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Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
MATHEMATICS RESEARCH DEVELOPMENTS
Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
TENSOR OPERATORS AND THEIR APPLICATIONS
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TENSOR OPERATORS AND THEIR APPLICATIONS
ARIF SALIMOV
New York
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Copyright © 2013 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com
NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.
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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Salimov, Arif, 1956- author. Tensor operators and their applications / Arif Salimov (Ataturk University, Faculty of Science, Dep. of Mathematics, Erzurum, Turkey). pages cm Includes bibliographical references and index.
ISBN: (eBook)
1. Tensor algebra. I. Title. QA200.S25 2012 515'.724--dc23
2012020865
Published by Nova Science Publishers, Inc. † New York
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To my teacher Professor Vladimir Vishnevskii
Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
Contents
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Preface 1 On Operators Applied to Pure Tensor Fields 1.1. Pure Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . 1.2. Tachibana Operators . . . . . . . . . . . . . . . . . . . . . . 1.2.1. ϕφ −operator Applied to a Tensor Field of Type (1,1) . 1.2.2. ϕφ −operator Applied to a Tensor Field of Type (1,s), s≥2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. ϕφ −operator Applied to a 1-form . . . . . . . . . . . 1.2.4. ϕφ −operator Applied to a Tensor Field of Type (0, s), s ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . 1.2.5. ϕφ −operator Applied to a Tensor Field of Type (r, s) . 1.3. Vishnevskii Operators . . . . . . . . . . . . . . . . . . . . . . 1.3.1. ψφ −operator Applied to a Tensor Field of Type (1,s), s≥0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. ψφ −operator Applied to a Tensor Field of Type (0,s) . 1.3.3. ψφ −operator Applied to a Tensor Field of Type (r,s), r>1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. ψφ −operator Applied to a Pure Connection . . . . . . . . . . 1.5. Tachibana Operators Applied to a Mixed Tensor Field . . . . . 1.5.1. Pure Tensor Fields of Mixed Kind on Submanifolds . . 1.5.2. ϕφ,φ˜ −operators . . . . . . . . . . . . . . . . . . . . . 1.5.3. ϕφ,φ˜ −operator Applied to Tensor Fields of Type (1, s, 0, q) and (0, s, 1, q) . . . . . . . . . . . . . . . . 1.5.4. ϕφ,φe −operator Applied to Tensor Fields of Type (0, s, 0, 0) and (0, 0, 0, q) . . . . . . . . . . . . . . . .
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. . .
1 2 4 6
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8 9
. 11 . 13 . 14 . 15 . 16 . . . . .
17 19 21 21 22
. 24 . 26
viii
Contents
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1.6. Yano-Ako Operators . . . . . . . . . . . . . . . . . . . . . . 1.6.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 1.6.2. ϕS −operator Applied to a Tensor Field of Type (1, s) . 1.6.3. ϕS −operator Applied to a Tensor Field of Type (0, s) . 1.7. ψS −operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1. ψS −operator Applied to a Tensor Field of Type (1, s), s≥0 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2. ψS −operator Applied to a Tensor Field of Type (0, s) . 1.8. Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 2 Algebraic Structures on Manifolds 2.1. Algebraic Theory . . . . . . . . . . . . . . . . . . 2.1.1. Associative Algebras . . . . . . . . . . . . 2.1.2. Commutative Algebras . . . . . . . . . . . 2.1.3. Holomorphic Functions . . . . . . . . . . 2.2. Algebraic Π−structures on Manifolds . . . . . . . 2.3. Integrable Regular Π−structure . . . . . . . . . . 2.4. Pure Tensors with Respect to the Regular Structure 2.5. A-holomorphic Tensors in Real Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . 2.6. Pure Connections . . . . . . . . . . . . . . . . . . 2.7. Torsion Tensors of Pure Π−connections . . . . . . 2.8. A−holomorphic Hypercomplex Connection . . . . 2.9. Some Properties of Pure Curvature Tensors . . . .
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28 28 29 30 32
. 33 . 34 . 35
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39 40 40 41 43 46 50 54
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57 58 61 62 64
3 Applications to the Norden Geometry 3.1. Hyper-K¨ahler-Norden Manifolds . . . . . . . . . . . . . . . . 3.2. Complex K¨ahler-Norden Manifolds . . . . . . . . . . . . . . 3.3. Almost Product Riemannian Manifolds . . . . . . . . . . . . 3.3.1. Decomposable Riemannian Manifolds . . . . . . . . . 3.3.2. Para-K¨ahler-Norden Manifolds . . . . . . . . . . . . . 3.3.3. Nonexistence of Para-K¨ahler-Norden Warped Metrics 3.4. Dual-K¨ahler-Norden Manifolds . . . . . . . . . . . . . . . . . 3.5. Norden–Hessian Structures . . . . . . . . . . . . . . . . . . . 3.6. Norden-Walker Manifolds with Proper Structures . . . . . . . 3.6.1. Almost Norden-Walker Metrics . . . . . . . . . . . . 3.6.2. Integrability of Proper Almost Complex Structures . .
. . . . . . . . . . .
