightlike submanifold of an indefinite Hermitian ¯ g, manifold ( M, ¯ J¯). Then M is said to be an invariant (complex) lightlike submanifold of M¯ if J¯ Rad(T M) = Rad(T M) and J¯(S(T M)) = S(T M). It is easy to see that for an invariant lightlike submanifold M of an indefinite Hermitian ¯ g, manifold ( M, ¯ J¯), we have J¯(tr (T M)) = tr (T M). Definition 3.5 ([16]) Let M be a lightlike submanifold of an indefinite Hermi¯ g, tian manifold ( M, ¯ J¯). Then M is called a screen real submanifold of M¯ if ¯ J Rad(T M) = Rad(T M) and J¯(S(T M)) ⊂ S(T M ⊥ ). Then it follows that for a screen real submanifold M, J¯(ltr (T M)) = ltr (T M) and ¯ J (μ) = μ, where μ is the complementary orthogonal vector subbundle to J¯(S(T M)) in S(T M ⊥ ). From Definition 3.3 of a slant lightlike submanifold, it is clear that a slant light¯ g, like submanifold M of an indefinite Hermitian manifold ( M, ¯ J¯) does not include invariant and screen real submanifolds. Moreover, it is known that a C R−lightlike submanifold ([14]) of an indefinite Hermitian manifold also does not include invariant and screen real submanifolds. Therefore, Sahin [37] derived relations between C R−lightlike submanifolds and slant lightlike submanifolds as below. Theorem 3.6 Let M be a q−lightlike submanifold of an indefinite Kaehler manifold ¯ g, ( M, ¯ J¯) of index 2q. Then any coisotropic C R−lightlike submanifold is a slant lightlike submanifold with θ = 0. In particular, a lightlike real hypersurface of an indefinite Hermitian manifold M¯ of index 2 is a slant lightlike submanifold with θ = 0. Moreover, any C R−lightlike submanifold of M¯ with D0 = {0} is a slant lightlike submanifold with θ = π2 . Therefore, coisotropic C R−lightlike submanifolds, lightlike real hypersurfaces and C R−lightlike submanifolds with D0 = {0} are also the examples of slant lightlike submanifolds. For any X ∈ (T M), we write

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J¯ X = T X + F X,

(9)

where T X is the tangential component of J¯ X and F X is the transversal component of J¯ X . Similarly, for any V ∈ (tr (T M)), we write J¯ V = BV + C V,

(10)

where BV is the tangential component of J¯ V and C V is the transversal component of J¯ V . Using the decomposition in (8), we denote by P1 , P2 , Q 1 and Q 2 the projection ¯ (T M) and D, respectively. Then on the distributions Rad(T M), J¯ Rad(T M), Jltr for any X ∈ (T M), we can write X = P1 X + P2 X + Q 1 X + Q 2 X.

(11)

On applying J¯ to (11), we obtain J¯ X = J¯ P1 X + J¯ P2 X + F Q 1 X + T Q 2 X + F Q 2 X.

(12)

Then using (9) and (10), we get J¯ P1 X = T P1 X ∈ ( J¯ Rad(T M)),

J¯ P2 X = T P2 X ∈ (Rad(T M)),

F P1 X = F P2 X = 0, T Q 2 X ∈ (D),

F Q 1 X ∈ (ltr (T M)),

and T X = T P1 X + T P2 X + T Q 2 X.

(13)

Lemma 3.7 Let M be a slant lightlike submanifold of an indefinite Kaehler manifold ¯ g, ( M, ¯ J¯) then F Q 2 X ∈ (S(T M ⊥ )), for any X ∈ (T M). Proof Since tr (T M) = S(T M ⊥ )⊥ltr (T M), then, it follows that F Q 2 X ∈ (S (T M ⊥ )) if and only if g(F Q 2 X, ξ ) = 0, for any ξ ∈ (RadT M). Hence g(F Q 2 X, ξ ) = g( J¯ Q 2 X − T Q 2 X, ξ ) = g( J¯ Q 2 X, ξ ) = −g(Q 2 X, J¯ξ ) = 0, 

and this gives the result.

Thus from Lemma 3.7, it follows that F(D p ) is a subspace of S(T M ⊥ ). Therefore, there exists an invariant subspace μ p of T p M¯ such that

and

S(T p M ⊥ ) = F(D p )⊥μ p ,

(14)

T p M¯ = S(T p M)⊥{Rad(T p M) ⊕ ltr (T p M)}⊥{F(D p )⊥μ p }.

(15)

Differentiating (12) and using (2)–(4), (9) and (10), for any X, Y ∈ (T M), we have

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(∇ X T )Y = A F Q 1 Y X + A F Q 2 Y X + Bh(X, Y ), and D s (X, F Q 1 Y ) + Dl (X, F Q 2 Y ) = F∇ X Y − h(X, T Y ) + Ch(X, Y ) − ∇ Xs F Q 2 Y − ∇ Xl F Q 1 Y.

(16)

Next, we recall existence theorems for a slant lightlike submanifold of an indefinite ¯ g, Hermitian manifold ( M, ¯ J¯), from [37]. Theorem 3.8 Let M be a q−lightlike submanifold of an indefinite Hermitian man¯ g, ifold ( M, ¯ J¯) of index 2q. Then M is a slant lightlike submanifold of M¯ if and only if the following conditions are satisfied: ¯ (T M) is a distribution on M. 1. Jltr 2. There exists a constant λ ∈ [−1, 0] such that (T Q 2 )2 X = λX,

(17)

for any X ∈ (T M). Moreover, in such case, λ = − cos2 θ . Corollary 3.9 Let M be a slant lightlike submanifold of an indefinite Hermitian ¯ g, manifold ( M, ¯ J¯), then g(T Q 2 X, T Q 2 Y ) = cos 2 θ g(Q 2 X, Q 2 Y ),

(18)

g(F Q 2 X, F Q 2 Y ) = sin 2 θ g(Q 2 X, Q 2 Y ),

(19)

for any X, Y ∈ (T M). Theorem 3.10 Let M be a q−lightlike submanifold of an indefinite Hermitian man¯ g, ifold ( M, ¯ J¯) of index 2q. Then M is a slant lightlike submanifold of M¯ if and only if the following conditions are satisfied: ¯ (T M) is a distribution on M. 1. Jltr 2. There exists a constant μ ∈ [−1, 0] such that B F Q 2 X = μQ 2 X, for any X ∈ (T M). Moreover, in this case, μ = − sin2 θ.

(20)

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4 Totally Umbilical Slant Lightlike Submanifolds Sahin [39] proved that there do not exist totally umbilical proper slant submanifolds in Kaehler manifolds other than totally geodesic proper slant submanifolds. It is known that a proper slant submanifold of a Kaehler manifold is even-dimensional, but this is not true for slant lightlike submanifold [37]. In this section, we study totally umbilical slant lightlike submanifolds of indefinite Kaehler manifolds. We prove that there do not exist totally umbilical proper slant lightlike submanifolds in indefinite Kaehler manifolds other than totally geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally umbilical proper slant lightlike submanifolds of indefinite Kaehler space forms. Definition 4.1 ([15]) A lightlike submanifold (M, g) of a semi-Riemannian man¯ g) ifold ( M, ¯ is said to be a totally umbilical in M¯ if there is a smooth transversal vector field H ∈ (tr (T M)) on M, called the transversal curvature vector field of M, such that, for X, Y ∈ (T M), h(X, Y ) = H g(X, ¯ Y ).

(21)

Using (2)–(4), it is clear that M is a totally umbilical, if and only if, on each coordinate neighborhood U there exist smooth vector fields H l ∈ (ltr (T M)) and H s ∈ (S(T M ⊥ )) such that h l (X, Y ) = H l g(X, Y ), h s (X, Y ) = H s g(X, Y ),

Dl (X, W ) = 0,

(22)

for X, Y ∈ (T M) and W ∈ (S(T M ⊥ )). A lightlike submanifold is said to be totally geodesic if h(X, Y ) = 0, for any X, Y ∈ (T M). Therefore, in other words, a lightlike submanifold is totally geodesic if H l = 0 and H s = 0. Theorem 4.2 ([33]) Let M be a totally umbilical slant lightlike submanifold of an ¯ g, indefinite Kaehler manifold ( M, ¯ J¯). Then at least one of the following statements is true: 1. M is an anti-invariant submanifold. 2. D = {0}. 3. If M is a proper slant submanifold, then H s ∈ (μ). Proof Let M be a totally umbilical slant lightlike submanifold of an indefinite ¯ then for any X = Q 2 X ∈ (D) using (21), we have Kaehler manifold M, h(T Q 2 X, T Q 2 X ) = g(T Q 2 X, T Q 2 X )H, therefore from (2), (18) and the above equation, we get ∇¯ T Q 2 X T Q 2 X − ∇T Q 2 X T Q 2 X = cos 2 θ g(Q 2 X, Q 2 X )H.

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Using (9) and the fact that M¯ is Kaehler manifold, we obtain J¯∇¯ T Q 2 X Q 2 X − ∇¯ T Q 2 X F Q 2 X − ∇T Q 2 X T Q 2 X = cos 2 θg(Q 2 X, Q 2 X )H. Then using (2)–(4), we get J¯∇T Q 2 X Q 2 X + J¯h l (T Q 2 X, X ) + J¯h s (T Q 2 X, X ) + A F Q 2 X T Q 2 X −∇Ts Q 2 X F Q 2 X − Dl (T Q 2 X , F Q 2 X ) − ∇T Q 2 X T Q 2 X = cos 2 θ g(Q 2 X, Q 2 X )H. Thus using (9), (10) and (22), we have T ∇T Q 2 X Q 2 X + F∇T Q 2 X Q 2 X + g(T Q 2 X, X ) J¯ H l + g(T Q 2 X, X )B H s +g(T Q 2 X, X )C H s + A F Q 2 X T Q 2 X − ∇Ts Q 2 X F Q 2 X −Dl (T Q 2 X , F Q 2 X ) − ∇T Q 2 X T Q 2 X = cos 2 θg(Q 2 X, Q 2 X )H. Equating the transversal components, we get F∇T Q 2 X Q 2 X + g(T Q 2 X, X )C H s − ∇Ts Q 2 X F Q 2 X −Dl (T Q 2 X , F Q 2 X ) = cos 2 θ g(Q 2 X, Q 2 X )H.

(23)

On the other hand, (19) holds for any X = Y ∈ (D) and by taking the covariant derivative with respect to T Q 2 X , we obtain g(∇Ts Q 2 X F Q 2 X, F Q 2 X ) = sin 2 θ g(∇T Q 2 X Q 2 X, Q 2 X ).

(24)

Now, taking the inner product in (23) with F Q 2 X , we obtain g(F∇T Q 2 X Q 2 X, F Q 2 X ) − g(∇Ts Q 2 X F Q 2 X, F Q 2 X ) = cos 2 θ g(Q 2 X, Q 2 X )g(H s , F Q 2 X ). Then using (19) and (24), we get cos 2 θ g(Q 2 X, Q 2 X )g(H s , F Q 2 X ) = 0.

(25)

Thus from (25), it follows that either θ = π/2 or Q 2 X = 0 or H s ∈ (μ). This completes the proof.  For a totally umbilical slant lightlike submanifold of an indefinite Kaehler manifold, using (2)–(5), we have the following important observation. Lemma 4.3 ([33]) Let M be a totally umbilical slant lightlike submanifold of an ¯ g, indefinite Kaehler manifold ( M, ¯ J¯) then g(∇ X X, J¯ξ ) = 0, for any X ∈ (D) and ξ ∈ (RadT M).

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Theorem 4.4 Every totally umbilical proper slant lightlike submanifold M of an ¯ g, indefinite Kaehler manifold ( M, ¯ J¯) is a totally geodesic slant lightlike submanifold. Proof Since M is a totally umbilical slant lightlike submanifold, we have h(T Q 2 X, T Q 2 X ) = g(T Q 2 X, T Q 2 X )H, for any X = Q 2 X ∈ (D). Using (18), we get h(T Q 2 X, T Q 2 X ) = cos 2 θ g(Q 2 X, Q 2 X )H.

(26)

For any X ∈ (D), using (16), we obtain F∇T Q 2 X X = h(T Q 2 X, T Q 2 X ) − Ch(T Q 2 X, Q 2 X ) + ∇Ts Q 2 X F Q 2 X + Dl (T Q 2 X, F Q 2 X ), since M is a totally umbilical slant lightlike submanifold, therefore Ch(T Q 2 X, X ) = g(T Q 2 X, X )C H = 0 and using (26), we get cos 2 θg(Q 2 X, Q 2 X )H = F∇T Q 2 X X − ∇Ts Q 2 X F Q 2 X − Dl (T Q 2 X, F Q 2 align X ).

(27)

Taking the scalar product of both sides to (27) with respect to F Q 2 X , we obtain cos 2 θ g(Q 2 X, Q 2 X )g(H ¯ s , F Q 2 X ) = g(F∇ ¯ T Q 2 X X, F Q 2 X ) − g(∇ ¯ Ts Q 2 X F Q 2 X, F Q 2 X ), using (19), we get cos 2 θ g(Q 2 X, Q 2 X )g(H ¯ s , F Q 2 X ) = sin 2 θ g(∇ ¯ T Q 2 X X, X ) − g(∇ ¯ Ts Q 2 X F Q 2 X, F Q 2 X ).

(28)

Since (19) holds for any X = Y ∈ (D) and by taking the covariant derivative with respect to ∇¯ T Q 2 X , we get g(∇ ¯ Ts Q 2 X F Q 2 X, F Q 2 X ) = sin 2 θ g(∇T Q 2 X Q 2 X, Q 2 X ).

(29)

Using (29) in (28), we obtain cos 2 θ g(Q 2 X, Q 2 X )g(H ¯ s , F Q 2 X ) = 0.

(30)

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Since M is a proper slant lightlike submanifold and g is a Riemannian metric on D, therefore, we have g(H ¯ s , F Q 2 X ) = 0. Thus, using Lemmas 3.7 and (14), we obtain H s ∈ (μ).

(31)

Now, since M¯ is an indefinite Kaehler manifold therefore for any X, Y ∈ (D), we have ∇¯ X J¯Y = J¯∇¯ X Y ; this implies that ∇ X T Q 2 Y + g(X, T Q 2 Y )H − A F Q 2 Y X + ∇ Xs F Q 2 Y + Dl (X, F Q 2 Y ) (32) = T ∇ X Y + F∇ X Y + g(X, Y ) J¯ H. Taking the scalar product of both sides to (32) with respect to J¯ H s and using (15) and (31), we obtain g(∇ ¯ Xs F Q 2 X, J¯ H s ) = g(X, Y )g(H s , H s ).

(33)

¯ we Since μ is an invariant subspace therefore using the Kaehlerian character of M, s s ¯ ¯ ¯ ¯ have ∇ X J H = J ∇ X H ; this implies that −A J¯ H s X + ∇ Xs J¯ H s + Dl (X, J¯ H s ) = −T A H s X − F A H s X + B∇ Xs H s + C∇ Xs H s + J¯ Dl (X, H s ). (34) On taking the scalar product of both sides to the above equation with respect to F Q 2 Y and using invariant character of μ, that is, C∇ Xs H s ∈ (μ), we get g(∇ ¯ Xs J¯ H s , F Q 2 Y ) = −g(F A H s X, F Q 2 Y ) = −sin 2 θ g(A H s X, Q 2 Y ).

(35)

Since ∇¯ is a metric connection therefore (∇¯ X g)(F Q 2 Y, J¯ H s ) = 0; this further ¯ Xs J¯ H s , F Q 2 Y ), therefore using (35), we implies that g(∇ ¯ Xs F Q 2 Y, J¯ H s ) = g(∇ obtain (36) g(∇ ¯ Xs F Q 2 Y, J¯ H s ) = −sin 2 θ g(A H s X, Q 2 Y ). From (33) and (36), we have g(X, Y )g(H s , H s ) = −sin 2 θ g(A H s X, Q 2 Y ),

(37)

using (5) in (37), we obtain g(X, Y )g(H s , H s ) = −sin 2 θ g(h ¯ s (X, Q 2 Y ), H s ) = −sin 2 θ g(X, Y )g(H s , H s ); this implies that (1 + sin 2 θ )g(X, Y )g(H s , H s ) = 0. Since g is a Riemannian metric on D therefore we obtain

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H s = 0.

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(38)

¯ we have ∇¯ X J¯ X = J¯∇¯ X X for any Furthermore, using the Kaehler character of M, X = Q 2 X ∈ (D); this implies that ∇ X T Q 2 X + h(X, T Q 2 X ) − A F Q 2 X X + ∇ Xs F Q 2 X +Dl (X, F Q 2 X ) = T ∇ X X + F∇ X X + Bh(X, X ) + Ch(X, X ). Since M is a totally umbilical slant lightlike manifold therefore using h(X, T Q 2 X ) = 0 in the above equation and then comparing the tangential components, we obtain ∇ X T Q 2 X − A F Q 2 X X = T ∇ X X + Bh(X, X ).

(39)

Taking the scalar product of both sides to (39) with J¯ξ ∈ ( J¯ Rad(T M)) and using Lemma 4.3, we get g(A F Q 2 X X, J¯ξ ) + g(h ¯ l (X, X ), ξ ) = 0.

(40)

Now using (5), we obtain g(h ¯ s (X, J¯ξ ), F Q 2 X ) + g( ¯ J¯ξ, Dl (X, F Q 2 X )) = g(A F Q 2 X X, J¯ξ ), further on using that M is a totally umbilical slant lightlike submanifold and (38), it implies that (41) g(A F Q 2 X X, J¯ξ ) = 0. Using (41) in (40), we obtain that g(h ¯ l (X, X ), ξ ) = 0; this implies that g(Q 2 X, Q 2 X ) l , ξ ) = 0. Since g is a Riemannian metric on D therefore g(H ¯ l , ξ ) = 0, then we g(H ¯ obtain that (42) H l = 0. Thus from (38) and (42), the proof is complete.



Taking into account that the connection ∇¯ is a metric connection on a semiRiemannian manifold M¯ and then using the Gauss and Weingarten formulae, we obtain ¯ l (X, Y ), Z ) + g(h ¯ l (X, Z ), Y ), (43) (∇ X g)(Y, Z ) = g(h for any X, Y, Z ∈ (T M). Contrary to general theory of submanifolds, (43) implies that in case of lightlike geometry, the induced linear connection on a lightlike submanifold is not a metric connection. In [14], Duggal and Bejancu derived a necessary and sufficient condition for an induced connection to be a metric connection as below. Theorem 4.5 Let M be an m-dimensional r −lightlike submanifold of an (m + ¯ g) n)−dimensional semi-Riemannian manifold ( M, ¯ such that r < {m, n} or M be a

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¯ g). coisotropic submanifold of ( M, ¯ Then ∇ is a metric connection if and only if h l vanishes identically on M. Thus from (22) and (42), we have the following observation immediately. Corollary 4.6 Let M be a totally umbilical proper slant lightlike submanifold of an ¯ g, indefinite Kaehler manifold ( M, ¯ J¯), then the induced connection ∇ is a metric connection on M. An indefinite complex space form is a connected indefinite Kaehler manifold ¯ g, ¯ ( M, ¯ J¯) of constant holomorphic sectional curvature c and is denoted by M(c). ¯ ¯ Then, the Riemannian curvature tensor R of M(c) is given by c ¯ ¯ Z )X − g(X, ¯ Z )Y + g(J ¯ Y, Z )J X R(X, Y )Z = {g(Y, 4 − g(J ¯ X, Z )J Y + 2g(X, ¯ J Y )J Z },

(44)

¯ Let M be a totally umbilical proper slant lightlike for X, Y, Z vector fields on M. ¯ submanifold of M(c) such that c = 0. Then using (44), for any X ∈ (D), Z ∈ ¯ (T M)) and ξ ∈ (Rad(T M)), we obtain ( Jltr c ¯ g( ¯ R(X, J¯ X )Z , ξ ) = − g(X, X )g( J¯ Z , ξ ). 2

(45)

On the other hand, using (22), (6) and (7), we obtain ¯ g( ¯ R(X, J¯ X )Z , ξ ) = −g(∇ X T Q 2 X, Z )g(H ¯ l , ξ ) − g(T Q 2 X, ∇ X Z )g(H ¯ l, ξ) + g(∇ J¯ X X, Z )g(H ¯ l , ξ ) + g(X, ∇ J¯ X Z )g(H ¯ l , ξ ).