69 70 76 79 79 80 83 86 89 92 93 95
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3.6.3. Holomorphic Norden-Walker (K¨ahler-Norden-Walker) Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4. Curvature Properties of Norden-Walker Manifolds . . 3.6.5. Isotropic K¨ahler-Norden-Walker Structures . . . . . . 3.6.6. Quasi-K¨ahler-Norden-Walker Structures . . . . . . . . 3.6.7. On the Goldberg Conjecture . . . . . . . . . . . . . . 3.7. Opposite Almost Complex Structure . . . . . . . . . . . . . . 3.8. Para-Norden-Walker Metrics . . . . . . . . . . . . . . . . . . 3.9. Some Notes Concerning Norden-Walker 8-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Applications to the Theory of Lifts 4.1. Tensor Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Horizontal and Complete Lifts of Vector Fields . . . . . . . . 4.2.1. Vertical Lifts of Tensor Fields and γ−operator . . . . . 4.2.2. Complete Lifts of Vector Fields . . . . . . . . . . . . 4.2.3. Horizontal Lifts of Vector Fields . . . . . . . . . . . . 4.2.4. Complete Lifts of Derivations . . . . . . . . . . . . . 4.2.5. Derivations DKX Y and Formulas on Lie Derivations . . 4.3. Cross-sections in the Tensor Bundle . . . . . . . . . . . . . . 4.4. Lifts of Affinor Fields . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Complete Lifts of Affinor Fields . . . . . . . . . . . . 4.4.2. Almost Complex Structures on Tqp (Mn ) . . . . . . . . 4.4.3. Almost Hyperholomorphic Pure Submanifolds in the Tensor Bundle . . . . . . . . . . . . . . . . . . . . . 4.4.4. Horizontal Lifts of Affinor Fields . . . . . . . . . . . 4.4.5. Diagonal Lifts of Affinor Fields . . . . . . . . . . . . 4.5. Lifts of Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Adapted Frames . . . . . . . . . . . . . . . . . . . . 4.5.2. Sasakian Metrics on the Tensor Bundles . . . . . . . . 4.5.3. Geodesics on Tensor Bundles . . . . . . . . . . . . . 4.5.4. Jacobi Tensor Fields . . . . . . . . . . . . . . . . . . 4.6. Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Para-Nordenian Structures in Cotangent Bundles . . . 4.6.2. Paraholomorphic Cheeger-Gromoll Metric in the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 4.6.3. On Almost Complex Structures in Tangent Bundles . .
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96 98 99 101 102 105 107
. 111
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117 118 120 120 121 122 122 124 126 128 128 133
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135 136 138 140 141 142 145 151 152 153
. 160 . 166
viii
Contents 4.7. Complete Lift of a Skew-Symmetric Tensor Field . . . . . . . . 168 175
Index
185
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References
Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
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Preface The notion of derivation of tensor fields is one of the central concepts and tools of differential geometry. Typical examples of these derivations include Lie and covariant differentiations with respect to a vector field on manifolds. A tensor operator applied to a pure tensor field with respect to a fixed tensor field of type (1, q), q > 0 is a generalization of Lie and covariant differentiations. An especially important class of tensor operators is the class of Tachibana operators associated with a considering fixed tensor field of type (1, 1). The study of Tachibana operators was started in the early 1960s by Tachibana [97], Tachibana and Koto [98] and Sato [88]. Shirokov [93] and Kruchkovich [36] developed the theory of Tachibana operators associated with a commutative hypercomplex structure. If affinors of commutaive hypercomplex structure are covariantly constant with respect to the torsion-free connection, then the class of Tachibana operators coincides with the class of Vishnevskii operators. This operator was first considered by Vishnevskii [103] and devoloped by Vishnevskii, Shirokov and Shurygin [107]. Lather the Tachibana and Vishnevskii operators was questioned in Norden geometry and in theory of complete and horizontal lifts by Salimov [64], [67]. Other important class of tensor operators is the class of Yano-Ako operators associated with a tensor field of type (1, 2). An affirmative answer to the application of Yano-Ako operators was obtained in [68], [86]. On the other hand, tensor operator can be applied to a pure connection and these operators were lately investigated in [71]. The author believe that differential geometric applications of tensor operators is a very fruitful research domain and povides many new problems in the study of modern differential geometry. However, in spite of its importance, tensor operators and related topics are not as yet so well-known, and there is no reference book covering this field. This was the motivation for publishing the present monograph. This research monograph is intended to provide a sys-
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Arif Salimov
tematic introduction to the theory of tensor operators. Let us first detailedly outline tensor operators and describe some of the areas in which they find applications. Our goal is to expound the recent developments in applications of tensor operators in hypercomplex, Norden geometry and theory of lifts. We plan to present all these developments in four chapters. Preliminaries needed in this monograph are intermediate level aspects of differential geometry. Instead of collecting them in a single chapter, we prefer to introduce them briefly as they are encountered in each section. In Chapter 1 we introduce tensor operators. We treat pure tensor fields with respect to the fixed tensor field, introduce several different families of tensor operators applied to pure tensor fields, and also treat the theory of tensor operators applied to pure connections. Commutative hypercomplex structures on manifolds defined by a fixed tensor field of type (1, 1) are discussed in Chapter 2. We introduce basic algebraic theory, hypercomplex structures and holomorphic manifold theory. Formulas are given for the holomorphic pure objects in the context of Tachibana operators associated with integrable commutative hypercomplex structures which are central to our subsequent discussion. Chapter 3 deals with special Riemannian metrics which is not necessarily positive definite. For example, complex Norden manifolds are defined by a pure metrics of neutral signature. We examine Kahler metrics of Norden type in the context of Tachibana operators. The second part of this chapter is devoted to a study of Kahler metrics of Norden type on the Walker manifolds. Chapter 4 treats the theory of complete, horizontal and diagonal lifts of tensor fields on a pure cross-sections in the tensor bundle. Various tensor operators are used for interpretation and construction of these lifts. This work was supported through the grants TBAG-108T590 and TBAG105T551 from TUBITAK (Turkey). I am very grateful to Professors V. N. Berestovskii and V. V. Shurygin for their valuable suggestions and encouragements.
Arif Salimov Ataturk University, Turkey January 2012
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Chapter 1
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On Operators Applied to Pure Tensor Fields In this chapter we introduce the tensor operators which are used throughout this book. In Section 1.1, we give the definition of pure tensor field with respect to the affinor structure (i.e. tensor structure of type (1, 1) ) and its pure product. In Section 1.2, we define the Tachibana operator applied to a pure tensor field. Explicit expressions of these operators for different pure tensors are presented. In Section 1.3, by using a linear connection, we define the Vishnevskii operator and we show that if the affinor structure is covariantly constant with respect to the torsion-free connection, then the Tachibana operator reduces to the Vishnevskii operator. Section 1.4 devoted to the study of tensor operators applied to a pure connection. By using these operators we discuss the purity conditons of curvature tensors. In Section 1.5, we discuss some tensor operators concerning pure tensor fields of mixed kind. In Section 1.6, we consider the case when the tensor structure on manifolds is a tensor structure of type (1, 2) and we define the Yano-Ako operator applied to a pure tensor field with respect to the tensor structure of type (1, 2). Also explicit expressions of Yano-Ako operators are presented. In Section 1.7, by using a linear connection, we define an operator associated with a given tensor structure of type (1, 2) and applied to an arbitrary tensor field. We prove that if the tensor structure of type (1, 2) is covariantly constant with respect to the torsion-free connection, then the Yano-Ako operator reduces to this operator. Section 1.8 deals with generalizations of some tensor operators.