(46)

Since M is a totally umbilical proper slant lightlike submanifold, using (42) in (46) and from (45), it follows that c g(X, X )g( J¯ Z , ξ ) = 0. 2 As g is a Riemannian metric on D and from (1), we have g( J¯ Z , ξ ) = 0 therefore c = 0. Thus, we have the following observation immediately. Theorem 4.7 There exists no totally umbilical proper slant lightlike submanifold of ¯ an indefinite complex space form M(c), c = 0.

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5 Minimal Slant Lightlike Submanifolds We recall the following theorem from [4]. Theorem 5.1 Let (M, g, S(T M), S(T M ⊥ )) be a lightlike submanifold of a semi¯ g). Riemannian manifold ( M, ¯ Then (i) h l = 0 on Rad(T M). (ii) h s = 0 (for Cases 2, 4) and h s = 0 on Rad(T M) if and only if LW g¯ = 0 on Rad(T M), for any W ∈ (S(T M ⊥ )) (for Cases 1, 3). In [14], a minimal lightlike submanifold M is defined only in a particular case, when M is a hypersurface of a 4-dimensional Minkowski space. Then using Theorem 5.1, Bejan and Duggal [4] defined the general notion of a minimal lightlike submanifold of a semi-Riemannian manifold M¯ as below. Definition 5.2 A lightlike submanifold (M, g, S(T M)) isometrically immersed in ¯ g) a semi-Riemannian manifold ( M, ¯ is minimal if (i) h s = 0 on Rad(T M) and (ii) trace h = 0, where trace is written with respect to g restricted to S(T M). From Theorem 5.1, it should be noted that the above definition is independent of S(T M) and S(T M ⊥ ), but it depends on the choice of transversal bundle tr (T M). As in the semi-Riemannian case, any totally geodesic submanifold is minimal. In [4], Bejan and Duggal have presented an example of proper lightlike minimal submanifold which is not totally geodesic. Moreover, Sahin [37] presented an example of a proper minimal slant lightlike submanifold which is not totally geodesic. Now, we recall an important lemma from [37]. Lemma 5.3 Let M be a proper slant lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯) such that dim(D) = dim(S(T M ⊥ )). If {e1 , . . . , em } is a local orthonormal basis of (D), then {csc θ Fe1 , . . . , csc θ Fem } is an orthonormal basis of S(T M ⊥ ). Lemma 5.3 is used to establish some necessary and sufficient conditions for a proper slant lightlike submanifold of an indefinite Kaehler manifold to be a minimal lightlike submanifold. Theorem 5.4 Let M be a proper slant lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯). Then M is a minimal lightlike submanifold if and only if trace A∗ξ j | S(T M) = 0, trace A Wα | S(T M) = 0, and g(D ¯ l (X, W ), Y ) = 0, ∀X, Y ∈ (Rad(T M)), ⊥ where {ξ j }rj=1 is a basis of Rad(T M), and {Wα }m α=1 is a basis of S(T M ).

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Theorem 5.5 Let M be a proper slant lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯) such that dim(D) = dim(S(T M ⊥ )). Then M is a minimal lightlike submanifold if and only if trace A∗ξ j | S(T M) = 0, trace A Fei | S(T M) = 0, and g(D ¯ l (X, Fei ), Y ) = 0, ∀X, Y ∈ (Rad(T M)), m where {ei }i=1 is a basis of D.

Definition 5.6 ([16]) A lightlike submanifold M of a semi-Riemannian manifold M¯ is called an irrotational lightlike submanifold if and only if ∇¯ X ξ ∈ (T M), for all X ∈ (T M) and ξ ∈ (Rad(T M)). Theorem 5.7 Let M be an irrotational slant lightlike submanifold of an indefinite ¯ g, Kaehler manifold ( M, ¯ J¯). Then M is a minimal lightlike submanifold if and only if trace A Wk | S(T M) = 0, trace A∗ξi | S(T M) = 0, where {Wk }lk=1 is a basis of S(T M ⊥ ) and {ξi }ri=1 is a basis of Rad(T M). Remark 1 Let M be a 2q−lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯) of index 2q. Then the submanifold M is said to be a hemi-slant lightlike submanifold [24] of M¯ if Rad(T M) is a distribution on M such that J¯(Rad(T M)) = ltr (T M) and for all x ∈ U ⊂ M and for each non-zero vector field X tangent to S(T M), the angle θ (X ) between J¯ X and the vector space S(T M) is constant. A hemi-slant lightlike submanifold is said to be proper if S(T M) = 0 and θ = 0, π2 . Haider et al. [24] proved characterization theorems for the existence of hemi-slant lightlike submanifolds of an indefinite Kaehler manifold. They also obtain necessary and sufficient conditions for hemi-slant submanifolds to be hemi-slant lightlike products. In [32], Sachdeva et al. further study the geometry of hemi-slant lightlike submanifolds of indefinite Kaehler manifolds and proved that there do not exist totally umbilical hemi-slant lightlike submanifolds of indefinite Kaehler manifolds other than totally geodesic hemi-slant lightlike submanifolds. They also obtain a characterization theorem on the non-existence of a totally umbilical hemi-slant lightlike submanifold of an indefinite complex space form and some characterization theorems on minimal hemi-slant lightlike submanifolds. Remark 2 The axiom of planes for Riemannian manifolds was introduced by Cartan in [6] as follows: A Riemannian manifold M¯ of dimension m ≥ 3 satisfies the axiom of k−planes if for each point p in M¯ and for every k−dimensional linear ¯ there exists a k−dimensional totally geodesic submanifold subspace T of T p ( M), M of M¯ containing p such that T p M = T . Then, Cartan [6] proved the following result.

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Theorem 5.8 A Riemannian manifold M¯ of dimension m ≥ 3 satisfies the axiom of k−planes for some k, 2 ≤ k < m if and only if it is a real space form. In the process of generalization, Leung and Nomizu [26] introduced the axiom of k−spheres as follows: A Riemannian manifold M¯ of dimension m ≥ 3 satisfies the axiom of k−spheres if for each point p in M¯ and for every linear subspace ¯ there exists a k−dimensional totally umbilical submanifold M of M¯ with T of T p M, parallel mean curvature vector field such that p ∈ M and T p M = T . Then, Leung and Nomizu [26] proved the following result. Theorem 5.9 A Riemannian manifold M¯ of dimension m ≥ 3 is a real space form if and only if it satisfies the axiom of k−spheres for some k, 2 ≤ k < m. Further, Graves and Nomizu [20] generalized these notions of axioms of planes and spheres for indefinite Riemannian manifolds. In [34], Sachdeva et al. introduced the axiom of indefinite hemi-slant 3−planes and 3−spheres for an indefinite Kaehler manifold as below. ¯ g) Definition 5.10 (Axioms of indefinite planes and spheres) Let ( M, ¯ be an indefinite ¯ almost Hermitian manifold with constant index 2q. Then M satisfies the axiom of indefinite hemi-slant 3−planes (indefinite hemi-slant 3−spheres, respectively) if for ¯ there exists a 3−dimensional each p ∈ M¯ and each hemi-slant 3-plane T in T p M, totally geodesic lightlike submanifold M (totally umbilical lightlike submanifold M with parallel transversal curvature vector field and an induced metric connection, respectively) such that p ∈ M and T p M = T . Then Sachdeva et al. [34] proved that if an indefinite Kaehler manifold satisfies the axioms of indefinite hemi-slant 3−planes and 3−spheres for some slant angle θ ∈ (0, π/2), then it is an indefinite complex space form. In particular, they proved the following. ¯ g) Theorem 5.11 ([34]) Let ( M, ¯ be an indefinite Kaehler manifold with index 2q. ¯ If M satisfies the axiom of indefinite hemi-slant 3−spheres for some θ ∈ (0, π/2), then M¯ is an indefinite complex space form. ¯ g) Theorem 5.12 ([34]) Let ( M, ¯ be an indefinite Kaehler manifold with index 2q. If ¯ M satisfies the axiom of indefinite hemi-slant 3−planes for some θ ∈ (0, π/2), then M¯ is an indefinite complex space form.

6 Warped Product Slant Lightlike Submanifolds Warped product manifolds are known to have applications in physics as they provide an excellent setting to model spacetime. Bishop and O’Neill [5] introduced the notion of warped product manifolds, in order to construct a large variety of manifolds of negative curvature. From a geometric point of view, this study got momentum, when

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the study of the warped product of C R−submanifolds of Kaehler manifolds was introduced by Chen [11, 12]. Following this field, many geometers started working along this line and presented numerous results. Sahin [36] obtained some results on warped product submanifolds of Kaehler manifolds with a slant factor. Although there are significant applications of warped product submanifolds in general theory of relativity, very limited specific information is available on its lightlike case. This motivated the geometers to carry out work on the geometry of warped product lightlike submanifolds. In this section, the theory of warped product submanifolds has been clubbed with slant lightlike submanifolds. In particular, we study warped product slant lightlike submanifolds of indefinite Kaehler manifolds and explore the non-existence of warped product slant lightlike submanifolds of indefinite Kaehler manifolds. Let M1 and M2 be two Riemannian manifolds with Riemannian metrics g M1 and g M2 , respectively, and f > 0 be a differentiable function on M1 . Assume that the product manifold M1 × M2 with its projection π : M1 × M2 → M1 and η : M1 × M2 → M2 . Then the warped product M = M1 × f M2 is the manifold M1 × M2 equipped with the Riemannian metric g, where g = g M1 + f 2 g M2 .

(47)

If X is tangent to M = M1 × f M2 at ( p, q) then using (47), we have X 2 = π∗ X 2 + f 2 (π(X ))η∗ X 2 . The function f is called the warping function of the warped product. For differentiable function f on M, the gradient ∇ f is defined by g(∇ f, X ) = X f , for all X ∈ T (M). Lemma 6.1 ([5]) Let M = M1 × f M2 be a warped product manifold. If X, Y ∈ T (M1 ) and U, Z ∈ T (M2 ), then ∇ X U = ∇U X =

Xf U = X (ln f )U. f

(48)

Corollary 6.2 On a warped product manifold M = M1 × f M2 , the manifolds M1 and M2 are totally geodesic and totally umbilical in M, respectively. Definition 6.3 ([35]) Let M be a lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯). Then M is said to be a transversal lightlike submanifold if the following conditions are satisfied: (i) Rad(T M) is transversal with respect to J¯, i.e., J¯(Rad(T M)) = ltr (T M). (ii) S(T M) is transversal with respect to J¯, that is, J¯(S(T M)) ⊆ S(T M ⊥ ). Definition 6.4 ([16]) Let M be a lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯). Then M is said to be a Screen Cauchy-Riemann (SC R)−lightlike submanifold of M¯ if the following conditions are satisfied:

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(ii) Rad(T M) is invariant with respect to J¯. (i) There exists real non-null distributions D ⊂ S(T M) such that

S(T M) = D ⊕ D ⊥ ,

J¯ D ⊥ ⊂ S(T M ⊥ ),

D ∩ D ⊥ = {0},

where D ⊥ is orthogonal complementary to D in S(T M). Theorem 6.5 A SC R−lightlike submanifold M of an indefinite Kaehler manifold M¯ is a holomorphic or complex (resp. screen real) lightlike submanifold if and only if D ⊥ = {0} (resp. D = {0}). Theorem 6.6 Let M¯ be an indefinite Kaehler manifold. Then there does not exist warped product submanifold M = Mθ × f MT of M¯ such that Mθ is a proper slant lightlike submanifold of M¯ and MT is a holomorphic Screen Cauchy-Riemann (SC R) ¯ lightlike submanifold of M. Proof Let X ∈ (S(T M)) be of a holomorphic SC R−lightlike submanifold MT and Z ∈ (D θ ) of a slant lightlike submanifold Mθ . Then using (48), we have g(∇ J¯ X Z , X ) = Z (ln f )g( J¯ X, X ) = 0, and further using (2) and (9), we get 0 = g( ¯ ∇¯ J¯ X Z , X ) = −g( ¯ J¯ Z , ∇¯ J¯ X J¯ X ) = g(∇ ¯ J¯ X T Z , J¯ X ) − g(F ¯ Z , h s ( J¯ X, J¯ X )). Furthermore, by virtue of (48), we obtain T Z (ln f )g(X, X ) = g(h ¯ s ( J¯ X, J¯ X ), F Z ). Then using polarization identity, we get T Z (ln f )g(X, Y ) = g(h ¯ s ( J¯ X, J¯Y ), F Z ),

(49)

for any X, Y ∈ (S(T M)) of a holomorphic SC R−lightlike submanifold MT and Z ∈ (D θ ) of a slant lightlike submanifold Mθ . On the other hand, using (4) and (48), we have ¯ , ∇¯ J¯ X Y ) − g(T ¯ Z , ∇¯ J¯ X J¯Y ) g(A F Z J¯ X, J¯Y ) = −g(∇¯ J¯ X F Z , J¯Y ) = g(Z = −g( ¯ ∇¯ J¯ X Z , Y ) + g( ¯ ∇¯ J¯ X T Z , J Y ) = −Z (ln f )g( J¯ X, Y ) + T Z (ln f )g(X, Y ).

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Now using (5), we have g(h ¯ s ( J¯ X, J¯Y ), F Z ) = g(A F Z J X, J¯Y ), therefore we obtain g(h ¯ s ( J¯ X, J¯Y ), F Z ) = −Z (ln f )g( J¯ X, Y ) + T Z (ln f )g(X, Y ).

(50)

Thus, (49) and (50) imply that Z (ln f )g( J¯ X, Y ) = 0, for any X, Y ∈ (S(T M)) of a holomorphic SC R−lightlike submanifold MT and Z ∈ (D θ ) of a slant lightlike submanifold Mθ . Since MT = {0} is a Riemannian and invariant therefore we obtain Zln f = 0; this shows that f is constant. Hence the proof is complete.  ¯ g, Theorem 6.7 Let ( M, ¯ J¯) be an indefinite Kaehler manifold. Then there does not exist warped product submanifold M = MT × f Mθ in M¯ such that MT is a holomorphic SC R−lightlike submanifold and Mθ is a proper slant lightlike submanifold ¯ of M. Proof Let X ∈ (S(T M)) be of a holomorphic SC R−lightlike submanifold MT and Z ∈ (D θ ) of a slant lightlike submanifold Mθ . Then using (48), we have g(∇T Z X, Z ) = X (ln f )g(T Z , Z ) = 0. This further using with (4), (5) and (18) implies that 0 = g( ¯ ∇¯ T Z X, Z ) = −g( ¯ J¯ X, ∇¯ T Z T Z ) − g( ¯ J¯ X, ∇¯ T Z F Z ) = g(∇T Z J¯ X, T Z ) + g( J¯ X, A F Z T Z ) = g(∇T Z J¯ X, T Z ) + g(h s ( J¯ X, T Z ), F Z ) = J¯ X (ln f )g(T Z , T Z ) + g(h ¯ s ( J¯ X, T Z ), F Z ) ¯ s ( J¯ X, T Z ), F Z ). = J¯ X (ln f ).cos 2 θ g(Z , Z ) + g(h On replacing X by J¯ X , it implies that ¯ s (X, T Z ), F Z ) = 0. X (ln f ).cos 2 θ g(Z , Z ) + g(h

(51)

Furthermore, on replacing Z by T Z and then using (17) and (18), we obtain g(h ¯ s (X, Z ), F T Z ) = X (ln f ).cos 2 θ g(Z , Z ).

(52)

On the other hand, using (2), (9), (17), (18) and (48), for any X ∈ (S(T M)) of a holomorphic SC R−lightlike submanifold MT and Y, Z ∈ (D θ ) of a slant lightlike submanifold Mθ , we have

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g(h ¯ s (T Z , X ), FY ) = −g(T ¯ Z , ∇¯ X J¯Y ) + g(T ¯ Z , ∇¯ X T Y ) 2 Z , ∇¯ X Y ) + g(F ¯ T Z , ∇¯ X Y ) + g(T ¯ Z , ∇X T Y ) = g(T ¯ = −cos 2 θ X (ln f )g(Z , Y ) + g(F ¯ T Z , h s (X, Y )) + X (ln f )g(T Z , T Y ) = g(F ¯ T Z , h s (X, Y )). On putting Y = Z , we get g(h ¯ s (T Z , X ), F Z ) = g(F ¯ T Z , h s (X, Z )).

(53)

Thus from (51)–(53), we have X (ln f )cos 2 θ g(Z , Z ) = 0. Since D θ is a proper slant and Z is non-null, we obtain X (ln f ) = 0. This proves our assertion.  ¯ g, Theorem 6.8 Let ( M, ¯ J¯) be an indefinite Kaehler manifold. Then there does not exist warped product submanifold M = M⊥ × f Mθ of M¯ such that M⊥ is a transversal lightlike submanifold and Mθ is a proper slant lightlike submanifold of ¯ M. Proof Assume that Z ∈ (D θ ) of a slant lightlike submanifold Mθ and X ∈ (S(T M)) of a transversal lightlike submanifold M⊥ then using (4), (9), (18) and (48), we have ¯ ∇¯ T Z X, J¯ Z ) = g(∇T Z X, T Z ) + g(h ¯ s (T Z , X ), F Z ) g(A J¯ X T Z , Z ) = g( = X (ln f )g(T Z , T Z ) + g(h ¯ s (T Z , X ), F Z ) = X (ln f )cos 2 θ g(Z , Z ) + g(h ¯ s (T Z , X ), F Z ). Using (5) on the left-hand side of the last equation, we obtain g(h ¯ s (T Z , Z ), J¯ X ) = X (ln f )cos 2 θ g(Z , Z ) + g(h ¯ s (T Z , X ), F Z ).

(54)

Replacing Z by T Z in (54) and then using (17) and (18), we get g(h ¯ s (Z , T Z ), J¯ X ) = −X (ln f )cos 2 θ g(Z , Z ) + g(h ¯ s (Z , X ), F T Z ). On the other hand, using (4), (9), (17) and (48), we have

(55)

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g(A F Z Y, T Z ) = −g( ¯ ∇¯ X F Z , T Z ) = g( ¯ ∇¯ X Z , J¯ T Z ) + g( ¯ ∇¯ X T Z , T Z ) 2 ¯ ∇¯ X Z , F T Z ) + g(∇ X T Z , T Z ) = g( ¯ ∇¯ X Z , T Z ) + g( = −cos 2 θ g(∇ X Z , Z ) + g(h ¯ s (X, Z ), F T Z ) + X (ln f )g(T Z , T Z ) ¯ s (X, Z ), F T Z ) = −cos 2 θ X (ln f )g(Z , Z ) + g(h + X (ln f )cos 2 θ g(Z , Z ) = g(h ¯ s (X, Z ), F T Z ). Hence using (5), we obtain g(h ¯ s (T Z , X ), F Z ) = g(h ¯ s (X, Z ), F T Z ).