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1.1. Pure Tensor Fields Let M be a C∞-manifold of finite dimension n. We denote by ℑrs (M) the module over F(M) of all C∞ -tensor fields of type (r, s) on M, i.e. of contravariant degree r and covariant degree s, where F(M) is the algebra of C∞ -functions on M. Definition 1. Let ϕ be an affinor field on M, i.e. ϕ ∈ ℑ11 (M). A tensor field t of type (r, s) is called pure tensor field with respect to ϕ if 1 2
1 2
r
r
t(ϕX1 , X2 , ..., Xs, ξ, ξ, ..., ξ) = t(X1 , ϕX2 , ..., Xs, ξ, ξ, ..., ξ) ... 1 2
r
= t(X1 , X2 , ..., ϕXs , ξ, ξ, ..., ξ) 1 2
r
= t(X1 , X2 , ..., Xs,´ϕξ, ξ, ..., ξ) 1
2
(1.1)
r
= t(X1 , X2 , ..., Xs, ξ,´ϕξ, ..., ξ) .. . Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
1 2
r
= t(X1 , X2 , ..., Xs, ξ, ξ, ...,´ϕξ) 1 2
r
for any X1 , X2 , ..., Xs ∈ ℑ10 (M) and ξ, ξ, ..., ξ ∈ ℑ01 (M), where ´ϕ is the adjoint operator of ϕ defined by (´ϕξ)(X) = ξ(ϕX) = (ξ ◦ ϕ)(X), X ∈ ℑ10 (M), ξ ∈ ℑ01 (M). Let x1 , x2 , ..., xn be a local coordinate system in M. ∂ , ..., Xs ∂xi1
1
By setting X1 =
r
= ∂x∂is and ξ = dx j1 , ..., ξ = dx jr in (1.1), we see that the condition j ... j of pure tensor fields may be expressed in terms of the components ϕij and ti11...is r as follows: jr m ... jr m j1 ... jr m ϕ = ϕ = ... = tij11i... ϕ = tij11m...i tmi 2 ...m is s i2 2 ...is i1 m j ... j
ti1 ...i2 s r ϕmj1
j m... j
(1.2)
j j ...m
= ti11...is r ϕmj2 = ... = ti11...i2 s ϕmjr .
We consider for convenience’s sake the vector, covector and scalar fields as pure tensor fields.
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3
Example 1. In particular, let now t ∈ ℑ11 (M) be a pure tensor field of type (1, 1). Then (1.1) may be written as: ϕ(tX) = t(ϕX). Thus if t ∈ ℑ11 (M) and ϕ ∈ ℑ11 (M) satisfies the commutativity condition ϕ ◦ t = t ◦ ϕ, C
(1.3)
C
where (ϕ ◦t)X = (ϕ ⊗ t)X = ϕ(tX) (⊗ is a tensor product with a contraction C), then t is a pure with respect to ϕ, and conversely ϕ is also a pure with respect to t. From (1.3) it follows easily that ϕ itself and the unit affinor field I are examples of the pure tensor field. Also, from (1.3) we have: if ϕ is a regular affinor field, i.e. det(ϕij ) 6= 0, then the affinor field ϕ−1 whose components are given by the elements of the inverse matrix of ϕ is also pure. Example 2. We consider the matrix equation Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
T
ϕg = gϕ,
(1.4)
where g = (gi j ), ϕ = (ϕij ), T ϕ is a transpose matrix. From (1.4) we obtain gim ϕmj = gm j ϕm i , which by virtue of (1.2) is a purity condition of a tensor field g ∈ ℑ02 (M). Remark 1. Note that, when g is a (pseudo-)Riemannian metric, a linear operator ϕ satisfying the purity condition g (ϕX,Y ) = g (X, ϕY ) is called a selfadjoint. For example, the Jacobi operator and the Ricci operator on a (pseudo-) Riemannian manifold, also an affinor of an almost complex structure on a Norden manifold are self-adjoint operators. Pure tensor fields have been studied in [19], [36], [37], [58], [66], [77], [88], [93], [97], [98], [105], [108], [112], [113] from different view points. From (1.1) we easily see that if K and L are both pure tensor fields of type (r, s), then K + L and f K ( f ∈ F(M)) are also pure tensor fields. We denote by ∗
ℑrs (M) the module of all pure tensor fields of type (r, s) on M with respect to
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Arif Salimov
the affinor field ϕ. If K and L are pure tensor fields of types (p1 , q1 ), p1 ≥ 1 and (p2 , q2 ), q2 ≥ 1 respectively, then the tensor product of K and L with contraction mi ...i r ...r 2 p 1 p K ⊗ L = K j1 ... jq 1 Lms2 ...s2q2 C
1
is also a pure tensor field. For simplicity, we shall prove only the case when ∗
∗
K ∈ ℑ11 (M) and L ∈ ℑ02 (M). In fact, C
(K ⊗ L)(ϕX,Y ) = K(L(ϕX,Y )) = K(L(X, ϕY )) C
= (K ⊗ L)(X, ϕY ), X,Y ∈ ℑ10 (M). ∗
∞
∗
We shall now make the direct sum ℑ(M) = ∑ ℑrs (M) into an alegebra over r,s=0
C
the real number R by defining the pure product (denoted by ⊗ or ” ◦ ”) of K ∈ ∗
∗
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ℑqp11 (M) and L ∈ ℑqp22 (M) as follows: mi2 ...i p r1 ...r p K j1 ... jq 1 Lms2 ...s2q2 , i f 1 i1 ...i p1 mr2 ...r p2 C C K L , if m j ... ⊗ : (K, L) → (K ⊗ L) = 2 jq1 s1 ...sq2 0 , if 0 , if
p1 , q2 ≥ 1, p2 , q1 ≥ 1, p1 , p2 = 0, q1 , q2 = 0.