(56)

Thus using (54)–(56), we get 2X (ln f )cos 2 θ g(Z , Z ) = 0. Since Mθ is a proper slant lightlike submanifold and D θ is Riemannian, therefore we obtain X (ln f ) = 0. Hence f is constant, which proves our assertion.  Remark 3 From Theorems 6.6, 6.7 and 6.8, it is clear that there do not exist warped product lightlike submanifolds of the following forms: • M = Mθ × f M T ; • M = M T × f Mθ ; • M = M⊥ × f Mθ . Now onwards, we call M = Mθ × f M⊥ a warped product slant lightlike submanifold, where Mθ is a proper slant lightlike submanifold and M⊥ is a transversal ¯ lightlike submanifold of an indefinite Kaehler manifold M. Theorem 6.9 Let M = Mθ × f M⊥ be a warped product slant lightlike submanifold ¯ g, of an indefinite Kaehler manifold ( M, ¯ J¯) such that M⊥ is a transversal lightlike ¯ Then submanifold and Mθ is a proper slant lightlike submanifold of M. g(h s (X, Y ), J¯ Z ) = −T X (ln f )g(Y, Z ), for any X ∈ (D θ ) of a slant lightlike submanifold Mθ and Y, Z ∈ (S(T M)) of transversal lightlike submanifold M⊥ . Proof For any X ∈ (D θ ) of a slant lightlike submanifold Mθ and Y, Z ∈ (S(T M)) of transversal lightlike submanifold M⊥ , using (4) and (9), we have g(h s (T X, Y ), J¯ Z ) = g(∇¯ Y T X, J¯ Z ) = g(∇Y X, Z ) + g(∇¯ Y J¯ F X, Z ). Since F(D θ ) ⊂ S(T M ⊥ ) and μ is invariant therefore using (10), we have J¯ F X = B F X and C F X = 0. Hence using (20) and (48), we obtain

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g(h s (T X, Y ), J¯ Z ) = X (ln f )g(Y, Z ) − sin 2 θg(∇¯ Y X, Z ). Again using (48), we have g(h s (T X, Y ), J¯ Z ) = (1 − sin 2 θ )X (ln f )g(Y, Z ) = cos 2 θ X (ln f )g(Y, Z ). Replacing X by T X and using (17), the assertion follows.



7 Pointwise Slant Lightlike Submanifolds Etayo in [18] extended the theory of slant submanifolds to quasi-slant submanifolds of almost Hermitian manifolds as natural extensions of anti-invariant submanifolds, invariant submanifolds, semi-slant submanifolds, slant submanifolds and C R−submanifolds. Later, quasi-slant submanifolds were studied by Chen and Garay [13] under the heading of pointwise slant submanifolds. Pointwise slant submersions were naturally extended from slant submersion by Lee and Sahin in [25]. Gulbahar et al. [21] investigated pointwise slant submanifolds of almost product Riemannian manifolds. In this section, we discuss pointwise slant lightlike submanifolds of an almost Hermitian manifold and characterization theorems for their existence. We study conditions for pointwise slant lightlike submanifold to become a slant lightlike submanifold. We provide some non-trivial examples of pointwise slant lightlike submanifolds and we discuss the condition for a totally umbilical pointwise slant lightlike submanifold to become a totally geodesic pointwise slant lightlike submanifold. Definition 7.1 ([22]) Suppose that M is a q−lightlike submanifold of an indefinite ¯ g, almost Hermitian manifold ( M, ¯ J¯) of index 2q. Then the submanifold M is said to be a pointwise slant lightlike submanifold of M¯ if (i) Rad(T M) ∩ J¯ Rad(T M) = {0}. (ii) At every point p ∈ U ⊂ M, the angle θ p (X ), called Wirtinger angle of X at a point p, between the vector space D p and J¯ X , is independent of X , for any non-null vector field X tangent to D at p ∈ U ⊂ M, where the distribution D ¯ (T M) in S(T M). It means that θ p (X ) is complementary to J¯ Rad(T M) ⊕ Jltr does not depend on the choice of X . The angle θ is a function on M, known as slant function of pointwise slant submanifold. If the slant angle θ p (X ) is independent of X ∈ D p and p ∈ U ⊂ M, then pointwise slant lightlike submanifold M is said to be a slant lightlike submanifold ¯ Pointwise slant lightlike submanifold M is called proper if D p = {0} and of M. θ p (X ) = 0, π/2 for each p ∈ U ⊂ M. Next, three theorems give us the necessary and sufficient conditions for a lightlike submanifold of an indefinite Kaehler manifold M¯ to be a pointwise slant lightlike submanifold.

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Theorem 7.2 ([22]) (Existence Theorem) Suppose that M is a q−lightlike submanifold of indefinite Kaehler manifold M¯ of index 2q. Then M becomes a pointwise slant submanifold if and only if ¯ (T M) is a distribution on M. (i) Jltr (ii) (T | D p )2 = −cos 2 θ p I , for a real-valued function θ p defined on T M. Corollary 7.3 Suppose that M is a pointwise slant lightlike submanifold of indefinite ¯ then Kaehler manifold M, g(T Q 2 X, T Q 2 Y ) = cos 2 θ p (Q 2 X )g(Q 2 X, Q 2 Y ),

(57)

g(F Q 2 X, F Q 2 Y ) = sin 2 θ p (Q 2 X )g(Q 2 X, Q 2 Y ). Theorem 7.4 ([22]) (Existence Theorem) Suppose that M is a q−lightlike submanifold of indefinite Kaehler manifold M¯ of index 2q. Then M becomes a pointwise slant lightlike submanifold if and only if ¯ (T M) is a distribution on M. (i) Jltr (ii) B F Q 2 X = −sin 2 θ p (Q 2 X ) Q 2 X , for vector field X on M. ¯ Then M is a pointTheorem 7.5 Suppose that M is a lightlike submanifold of M. ¯ if and only if wise slant lightlike submanifold of indefinite Kaehler manifold M, endomorphism T preserves orthogonality. Example 2 Suppose that M is a submanifold of (R82 , g) ¯ such that x 1 = sinx 4 , x 2 = x 1 , x 3 = x 5 , x 4 = x 3 , x 5 = 0, x 6 = cosx 4 , x 7 = x 2 , x 8 = x 5 , where the signature of g¯ is (+, +, +, +, +, +, −, −) with respect to basis {∂ x1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ x7 , ∂ x8 }. Then T M = span{Z 1 = ∂ x2 , Z 2 = ∂ x7 , Z 3 = ∂ x4 , Z 4 = cosx4 ∂ x1 − sinx4 ∂ x6 , Z 5 = ∂ x3 + ∂ x8 }, hence, M becomes a lightlike submanifold of (R82 , g), ¯ where Rad(T M) = span{Z 5 } and J¯ Rad(T M) = span{Z 3 − Z 2 }. Choose D = span{Z 1 , Z 4 } which is Riemannian. Then clearly M becomes a pointwise slant lightlike submanifold with slant angle cosx4 such that lightlike transversal bundle spanned by N = 21 {∂ x3 − ∂ x8 } and S(T M ⊥ ) = span{W1 = sinx4 ∂ x1 + cosx4 ∂ x6 , W2 = ∂ x5 }, which is also Riemannian.

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Example 3 Suppose that M is a submanifold of (R82 , g) ¯ such that X (u, v, a, t, s) = (u, v, sina, cosa, −a 2 sint, −a 2 cost, u, s), where the signature of g¯ is (−, −, +, +, +, +, +, +) w.r.t. the basis {∂ x1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ x7 , ∂ x8 }. Then T M = span{Z 1 = ∂ x1 + ∂ x7 , Z 2 = ∂ x2 , Z 3 = cosa∂ x3 − sina∂ x4 − 2asint∂ x5 − 2acost∂ x6 , Z 4 = −a 2 cost∂ x5 + a 2 sint∂ x6 , Z 5 = ∂ x8 }, hence, M becomes a lightlike submanifold of (R82 , g), ¯ where Rad(T M) = span{Z 1 } and J¯ Rad(T M) = span{Z 5 + Z 2 }. Choose D = span{Z 3 , Z 4 } which is Rieman2θ nian. Then M becomes a pointwise slant lightlike submanifold with slant angle √1+θ 2

such that ltr (T M) = span{N = 21 {−∂ x1 + ∂ x7 }} and

S(T M ⊥ ) = span{W1 = sina∂ x3 + cosa∂ x4 , W2 = 2aseca∂ x3 + sint∂ x5 + cost∂ x6 }, which is also Riemannian. Now, we discuss conditions for a pointwise slant lightlike submanifold to be a slant lightlike submanifold. Theorem 7.6 ([22]) Suppose that M is a pointwise slant lightlike submanifold of ¯ g, an indefinite Kaehler manifold ( M, ¯ J¯). Then M is a slant lightlike submanifold, if and only if g(A F X ∗ X, Y ) = g(A F X X ∗ , Y ), for a unit tangent vector X ∈ (D p ) and Y ∈ (T M). ¯ Proof Let θ be a slant function on a pointwise slant lightlike submanifold M of M. Then for unit vector X ∈ (D p ), we have T X = cosθ p (X ).X ∗ ,

(58)

where X ∗ is a unit tangent vector in D p and orthogonal to X . Then for unit vector field X ∈ (D) and Y ∈ (T M), with (2) and (4), we derive ∇¯ Y J¯ X = cosθ p (X ){∇Y X ∗ + h l (Y, X ∗ ) + h s (Y, X ∗ )} − sinθ p (X )Y (θ p (X ))X ∗ − A F X Y + ∇Ys F X + Dl (Y, F X ), and

(59)

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J¯∇¯ Y X = T ∇Y X + F∇Y X + Bh l (Y, X ) + Bh s (Y, X ) + Ch s (Y, X ).

(60)

On expanding the Kaehlerian property ∇¯ Y J¯ X = J¯∇¯ Y X and then on comparing tangential components of (59) and (60), it yields sinθ p (X )Y (θ p (X ))X ∗ = cosθ p (X )∇Y X ∗ − A F X Y − T ∇Y X − Bh l (Y, X ) − Bh s (Y, X ). Taking inner product of the above expression w.r.t. X ∗ , it further leads to sinθ p (X )Y (θ p (X )) = cosθ p (X )g(∇Y X ∗ , X ∗ ) − g(Bh l (Y, X ), X ∗ ) − g(Bh s (Y, X ), X ∗ ) − g(A F X Y, X ∗ ) − g(T ∇Y X, X ∗ ).