In particular, let K = X ∈ ℑ10 (M), and L ∈ Λq (M) be a q−form. Then the pure C
product X ⊗ L coincides with the usual interior product ιX L.
1.2. Tachibana Operators r Definition 2. [71], [97] Let ϕ ∈ ℑ11 (M), and ℑ(M) = ∑∞ r,s=0 ℑs (M) be a tensor ∗
algebra over R. A map φϕ |r+s>0 : ℑ(M) → ℑ(M) is called a Tachibana operator or φϕ −operator on M if (a) φϕ is linear with respect to constant coefficients, ∗
(b) φϕ : ℑrs (M) → ℑrs+1 (M) for all r and s, C
C
C
∗
(c) φϕ (K ⊗ L) = (φϕ K) ⊗ L + K ⊗ φϕ L for all K, L ∈ ℑ(M).
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(d) φϕX Y = −(LY ϕ)X for all X,Y ∈ ℑ10 (M), where LY is the Lie derivation with respect to Y . (e) (φϕX ω)Y = (d(ιY ω))(ϕX)−(d(ιY (ω◦ϕ)))X +ω((LY ϕ)X) = (ϕX)(ιY ω) −X(ιϕY ω) + ω((LY ϕ)X) for all ω ∈ ℑ01 (M) and X,Y ∈ ℑ10 (M), where ιY ω = C
ω(Y ) = ω ⊗ Y . Remark 2. If r = s = 0, then from (c), (d) and (e) of Definition 2 we have φϕX (ιY ω) = (ϕX)(ιY ω) − X(ιϕY ω) for ιY ω ∈ ℑ00 (M), which is not well-defined φϕ −operator. Different choices of Y and ω leading to same function f = ιY ω do 2 get the same values. Consider M = R with standard coordinates x, y. Let ϕ = 0 1 . Consider the function f ≡ 1. This may be written in many different 1 0 ∂ ∂ ∂ ways as ιY ω. Indeed taking ω = dx, we may choose Y = ∂x or Y = ∂x + x ∂y . Now the right-hand side of φϕX (ιY ω) = (ϕX)(ιY ω) − X(ιϕY ω) is (ϕX)1 − 0 = 0 ∂ in the first case, and (ϕX)1 − Xx = −Xx in the second case. For X = ∂x , the latter expression is −1 6= 0. Therefore, we put r + s > 0.
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Remark 3. From (d) of Definition 2 we have φϕX Y = [ϕX,Y ] − ϕ [X,Y ] . By virtue of [ f X, gY ] = f g [X,Y ] + f (Xg)Y − g (Y f ) X for any f , g ∈ F(M), we see that φϕX Y is linear in X, but not Y. We have in fact φϕ( f X)Y
= [ f ϕX,Y ] − ϕ [ f X,Y ] =
f [ϕX,Y ] − (Y f )ϕX − ϕ( f [X,Y ]) + ϕ(Y f )X
=
f ([ϕX,Y ] − ϕ [X,Y ])
=
f φϕX Y,
φϕX (gY ) = g φϕX Y + ((ϕX) g)Y − (Xg)ϕY. In the sequel we write φϕY X for φϕX Y .
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1.2.1.
φϕ−operator Applied to a Tensor Field of Type (1,1) ∗
Let t ∈ ℑ11 (M), i.e. t ◦ ϕ = ϕ ◦ t (see Example 1). Then tY ∈ ℑ10 (M) for any Y ∈ ℑ10 (M) and by (c) of Definition 2 we have φϕtY X
C C φϕt X ⊗ Y + t ⊗ φϕY X = φϕt (X,Y ) + t φϕY X . =
(1.5)
Using (1.5), we have from (d) of Definition 2
φϕt (X,Y ) = φϕtY X − t φϕY X
= (−LtY ϕ + t (LY ϕ))X = [ϕX,tY ] − ϕ [X,tY ] − t [ϕX,Y ] + ϕt [X,Y ] = Qϕ,t (X,Y ) .
(1.6)
The tensor field Qϕ,t ∈ ℑ12 (M) is called the Nijenhuis-Shirokov tensor field [35], [36]. Thus we have Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
∗
Theorem 1. Let t ∈ ℑ11 (M). Then φϕt is the Nijenhuis-Shirokov tensor field. Since t ◦ ϕ = ϕ ◦ t is trivially satisfied for t = ϕ, we from (1.6) obtain φϕ ϕ (X,Y ) =
−LϕY ϕ + ϕ (LY ϕ) X
= [ϕX, ϕY ] − ϕ [X, ϕY ] − ϕ [ϕX,Y ] + ϕ2 [X,Y ] = Nϕ (X,Y ),
where Nϕ is the Nijenhuis tensor field constructed from ϕ [57]. Thus we have Theorem 2. If Nϕ is the Nijenhuis tensor field of ϕ, then Nϕ = φϕ ϕ.
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∗
Let now t ◦ ϕ 6= ϕ ◦ t , i.e. t ∈ / ℑ11 (M). In this case φϕt is not a tensor field. We consider the expression φϕt + φt ϕ (X,Y ) = [ϕX,tY ] + [tX, ϕY ] + ϕt [X,Y ] +tϕ [X,Y ] − ϕ [X,tY ] − ϕ [tX,Y ] −t [X, ϕY ] − t [ϕX,Y ], which is nothing but the torsion tensor field Sφ,t of φ and t [57], [33, p.38]. Thus we have Theorem 3. Let ϕ,t ∈ ℑ11 (M). The expression φϕt + φt ϕ without the purity condition t ◦ϕ = ϕ◦t defines a tensor field Sϕ,t of type (1, 2), where Sϕ,t is the torsion tensor field of ϕ and t. From the expression (1.6) and from Theorem 3 we have
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Corollary 4. Let t be a tensor field of type (1, 1) without the purity condition. Then Sϕ,t (X,Y ) = Qϕ,t (X,Y ) − Qϕ,t (Y, X) + (ϕt − tϕ) [X,Y ] . By a straightforward computation, we have (a) φϕ I = 0, (b) φϕX (tY ) = Qϕ,t (X,Y ) + t ◦ φϕY X, (c) φϕt (X,Y ) = Qϕ,t (X,Y ) = −Qt,ϕ (Y, X) = − (φt ϕ) (Y, X), (d) Qϕ,t1 ◦t2 = Qϕ,t1 ◦ t2 + t1 ◦ Qϕ,t2 , ∗
(e) Qϕ,ϕ◦t2 = Nϕ ◦ t2 + ϕ ◦ Qϕ,t2 for any t,t1,t2 ∈ ℑ11 (M) and X,Y ∈ ℑ10 (M), where I is an identity affinor and C Qϕ,t Y X = (Qϕ,t ⊗Y )X = Qϕ,t (X,Y ) , C t1 ◦ Qϕ,t2 (X,Y ) = (t1 ⊗ Qϕ,t2 ) (X,Y ) = Qϕ,t2 (X,t1Y ) , Qϕ,t1 ◦ t2 (X,Y ) = t2 (Qϕ,t1 (X,Y )).