(61)

Using (43) and (57), we have g(∇Y X ∗ , X ∗ ) = 0 and g(T ∇Y X, X ∗ ) = 0, respectively, then from (61), it follows that sinθ p (X )Y (θ p (X )) = −g(A F X Y, X ∗ ) + g(h s (Y, X ), J¯ X ∗ ).

(62)

Further from (5), the above expression becomes sinθ p (X )Y (θ p (X )) = −g(A F X Y, X ∗ ) + g(A F X ∗ Y, X ) = g(A F X ∗ X, Y ) − g(A F X X ∗ , Y ), and hence the result is proved.

(63) 

For X, Y ∈ (D p ), using (5) in Eq. (62), we have sinθ p (X )Y (θ p (X )) = −g(h s (Y, X ∗ ), F X ) + g(h s (Y, X ), F X ∗ ), thus, we have the following observation immediately. Theorem 7.7 ([22]) Every totally geodesic pointwise slant lightlike submanifold of an indefinite Kaehler manifold is always a slant lightlike submanifold. Assume that there exists a transversal vector bundle tr (T M) which is parallel along D with respect to the metric connection ∇¯ on an indefinite Kaehler manifold ¯ Then for W ∈ (S(T M ⊥ )) ⊂ (tr (T M)) and X ∈ (D), it follows that ∇¯ X W ∈ M. (tr (T M)). Further using this result with (4), we have A W X = 0 and then using this fact in (63), we obtain sinθ p (X )Y (θ p (X )) = 0. Hence, we have the following assertion. Theorem 7.8 ([22]) Assume that there exists a transversal vector bundle tr (T M) which is parallel along D with respect to the metric connection ∇¯ on an indefinite

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¯ Then a pointwise slant lightlike submanifold M of M¯ becomes Kaehler manifold M. a slant lightlike submanifold. Sahin [39] proved that there do not exist totally umbilical proper slant submanifolds in Kaehler manifolds other than totally geodesic proper slant submanifolds. This result is also valid for totally umbilical proper pointwise slant lightlike submanifolds of indefinite Kaehler manifolds analogously. In particular, we have the following observation. Theorem 7.9 ([22]) Every totally umbilical proper pointwise slant lightlike submanifold M of an indefinite Kaehler manifold M¯ is a totally geodesic pointwise ¯ slant lightlike submanifold of M.

8 Semi-slant Lightlike Submanifolds Definition 8.1 ([40]) Let M be a q−lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯) of index 2q such that 2q < dim(M). Then the lightlike submanifold M is said to be a semi-slant lightlike submanifold of M¯ if the following conditions are satisfied: (i) J¯ Rad(T M) is a distribution on M such that Rad(T M) ∩ J¯ Rad(T M) = {0}. (ii) There exist non-degenerate orthogonal distributions D1 and D2 on M such that ¯ (T M)) ⊕or th D1 ⊕or th D2 . S(T M) = ( J¯ Rad(T M) ⊕ Jltr (iii) The distribution D1 is an invariant distribution with respect to J¯, that is, J¯ D1 = D1 . (iv) The distribution D2 is slant with angle θ ( = 0), that is, for each p ∈ M and each non-zero vector X ∈ (D2 ) p , the angle θ between J¯ X and the vector subspace (D2 ) p is a non-zero constant, which is independent of the choice of p ∈ M and X ∈ (D2 ) p . This constant angle θ is called the slant angle of the distribution D2 . A semi-slant lightlike submanifold is said to be proper if D1 = {0}, D2 = {0} and θ = π2 . Hence, from the definition of a semi-slant lightlike submanifold of an indefinite Kaehler ¯ we have the following decomposition manifold M, ¯ (T M)) ⊕or th D1 ⊕or th D2 . T M = Rad(T M) ⊕or th ( J¯ Rad(T M) ⊕ Jltr In particular, we have (i) If D1 = 0, then M is a slant lightlike submanifold. (ii) If D1 = {0} and θ = π/2, then M is a C R−lightlike submanifold. Thus, the class of semi-slant lightlike submanifolds of indefinite Kaehler manifolds include slant lightlike submanifolds and C R−lightlike submanifolds as its sub-cases.

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Example 4 Let (R12 ¯ J¯) be an indefinite Kaehler manifold, where g¯ is of sig2 , g, nature (−, +, +, +, +, +, −, +, +, +, +, +) with respect to the canonical basis {∂ x 1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ y1 , ∂ y2 , ∂ y3 , ∂ y4 , ∂ y5 , ∂ y6 }. Suppose M is a submanifold of R12 2 and is given by x 1 = y2 = u1, x 2 = u2, x 5 = u6,

y 1 = u 3 , x 3 = y 4 = u 4 , x 4 = −y 3 = u 5 ,

y 5 = u 7 , x 6 = k cos u 7 ,

y 6 = k sin u 7 ,

where k is any constant. Then, the local frame of T M is given by {Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 , Z 7 }, where Z 1 = 2(∂ x1 + ∂ y2 ), Z 4 = 2(∂ x3 + ∂ y4 ),

Z 2 = 2∂ x2 ,

Z 3 = 2∂ y1 ,

Z 5 = 2(∂ x4 − ∂ y3 ),

Z 6 = 2(∂ x5 ),

Z 7 = 2(∂ y5 − k sin u 7 ∂ x6 + k cos u 7 ∂ y6 ). Hence, Rad(T M) = span{Z 1 } and S(T M) = span{Z 2 , Z 3 , Z 4 , Z 5 , Z 6 , Z 7 }. Then ltr (T M) is spanned by N1 = −∂ x1 + ∂ y2 , and S(T M ⊥ ) is spanned by W1 = 2(∂ x3 − ∂ y4 ), W2 = 2(∂ x4 + ∂ y3 ), W3 = 2(k cos u 7 ∂ x6 + k sin u 7 ∂ y6 ), W4 = 2(k 2 ∂ y5 + k sin u 7 ∂ x6 − k cos u 7 ∂ y6 ). It follows that J¯ Z 1 = Z 2 − Z 3 and J¯ N = 1/2(Z 2 + Z 3 ) imply J¯ Rad(T M) and ¯ (T M) are distributions on M. On the other hand, we can see that D1 = Jltr span{Z 4 , Z 5 } such that J¯ Z 4 = Z 5 implies D1 is invariant with respect√to J¯ and D2 = span{Z 6 , Z 7 } is a slant distribution with slant angle θ = arccos(1/ 1 + k 2 ). Hence, M is a semi-slant 2−lightlike submanifold of R12 2 . Shukla and Yadav [40] established necessary and sufficient conditions for a lightlike submanifold of an indefinite Kaehler manifold to be a semi-slant lightlike submanifold. Theorem 8.2 Let M be a q−lightlike submanifold of an indefinite Kaehler manifold ¯ g, ( M, ¯ J¯) of index 2q. Then M is a semi-slant lightlike submanifold of M¯ if and only if (i) J¯ Rad(T M) is a distribution on M such that Rad(T M) ∩ J¯ Rad(T M) = {0}.

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(ii) The distribution D1 is an invariant distribution with respect to J¯, that is, J¯ D1 = D1 . (iii) There exists a constant λ ∈ [0, 1) such that T 2 X = −λX . Moreover, there also exists a constant μ ∈ (0, 1] such that B F X = −μX , for all X ∈ (D2 ), where D1 and D2 are non-degenerate orthogonal distributions on M such ¯ (T M)) ⊕or th D1 ⊕or th D2 and λ = cos2 θ, θ is that S(T M) = ( J¯ Rad(T M) ⊕ Jltr a slant angle of D2 . Proof Let M be a semi-slant lightlike submanifold of an indefinite Kaehler manifold ¯ Then, the distribution D1 is an invariant distribution with respect to J¯, and M. ¯ J Rad(T M) is a distribution on M such that Rad(T M) ∩ J¯ Rad(T M) = {0}. Now, for any X ∈ (D2 ), we have |P X | = | J¯ X | cos θ; this implies cos θ =

|P X | . | J¯ X |

(64)

In view of (64), we get cos2 θ = this further gives

g(X, P 2 X ) |P X |2 g(P X, P X ) = ; = | J¯ X |2 g( J¯ X, J¯ X ) g(X, J¯2 X )

g(X, P 2 X ) = cos2 θ g(X, J¯2 X ).

(65)

Since M is a semi-slant lightlike submanifold therefore cos2 θ = λ(constant) ∈ [0, 1), then from (65), we get g(X, P 2 X ) = λg(X, J¯2 X ) = g(X, λ J¯2 X ); this implies

g(X, (P 2 − λ J¯2 )X ) = 0.

(66)

Since (P 2 − λ J¯2 )X ∈ (D2 ) and the induced metric g = g| D2 ×D2 is non-degenerate (positive definite) therefore from (66), we have (P 2 − λ J¯2 )X = 0; this implies P 2 X = λ J¯2 X = −λX.

(67)

Now, for any vector field X ∈ (D2 ), we have J¯ X = P X + F X,

(68)

where P X and F X are tangential and transversal parts of J¯ X , respectively. On applying J¯ to (68) and taking the tangential components on both sides, we get − X = P 2 X + B F X.