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Arif Salimov
φϕ−operator Applied to a Tensor Field of Type (1,s), s ≥ 2
1.2.2.
∗
Let t ∈ ℑ1s (M), that is ϕ(t(Y1,Y2 , ...,Ys)) = t(ϕY1 ,Y2 , ...,Ys) .. . = t(Y1,Y2 , ..., ϕYs). Using (c) and (d) of Definition 2 we have φϕt (X,Y1 ,Y2 , ...,Ys) = φϕ (t (Y1 ,Y2 , ...,Ys)) X s − ∑ t Y1 ,Y2 , ..., φϕYλ X, ...,Ys λ=1
= − Lt(Y1 ,Y2 ,...,Ys ) ϕ X
(1.7)
s
+ ∑ t Y1 ,Y2 , ..., LYλ ϕ X, ...,Ys . λ=1
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The expression (1.7) has components h φϕ t
s
k j1 ... js
h h m m h h m = ϕm k ∂m t j1 ... js − ϕm ∂k t j1 ... js − t j1 ... js ∂m ϕk + ∑ t j1 ...m... js ∂ jλ ϕk (1.8) λ=1
with respect to natural coordinate system in M, which are components of a tensor field of type (1, s + 1) with the purity condition of t. Remark 4. In particular, let t ∈ ℑ12 (M). Using the alternation of φϕt k j1 j2
with respect to k, j1 and j2 , we can check by an elementary tensor computation h that φϕt are components of a tensor field of type (1, 3) without the purity [k j1 j2 ]
condition of t [113].
h Willmore [111] proved that the tensor field with components φϕt
[k j1 j2 ]
reduces to the tensor field introduced by Slebodzinski [95], when t = Nφ , ϕ2 = −id, and the Slebodzinski tensor field is identically zero.
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1.2.3.
9
φϕ−operator Applied to a 1-form
Let ω ∈ ℑ01 (M). Using Definition 2, we have (φϕ ω)(X,Y) = (φϕX ω)Y
(1.9)
= (ϕX) (ιY ω) − X ιϕY ω + ω ((LY ϕ)X) = (LϕX ω − LX (ω ◦ ϕ))Y
for any X,Y ∈ ℑ10 (M), where the 1-form ω ◦ ϕ is defined by C
(ω ◦ ϕ)Y = (´ϕω)Y = (ω ⊗ ϕ)Y = ω (ϕY ) . From (1.9) we see that the tensor field φϕ ω ∈ ℑ02 (M) has components m m φϕ ω = ϕm i ∂m ω j − ∂i (ωm ϕ j ) + ωm ∂ j ϕi ij
with respect to the natural frame. The φϕ ω is not skew-symmetric in i ij and j in general. The the tensor field φϕ ω coincides with the tensor field
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[i j]
introduced by Frolicher and Nijenhuis [29], [30].
Theorem 5. Let ω be an exact 1-form. Then ω ∈ Kerφϕ if and only if an associated 1-form ω ◦ ϕ is a closed 1-form. Proof. Let ω ∈ ℑ01 (M). Using [33, p.36] 1 (dω)(X,Y) = {X(ω(Y )) −Y ω(X) − ω([X,Y])} 2 for any X,Y ∈ ℑ10 (M) and ω ∈ ℑ01 (M),we have 1 {Y (ω(ϕX)) − (ϕX)(ω(Y )) − ω([Y, ϕX])} 2 1 = {Y (ω(ϕX)) − (ϕX)(ω(Y )) + ω([ϕX,Y ])} 2 1 = {Y (ω(ϕX)) − (ϕX)(ω(Y )) + ω([ϕX,Y ] 2 −ϕ[X,Y ]) + ω(ϕ[X,Y ])}. (1.10)
(dω)(Y, ϕX) =
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From (1.9), we have (φϕ ω)(X,Y ) = (ϕX)(ω(Y )) − X(ω(ϕY )) − ω([ϕX,Y ] − ϕ[X,Y ]).
(1.11)
Substituting (1.11) into (1.10), we obtain 1 (dω)(Y, ϕX) = {− φϕ ω (X,Y ) +Y (ω(ϕX)) − X(ω(ϕY )) + ω(ϕ[X,Y ])} 2 1 = {− φϕ ω (X,Y ) +Y ((ω ◦ ϕ)(X)) 2 −X((ω ◦ ϕ)(Y )) − (ω ◦ ϕ)([Y, X])} 1 = − φϕ ω (X,Y ) + (d(ω ◦ ϕ))(Y, X). 2
From this we see that equation φϕ ω = 0 is equivalant to
(d(ω ◦ ϕ))(Y, X) = (dω)(Y, ϕX), which for ω = d f turns into the following simple form: (d(d f ◦ ϕ))(Y, X) = (d 2 f )(Y, ϕX) = 0, Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
i.e. d f ◦ ϕ is a closed 1-form. Theorem 6. Let ω ∈ ℑ01 (M) and ϕ2 = −id. Then ω ◦ ϕ ∈ Ker φϕ if and only if ω ∈ Ker φϕ . Proof. If we substitute ω ◦ ϕ into ω and ϕX into X, then the equation (1.9) may be written as (φϕ (ω ◦ ϕ))(ϕX,Y ) = (Lϕ2 X (ω ◦ ϕ) − LϕX (ω ◦ ϕ2 ))Y = −(LX (ω ◦ ϕ) + LϕX ω)Y = −(φϕ ω)(X,Y ) or ((φϕ (ω ◦ ϕ)) ◦ ϕ)(X,Y ) = −(φϕ ω)(X,Y), from which by virtue of detϕ 6= 0, we see that φϕ (ω ◦ ϕ) = 0 if and only if φϕ ω = 0. Corollary 7. Let ω ∈ ℑ01 (M), and ω ∈ Kerφϕ . If ϕ2 = −id, then ω ◦ Nϕ = 0, where ω ◦ Nϕ (X,Y ) = ω Nϕ (X,Y ) . Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
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Proof. It follows immediately from Theorem 6 and the following formula: φϕ (ω ◦ ϕ) = (φϕ ω) ◦ ϕ + ω ◦ Nϕ .
1.2.4.
φϕ−operator Applied to a Tensor Field of Type (0, s), s ≥ 2 ∗
Theorem 8. Let ω ∈ ℑ0s (M). Then (φϕ ω) (X,Y1 , ...,Ys) = (ϕX) (ω (Y1 , ...,Ys)) − X (ω (ϕY1 , ...,Ys)) s
+ ∑ ω(Y1 , ..., (LYλ ϕ)X, ...,Ys). λ=1
Proof. For simplicity, let s = 2 and ωY1 be an 1-form such that ωY1 (Y2 ) = ω (Y1 ,Y2 ) . By (e) of Definition 2 we have (φϕX ωY1 )Y2 = (ϕX)(ωY1 (Y2 )) − X(ωY1 (ϕY2 )) + ωY1 ((LY2 ϕ)X) Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
= (ϕX)(ω (Y1 ,Y2 )) − X(ω (Y1 , ϕY2 )) + ω(Y1 , (LY2 ϕ)X), from which, by virtue of C
C
C
(φϕX (ω ⊗Y1 ))Y2 = ((φϕX ω) ⊗ Y1 + ω ⊗ (φϕX Y1 ))Y2 = (φϕX ω) (Y1 ,Y2 ) + ω(φϕX Y1 ,Y2 ) it follows that (φϕX ω) (Y1 ,Y2 ) = (ϕX)(ω(Y1 ,Y2 )) − X(ω (ϕY1 ,Y2 )) +ω(Y1 , (LY2 ϕ)X) + ω((LY1 ϕ)X,Y2 ) Thus the proof is completed. ∗
Let now ω ∈ ℑ02 (M), i.e. ω(ϕX,Y ) = ω(X, ϕY ).
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(1.12)
12
Arif Salimov
Taking account of (1.12), Theorem 8, and φϕX Y = −(LY ϕ)X we now have (φϕ ω)(X,Y, Z) = (ϕX)(ω(Y, Z)) − X(ω(ϕY, Z))
(1.13)
+ω((LY ϕ)X, Z) + ω(Y, (LZ ϕ)X) = (LϕX ω − LX (ω ◦ ϕ))(Y, Z), where the tensor field ω ◦ ϕ is defined by (ω ◦ ϕ)(X,Y ) = ω(ϕX,Y ). ∗
Remark 5. We note that if ω ∈ ℑ02 (M) is a symmetric (skew-symmetric) pure ∗
tensor field, then ω ◦ ϕ ∈ ℑ02 (M) is also symmetric (skew-symmetric) pure tensor field. From (1.13) we see that the tensor field φϕ ω ∈ ℑ03 (M) has components m m φϕ ω = ϕm i ∂m ωi j − ∂k (ω ◦ ϕ)i j + ωm j ∂i ϕk + ωim ∂ j ϕk ki j
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with respect to the natural frame, where
m (ω ◦ ϕ)i j = ωm j ϕm (1.14) i = ωim ϕ j . The expression φϕ ω are components of a tensor field of type (0,3) [ki j] without the purity condition (1.14) , and if ω is a 2-form, then φϕ ω coin[ki j]
cides with the tensor field introduced by Frolicher and Nijenhuis [29], [30]. ∗
If ω ∈ ℑ0s (M), s ≥ 2, then taking account of Theorem 8, we have φϕ ω (X,Y1 , ...,Ys) = (ϕX) (ω (Y1 , ...,Ys)) − X (ω (ϕY1 , ...,Ys)) s + ∑ ω Y1 , ..., LYλ ϕ X, ...,Ys λ=1
= where ω ◦ ϕ is defined by
LϕX ω − LX (ω ◦ ϕ) (Y1 , ...,Ys) ,
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(1.15)
On Operators Applied to Pure Tensor Fields
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(ω ◦ ϕ)(Y1 , ...,Ys) = ω (ϕY1 ,Y2 , ...,Ys) .. . = ω (Y1 ,Y2 , ..., ϕYs ). Using (1.15), in a way similar to that of the proof of Theorem 6, we can prove ∗
Theorem 9. Let ω ∈ ℑ0s (M), s ≥ 2, and ϕ2 = −id. Then ω ◦ ϕ ∈ Kerφϕ if and only if ω ∈ Kerφϕ . ∗
Corollary 10. Let ω ∈ ℑ0s (M), s ≥ 2, ω ∈ Kerφϕ . If ϕ2 = −id, then ω ◦ Nϕ = 0, where ω ◦ Nϕ (X,Y1 , ...,Ys) = ω Nϕ (X,Y1 )Y2 , ...,Ys .
φϕ−operator Applied to a Tensor Field of Type (r, s)
1.2.5.
∗
Let t ∈ ℑrs (M), r > 1, s ≥ 1. We now define a pure tensor field of type ∗
(0,s) t 1 2
r
ξ,ξ,...,ξ Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
t1 2
r
1 2
∈ ℑ0s (M) by t 1 2
r
ξ,ξ,...,ξ
r
(Y1 ,Y2 , ...,Ys) = t(Y1 ,Y2 , ...,Ys, ξ, ξ, ..., ξ), where
has components of the form:
ξ,ξ,...,ξ 1 2
(t 1 2
r
ξ,ξ,...,ξ
r
r ) j1 j2 ... js = t ij11...i ... js ξi1 ξi2 ...ξir .