(69)

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From (67) and (69), we obtain B F X = −μX,

(70)

where 1 − λ = μ(constant) ∈ (0, 1] and this proves (iii). Conversely, suppose that conditions (i), (ii) and (iii) are satisfied. From (69), for any X ∈ (D2 ), we have − X = P 2 X − μX ;

(71)

P 2 X = −λX,

(72)

this implies where 1 − μ = λ(constant) ∈ [0, 1). Now cos θ =

g(X, J¯ P X ) g(X, J¯2 X ) g( J¯ X, J¯ X ) g( J¯ X, P X ) =− = −λ =λ . | J¯ X ||P X | | J¯ X ||P X | | J¯ X ||P X | | J¯ X ||P X |

From the above equation, we obtain cos θ = λ

| J¯ X | . |P X |

(73)

Therefore, (64) and (73) give cos2 θ = λ(constant). Hence, M is a semi-slant lightlike submanifold.  Later, Shukla and Yadav [40] derived conditions for D1 and the radical distribution to be integrable and to define totally geodesic foliation in M. In [41], Shukla and Yadav defined radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds as below. Definition 8.3 Let M be a 2q−lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯) of index 2q such that 2q < dim(M). Then the lightlike submanifold M is said to be a radical transversal screen semi-slant lightlike submanifold of M¯ if the following conditions are satisfied: (i) J¯ Rad(T M) = ltr (T M). (ii) There exist non-degenerate orthogonal distributions D1 and D2 on M such that S(T M) = D1 ⊕or th D2 , where the distribution D1 is invariant with respect to J and the distribution D2 is slant with angle θ ( = 0). A radical transversal screen semi-slant lightlike submanifold is said to be proper if D1 = {0}, D2 = {0} and θ = π2 . Hence, we have the following decomposition: T M = Rad(T M) ⊕or th D1 ⊕or th D2 .

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Example 5 Let (R12 ¯ J¯) be an indefinite Kaehler manifold, where g¯ is of sig2 , g, nature (−, +, +, +, +, +, −, +, +, +, +, +) with respect to the canonical basis {∂ x 1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ y1 , ∂ y2 , ∂ y3 , ∂ y4 , ∂ y5 , ∂ y6 }. Suppose M is a submanifold of R12 2 and is given by x 1 = −y 2 = u 1 , x 2 = −y 1 = u 2 , x 3 = u 3 cos β, x 4 = u 4 sin β,

y 4 = u 3 sin β, x 5 = u 5 sin u 6 , x 6 = sin u 5 ,

y 3 = −u 4 cos β, y 5 = u 5 cos u 6 ,

y 6 = cos u 5 .

Then the local frame of T M is given by {Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 }, where Z 1 = 2(∂ x1 − ∂ y2 ), Z 3 = 2(cos β∂ x3 + sin β∂ y4 ),

Z 2 = 2(∂ x2 − ∂ y1 ), Z 4 = 2(sin β∂ x4 − cos β∂ y3 ),

Z 5 = 2(sin u 6 ∂ x5 + cos u 6 ∂ y5 + cos u 5 ∂ x6 − sin u 5 ∂ y6 ), Z 6 = 2(u 5 cos u 6 ∂ x5 − u 5 sin u 6 ∂ y5 ). Hence, Rad(T M) = span{Z 1 , Z 2 } and S(T M) = span{Z 3 , Z 4 , Z 5 , Z 6 }. Then ltr (T M) is spanned by N1 = −∂ x1 − ∂ y2 , N2 = −∂ x2 − ∂ y1 and S(T M ⊥ ) is spanned by W1 = 2(sin β∂ x3 − cos β∂ y4 ), W2 = 2(cos β∂ x4 + sin β∂ y3 ), W3 = 2(sin u 6 ∂ x5 + cos u 6 ∂ y5 − cos u 5 ∂ x6 + sin u 5 ∂ y6 ), W4 = 2(u 5 sin u 5 ∂ x6 + u 5 cos u 5 ∂ y6 ). Thus, it follows that J¯ Z 1 = 2N2 and J¯ Z 2 = 2N1 imply J¯ Rad(T M) = ltr (T M). On the other hand, D1 = span{Z 3 , Z 4 } such that J¯ Z 3 = Z 4 ; this implies that D1 is invariant with respect to J¯ and D2 = span{Z 5 , Z 6 } is a slant distribution with slant angle π/4. Hence, M is a radical transversal screen semi-slant 2−lightlike submanifold of R12 2 . Further, Shukla and Yadav [41] derived necessary and sufficient conditions for a lightlike submanifold of an indefinite Kaehler manifold to be a radical transversal screen semi-slant lightlike submanifold. Theorem 8.4 Let M be a 2q−lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯). Then the lightlike submanifold M is a radical transversal screen semi-slant lightlike submanifold if and only if ¯ (T M) is a distribution on M such that Jltr ¯ (T M) = Rad(T M). (i) Jltr

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(ii) The distribution D1 is invariant with respect to J¯, that is, J¯ D1 = D1 . (iii) There exists a constant λ ∈ [0, 1) such that T 2 X = −λX . Moreover, there also exists a constant μ ∈ (0, 1] such that B F X = −μX , for all X ∈ (D2 ), where D1 and D2 are non-degenerate orthogonal distributions on M such that S(T M) = D1 ⊕or th D2 and λ = cos2 θ , θ is slant angle of D2 . Shukla and Yadav also discussed the integrability and totally geodesic foliations of the distributions of radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds in [41].

9 Screen Pseudo-slant Lightlike Submanifolds In [42], Shukla and Yadav defined screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds as below. Definition 9.1 Let M be a 2q−lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯) of index 2q such that 2q < dim(M). Then the lightlike submanifold M is said to be a screen pseudo-slant lightlike submanifold of M¯ if the following conditions are satisfied: (i) The radical distribution Rad(T M) is an invariant distribution with respect to J¯, that is, J¯ Rad(T M) = Rad(T M). (ii) There exist non-degenerate orthogonal distributions D1 and D2 on M such that S(T M) = D1 ⊕or th D2 , where J¯ D1 ⊂ S(T M ⊥ ) and the distribution D2 is slant with angle θ ( = π/2). A screen pseudo-slant lightlike submanifold is said to be proper if D1 = {0}, D2 = {0} and θ = 0. In particular, we have (i) (ii) (iii) (iv)

If If If If

D1 D2 D1 D1

= 0, then M is a screen slant lightlike submanifold. = 0, then M is a screen real lightlike submanifold. = 0 and θ = 0, then M is an invariant lightlike submanifold.

= 0 and θ = 0, then M is a screen C R−lightlike submanifold.

Thus, the class of screen pseudo-slant lightlike submanifolds of an indefinite Kaehler manifold includes invariant, screen slant, screen real and SC R−lightlike submanifolds as its sub-cases. ¯ J¯) be an indefinite Kaehler manifold, where g¯ is of Example 6 ([42]) Let (R12 2 , g, signature (−, +, +, +, +, +, −, +, +, +, +, +) with respect to the canonical basis {∂ x 1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ y1 , ∂ y2 , ∂ y3 , ∂ y4 , ∂ y5 , ∂ y6 }. Suppose M is a submanifold of R12 2 given by x 1 = y 2 = u 1 , x 2 = −y 1 = u 2 , x 3 = u 3 cos β,

y 3 = u 3 sin β,

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x 4 = u 4 sin β,

y 4 = u 4 cos β, x 5 = u 5 cos θ, x 6 = u 6 sin θ,

379

y 5 = u 6 cos θ,

y 6 = u 5 sin θ.

Then local frame of T M is given by {Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 }, where Z 1 = 2(∂ x1 + ∂ y2 ),

Z 2 = 2(∂ x2 − ∂ y1 ),

Z 3 = 2(cos β∂ x3 + sin β∂ y3 ),

Z 4 = 2(sin β∂ x4 + cos β∂ y4 ),

Z 5 = 2(cos θ ∂ x5 + sin θ ∂ y6 ),

Z 6 = 2(sin θ ∂ x6 + cos θ ∂ y5 ).

Hence, Rad(T M) = span{Z 1 , Z 2 } and S(T M) = span{Z 3 , Z 4 , Z 5 , Z 6 }. Then ltr (T M) is spanned by N1 = −∂ x1 + ∂ y2 ,

N2 = −∂ x2 − ∂ y1 ,

and S(T M ⊥ ) is spanned by W1 = 2(sin β∂ x3 − cos β∂ y3 ), W2 = 2(cos β∂ x4 − sin β∂ y4 ), W3 = 2(sin θ ∂ x5 − cos θ ∂ y6 ), W4 = 2(cos θ ∂ x6 − sin θ ∂ y5 ). It follows that J¯ Z 1 = Z 2 ; this implies that Rad(T M) is invariant and on the other hand, D1 = span{Z 3 , Z 4 } such that J¯ Z 3 = W1 and J¯ Z 4 = W2 ; this implies J¯ D1 ⊂ S(T M ⊥ ). It should be noted that D2 = span{Z 5 , Z 6 } is a slant distribution with slant angle 2θ. Thus, M is a screen pseudo-slant 2−lightlike submanifold of R12 2 . Let the projections on RadT M, D1 and D2 in T M be denoted by P1 , P2 and P3 , respectively, and the projections of tr (T M) on ltr (T M), J¯(D1 ) and D  be denoted by Q 1 , Q 2 and Q 3 , respectively, where D  is a non-degenerate orthogonal complementary subbundle of J¯(D1 ) in S(T M ⊥ ). Then, for any X ∈ (T M), we can write X = P1 X + P2 X + P3 X ; on applying J¯ to the above equation, we have J¯ X = J¯ P1 X + J¯ P2 X + J¯ P3 X ; this further gives J¯ X = J¯ P1 X + J¯ P2 X + f P3 X + F P3 X,

(74)

where f P3 X (resp. F P3 X ) denotes the tangential (resp. transversal) component of J¯ P3 X . Thus, we get

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J¯ P1 X ∈ (RadT M),

J¯ P2 X ∈ ( J¯(D1 )) ⊂ (S(T M ⊥ )),

and f P3 X ∈ (D2 ),

F P3 X ∈ (D  ).

Also, for any W ∈ (tr (T M)), we can write W = Q1 W + Q2 W + Q3 W ; on applying J¯ to the above equation, we obtain J¯ W = J¯ Q 1 W + J¯ Q 2 W + J¯ Q 3 W ; this further gives J¯ W = J¯ Q 1 W + J¯ Q 2 W + B Q 3 W + C Q 3 W,

(75)

where B Q 3 W (resp. C Q 3 W ) denotes the tangential (resp. transversal) component of J¯ Q 3 W . Therefore, we have J¯ Q 1 W ∈ (ltr (T M)), and

J¯ Q 2 W ∈ (D1 ),

B Q 3 W ∈ (D2 ),

C Q 3 W ∈ (D  ).

Then, Shukla and Yadav derived necessary and sufficient conditions for a lightlike submanifold of an indefinite Kaehler manifold to be a screen pseudo-slant lightlike submanifold in [42], as below. Theorem 9.2 ([42]) Let M be a 2q−lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯). Then the lightlike submanifold M is a screen pseudo-slant lightlike submanifold of M¯ if and only if (i) The lightlike transversal bundle ltr (T M) is invariant with respect to J¯ and J¯ D1 ⊂ S(T M ⊥ ). (ii) There exists a constant λ ∈ (0, 1] such that T 2 X = −λX . Moreover, there also exists a constant μ ∈ [0, 1) such that B F X = −μX , for all X ∈ (D2 ), where D1 and D2 are non-degenerate orthogonal distributions on M such that S(T M) = D1 ⊕or th D2 and λ = cos2 θ , θ is slant angle of D2 . Proof Let M be a screen pseudo-slant lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯). Then by definition, the distribution D1 satisfies J¯ D1 ⊂ S(T M ⊥ ) and Rad(T M) is an invariant distribution with respect to J¯. For any N ∈ (ltr (T M)) and X ∈ (S(T M)), using (74), we get g( ¯ J¯ N , X ) = −g(N ¯ , J¯ X ) = −g(N ¯ , J¯ P2 X + f P3 X + F P3 X ) = 0.

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Thus J¯ N does not belong to (S(T M)). For any N ∈ (ltr (T M)) and W ∈ (S(T M ⊥ )), from (75), we have g( ¯ J¯ N , W ) = −g(N ¯ , J¯ W ) = −g(N ¯ , J¯ Q 2 W + B Q 3 W + C Q 3 W ) = 0. Hence, we conclude that J¯ N does not belong to (S(T M ⊥ )). Now, suppose that J¯ N ∈ (Rad(T M)), then J¯( J¯ N ) = J¯2 N = −N ∈ (ltr T M); this contradicts that RadT M is invariant. Thus, ltr (T M) is invariant with respect to J¯. Next, for any X ∈ (D2 ), we have |P X | = | J¯ X | cos θ, which implies cos θ =

|P X | . | J¯ X |

(76)

In view of (76), we get cos2 θ = this further gives

g(X, P 2 X ) |P X |2 g(P X, P X ) = ; = | J¯ X |2 g( J¯ X, J¯ X ) g(X, J¯2 X )

g(X, P 2 X ) = cos2 θ g(X, J¯2 X ).

(77)

Since M is a screen pseudo-slant lightlike submanifold therefore cos2 θ = λ ∈ (0, 1] and hence from (77), we obtain g(X, P 2 X ) = g(X, λ J¯2 X ); this implies g(X, (P 2 − λ J¯2 )X ) = 0.

(78)

Since (P 2 − λ J¯2 )X ∈ (D2 ) and the induced metric g = g| D2 ×D2 is non-degenerate (positive definite), therefore from (78), we have (P 2 − λ J¯2 )X = 0, which further implies (79) P 2 X = λ J¯2 X = −λX. Now, for any vector field X ∈ (D2 ), we have J¯ X = P X + F X,

(80)

where P X and F X are tangential and transversal parts of J¯ X , respectively. On applying J¯ to (80) and taking the tangential components of both sides, we get

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− X = P 2 X + B F X,

(81)

and from (79) and (81), we get B F X = −μX, ∀X ∈ (D2 ),

(82)

where 1 − λ = μ(constant) ∈ [0, 1) and this proves the assertion (ii). Conversely, suppose that conditions (i) and (ii) are satisfied. We can show that Rad(T M) is invariant in a similar way that ltr (T M) is invariant. From (81), for any X ∈ (D2 ), we get (83) − X = P 2 X − μX ; this implies P 2 X = −λX,

(84)

where 1 − μ = λ(constant) ∈ (0, 1]. Next cos θ =

g(X, P 2 X ) g(X, J¯2 X ) g( J¯ X, J¯ X ) g( J¯ X, P X ) =− = −λ =λ . | J¯ X ||P X | | J¯ X ||P X | | J¯ X ||P X | | J¯ X ||P X |

From the above equation, we get cos θ = λ

| J¯ X | . |P X |

(85)

Therefore, (76) and (85) give cos2 θ = λ(constant) and hence, M is a screen pseudoslant lightlike submanifold.  Later, Shukla and Yadav [42] derived conditions for the distributions of screen pseudo-slant lightlike submanifolds to be integrable and to be defined totally geodesic foliations in M. It is well known that the induced connection on a lightlike submanifold of a semi-Riemannian is not a metric connection. Shukla and Yadav [42] also obtained conditions for the induced connection on screen pseudo-slant lightlike submanifolds to be a metric connection. In [43], Shukla and Yadav defined radical transversal screen pseudo-slant lightlike submanifolds of indefinite Kaehler manifolds. Definition 9.3 Let M be a 2q−lightlike submanifold of an indefinite Kaehler man¯ g, ifold ( M, ¯ J¯) of index 2q such that 2q < dim(M). Then the lightlike submanifold M is said to be a radical transversal screen pseudo-slant lightlike submanifold of M¯ if the following conditions are satisfied: (i) J¯ Rad(T M) = ltr (T M). (ii) There exist non-degenerate orthogonal distributions D1 and D2 on M such that S(T M) = D1 ⊕or th D2 , where J¯ D1 ⊂ S(T M ⊥ ) and the distribution D2 is slant with angle θ ( = π/2).

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A radical transversal screen pseudo-slant lightlike submanifold is said to be proper if D1 = {0}, D2 = {0} and θ = 0. Hence, we have T M = Rad(T M) ⊕or th D1 ⊕or th D2 .

(86)

Example 7 Let (R12 ¯ J¯) be an indefinite Kaehler manifold, where g¯ is of sig2 , g, nature (−, +, +, +, +, +, −, +, +, +, +, +) with respect to the canonical basis {∂ x 1 , ∂ x2 , ∂ x3 , ∂ x4 , ∂ x5 , ∂ x6 , ∂ y1 , ∂ y2 , ∂ y3 , ∂ y4 , ∂ y5 , ∂ y6 }. Suppose M is a submanifold of R12 2 and is given by x 1 = y 2 = u 1 , x 2 = y 1 = u 2 , x 3 = u 3 cos β, x 4 = u 4 sin β,

y 4 = u 4 cos β, x 5 = u 5 ,

x 6 = k cos u 6 ,

y 3 = u 3 sin β, y5 = u6,

y 6 = k sin u 6 ,

where k is any constant. Then the local frame of T M is given by {Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 }, where Z 1 = 2(∂ x1 + ∂ y2 ), Z 3 = 2(cos β∂ x3 + sin β∂ y3 ), Z 5 = 2(∂ x5 ),

Z 2 = 2(∂ x2 + ∂ y1 ), Z 4 = 2(sin β∂ x4 + cos β∂ y4 ),

Z 6 = 2(∂ y5 − k sin u 6 ∂ x6 + k cos u 6 ∂ y6 ).

Hence Rad(T M) = span{Z 1 , Z 2 } and S(T M) = span{Z 3 , Z 4 , Z 5 , Z 6 }. Then ltr (T M) is spanned by N1 = −∂ x1 + ∂ y2 ,

N2 = −∂ x2 + ∂ y1 ,

and S(T M ⊥ ) is spanned by W1 = 2(sin β∂ x3 − cos β∂ y3 ), W2 = 2(cos β∂ x4 − sin β∂ y4 ), W3 = 2(k cos u 6 ∂ x6 + k sin u 6 ∂ y6 ), W4 = 2(k 2 ∂ y5 + k sin u 6 ∂ x6 − k cos u 6 ∂ y6 ). It follows that J¯ Z 1 = −2N2 , J¯ Z 2 = −2N1 ; this implies that J¯ Rad(T M)=ltr (T M). On the other hand, we can see that D1 = span{Z 3 , Z 4 } such that J¯ Z 3 = W1 , ⊥ J¯ Z 4 = W2 imply J¯ D1 ⊂ S(T M √ ) and D2 = span{Z 5 , Z 6 } is a slant distribution with slant angle θ = arccos(1/ 1 + k 2 ). Hence, M is a radical transversal screen pseudo-slant 2−lightlike submanifold of R12 2 .

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Theorem 9.4 ([43]) Let M be a 2q−lightlike submanifold of an indefinite Kaehler ¯ g, manifold ( M, ¯ J¯). Then the lightlike submanifold M is a radical transversal screen pseudo-slant lightlike submanifold of M¯ if and only if ¯ (T M) is a distribution on M such that Jltr ¯ (T M) = Rad(T M) and J¯ D1 ⊂ (i) Jltr ⊥ S(T M ). (ii) There exists a constant λ ∈ (0, 1] such that T 2 X = −λX . Moreover, there also exists a constant μ ∈ [0, 1) such that B F X = −μX , for all X ∈ (D2 ), where D1 and D2 are non-degenerate orthogonal distributions on M such that S(T M) = D1 ⊕or th D2 and λ = cos2 θ , θ is slant angle of D2 . Finally, Shukla and Yadav [43] also obtained conditions for the distributions of radical transversal screen pseudo-slant lightlike submanifold of indefinite Kaehler manifolds to be integrable and to define totally geodesic foliation in M. Remark 4 It is well known that contact geometry has a vital role in the theory of differential equations, optics and phase spaces of a dynamical system (for details, see [1, 28, 31]). Fritelli et al. [19] gave a self-contained presentation of the null surface formulation of the Einstein field equations of general relativity, based on the contact geometry of differential equation. Hence, due to potential applications of contact geometry in mathematical physics, many geometers are also working on the geometry of slant lightlike submanifolds of indefinite contact manifolds these days. Acknowledgements The authors would like to express their sincere gratitude to the referees for their valuable suggestions, which definitely improved our manuscript.

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