According to Theorem 8, we find 1 2
r
r
1
µ
r
φϕX t(Y1 , ...,Ys, ξ, ξ, ..., ξ)+ ∑ t(Y1, ...,Ys, ξ, ..., φϕX ξ, ..., ξ) µ=1
= (φϕX t 1 2
r
−X(t 1 2
r
ξ,ξ,...,ξ
) (Y1 , ...,Ys) = (ϕX) (t 1 2
r
ξ,ξ,...,ξ
s ξ,ξ,...,ξ
(ϕY1 , ...,Ys))+ ∑ t 1 2
r
λ=1 ξ,ξ,...,ξ
(Y1 , ...,Ys))
Y1 , ..., (LYλ ϕ)X, ...,Ys .
µ µ µ Then, using φϕX ξ = LϕX ξ − LX ξ ◦ ϕ (see (1.9)), we see that φϕt for t ∈ ∗
ℑrs (M), r > 1, s ≥ 1, is by definition, a tensor field of type (r, s + 1) given by
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Arif Salimov 1 2 r 1 2 r φϕ t X,Y1 , ...,Ys, ξ, ξ, ..., ξ = φϕX t Y1 , ...,Ys, ξ, ξ, ..., ξ 1 2 r 1 2 r = (ϕX)t Y1 , ...,Ys, ξ, ξ, ..., ξ − Xt ϕY1 , ...,Ys, ξ, ξ, ..., ξ s 1 2 r + ∑ t Y1 , ..., LYλ ϕ X, ...,Ys, ξ, ξ, ..., ξ λ=1 r
µ µ 1 r − ∑ t Y1 , ...,Ys, ξ, ..., LϕX ξ − LX ξ ◦ ϕ , ..., ξ .
(1.16)
µ=1
µ
By setting X = ∂k , Yλ = ∂ jλ , ξ = dxiµ , λ = 1, ..., s; µ = 1, ..., r in the equa i1 ...ir tion (1.16), we see that the components φϕt of φϕt with respect to local j1 ... js
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coordinate system x1 , ..., xn may be expressed as follows:
φϕt
i1 ...ir
k j1 ... js
i1 ...ir i1 ...ir = ϕm k ∂m t j1 ... js − ∂k (t ◦ ϕ) j1 ... js
(1.17)
s r i1 ...ir iµ iµ r + ∑ ∂ jλ ϕm t + ∂ ϕ − ∂ ϕ , ∑ k m m k t ij11...m...i k j1 ...m... js ... js λ=1
µ=1
where i1 ...ir m i1 ...ir m m...ir i1 i1 ...m ir r (t ◦ ϕ)ij11...i ... js = tm... js ϕ j1 = ... = t j1 ...m ϕ js = t j1 ... js ϕm = ... = t j1 ... js ϕm .
The operator (1.17) first introduced by Tachibana [97].
1.3. Vishnevskii Operators Suppose now that ∇ is a linear connection on M, and let ϕ ∈ ℑ11 (M).We can replace the condition (d) of Definition 2 by 0
(d ) ψϕX Y = ∇ϕX Y − ϕ (∇X Y ) for any X,Y ∈ ℑ10 (M). Then we can consider a new operator as follows.
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Definition 3. By a Vishnevskii operator or ψϕ −operator on M, we shall mean a ∗
map ψϕ : ℑ(M) → ℑ(M), which satisfies conditions (a), (b), (c), (e) of Definition 0 2 and the condition (d ). Remark 6. Taking account of definition of connection ∇, we easily see that ψϕX Y is linear in X, but not Y . In the sequal we write ψϕY X for ψϕX Y .
1.3.1.
ψϕ −operator Applied to a Tensor Field of Type (1,s), s ≥ 0 ∗
Let t ∈ ℑ1s (M). Since t (Y1 ,Y2 , ...,Ys) ∈ ℑ10 (M), then using Definition 3, we have ψϕX t (Y1 ,Y2 , ...,Ys) = ψϕt (X,Y1 ,Y2 , ...,Ys) s = ψϕX t (Y1 ,Y2 , ...,Ys) − ∑ t Y1 , ..., ψϕYλ X, ...,Ys λ=1
= ∇ϕX t (Y1 ,Y2 , ...,Ys) − ϕ∇X (t (Y1 ,Y2 , ...,Ys)) s − ∑ t Y1 , ..., ψϕYλ X, ...,Ys ,
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λ=1
ψϕX t (Y1 ,Y2 , ...,Ys) =
∇ϕX t (Y1 , ...,Ys) +
s
∑t
Y1 , ..., ∇ϕX Yλ , ...,Ys
λ=1 s
−ϕ (∇X t) (Y1 , ...,Ys) − ϕ( ∑ t (Y1 , ..., ∇X Yλ , ...,Ys)) λ=1
− ∑ t Y1 , ..., ψϕYλ X, ...,Ys . s
(1.18)
λ=1
Since t is a pure tensor field, we have s
ϕ( ∑ t (Y1 , ..., ∇X Yλ , ...,Ys)) = λ=1
s
∑ t (Y1, ..., ϕ(∇X Yλ) , ...,Ys) .
λ=1
Substituting (1.19) into (1.18), we obtain ψϕ t (X,Y1 , ...,Ys) = ∇ϕX t − ϕ (∇X t) (Y1 , ...,Ys)
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(1.19)
16
Arif Salimov s s + ∑ t Y1 , ..., ∇ϕX Yλ − ϕ (∇X Yλ ) , ...,Ys − ∑ t Y1 , ..., ψϕYλ X, ...,Ys λ=1
∇ϕX t − ϕ (∇X t) (Y1 , ...,Ys).
= Thus
λ=1
ψϕt (X,Y1 , ...,Ys) = ∇ϕX t − ϕ (∇X t) (Y1 , ...,Ys) .
(1.20)
From (1.20) we have
∗
Theorem 11. Let t ∈ ℑ1s (M). If ∇t = 0, then t ∈ Kerψϕ .
1.3.2.
ψϕ −operator Applied to a Tensor Field of Type (0,s)
Let ω ∈ ℑ01 (M). Using Definition 3, we have
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ψϕ ω (X,Y ) = (ψϕX ω)Y
(1.21)
= (ϕX)(ιY ω) − X(ιϕY ω) − ω ∇ϕX Y − ϕ (∇X Y ) = ∇ϕX ω − ∇X (ω ◦ ϕ) Y
for any X,Y ∈ ℑ10 (M),where (ω ◦ ϕ)Y = ω (ϕY ). From (1.21) we see that ψϕX ω = ∇ϕX ω − ∇X (ω ◦ ϕ) is a 1−form. ∗
Let now ω ∈ ℑ0s (M), s > 1. Using Theorem 8, by similar devices we have ψϕ ω (X,Y1 , ...,Ys) = ψϕX ω (Y1 , ...,Ys) s
= (ϕX)(ω (Y1 , ...,Ys)) − Xω (ϕY1 , ...,Ys) − ∑ (ω Y1 , ..., ∇ϕX Yλ , ...,Ys λ=1
− ω (Y1 , ..., ϕ(∇X Yλ ) , ...,Ys)) = ∇ϕX ω (Y1 , ...,Ys) − Xω (ϕY1 ,Y2 , ...,Ys) +
= +
s
∑ ω (Y1 , ..., ϕ∇X Yλ, ...,Ys) λ=1
∇ϕX ω (Y1 , ...,Ys) − Xω (ϕY1 ,Y2 , ..,Ys) + ω (ϕ(∇X Y1 ),Y2, ...,Ys)
ω (ϕY1 , ∇X Y2 , ...,Ys) + ... + ω (ϕY1 ,Y2 , ..., ∇X Ys )
= (∇ϕX ω − ∇X (ω ◦ ϕ) (Y1 , ...,Ys) .
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Thus ψϕ ω (X,Y1 , ...,Ys) = (∇ϕX ω − ∇X (ω ◦ ϕ) (Y1 , ...,Ys) .
(1.22)
From (1.22) we have
∗
/ Kerψϕ . Theorem 12. Let ω ∈ ℑ0s (M). If ∇ω = 0, then ω ∈
1.3.3.
ψϕ −operator Applied to a Tensor Field of Type (r,s), r > 1 ∗
Let t ∈ ℑrs (M). By similar devices (see Section 1.2.5) we have
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1 2 r ψϕt X,Y1 , ...,Ys, ξ, ξ, ..., ξ 1 2 r 1 2 r = (ϕX)t Y1 , ...,Ys, ξ, ξ, ..., ξ − Xt ϕY1 ,Y2 , ...,Ys, ξ, ξ, ..., ξ s 1 2 r − ∑ t Y1 , ..., ∇ϕX Yλ − ϕ (∇X Yλ ) , ...,Ys, ξ, ξ, ..., ξ λ=1 r
µ µ 1 r ∑ t Y1 , ...,Ys, ξ, ..., ∇ϕX ξ − ∇X ξ ◦ ϕ , ..., ξ .
−
µ=1
Substituting µ µ 1 r t Y1 , ...,Ys, ξ, ..., ∇ϕX ξ − ∇X ξ ◦ ϕ , ..., ξ µ µ µ 1 r = t Y1 , ...,Ys, ξ, ..., ∇ϕX ξ − ∇X ξ ◦ ϕ − ξ ◦ ∇X ϕ, ..., ξ µ µ r 1 0 = t Y1 , ...,Ys, ξ, ..., ∇ϕX ξ − ϕ∇X ξ, ..., ξ µ 1 r − t Y1 , ...,Ys, ξ, ..., ξ◦ ∇X ϕ, ..., ξ into (1.23), we have
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(1.23)
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Arif Salimov
1 2 r ψϕt X,Y1 , ...,Ys, ξ, ξ, ..., ξ =
1 2 r ∇ϕX t − ∇X (t ◦ ϕ) Y1 , ...,Ys, ξ, ξ, ..., ξ µ 1 r r + ∑µ=1 t Y1 , ...,Ys, ξ, ..., ξ ◦ ∇X ϕ, ..., ξ . (1.24)
Let now ∇ϕ = 0. Then from (1.24), we have ψφX t = ∇ϕX t − (∇X t) ◦ ϕ.
(1.25)
From (1.25) we easily see that ψϕt has components i1 ...ir i1 ...ir mi2 ...ir j1 ψϕt = ϕm k ∇m t j1 ... js − ϕm ∇k t j1 ... js k j1 ... js
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with respect to the natural frame {∂i } . The operator (1.25), first introduced by Vishnevskii [103] for an integrable ϕ−structures. In a manifold with an affinor field ϕ, a connection ∇ is called a ϕ−connection if ∇ϕ = 0. Theorem 13. If a torsion tensor of ϕ−connection ∇ is pure, then φϕX t = ψϕX t ∗
for any t ∈ ℑrs (M). Proof. Let ∇ϕ = 0 and ϕT (X,Y ) = T (ϕX,Y ) = T (X, ϕY ), where T (X,Y ) = ∇X Y − ∇Y X − [X,Y ]. From (d) of Definition 2 and (d´), we obtain φϕX Y
= −(LY ϕ)X = [ϕX,Y ] − ϕ[X,Y ] = ∇ϕX Y − ∇Y ϕX − T (ϕX,Y ) − ϕ(∇X Y − ∇Y X − T [X,Y ]) = ∇ϕX Y − ϕ(∇X Y ) − (∇ϕ)(Y, X) + ϕT (X,Y ) − T (ϕX,Y ) = ∇ϕX Y − ϕ(∇X Y ) = ψϕX Y.
Remark 7. Since a zero torsion tensor field is pure, it follows that φϕX Y = ψϕX Y for any torsion-free ϕ−connection.
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1.4.
19
ψϕ −operator Applied to a Pure Connection
Let ϕ ∈ ℑ11 (M), and ∇ be a torsion-free ϕ−connection, i.e. ∇ϕ = 0. Then, it is i well known that ϕ is an integrable [92], i.e. the matrix ϕ = ϕ j is reduced to constant form in a certain (holonomic) natural frame in a neighbourhood Ux of every point x ∈ M. If ∇ is a torsion-free ϕ−connection, then from ∇ϕ = 0 we have k m m k Γkm j ϕm i = Γim ϕ j = Γi j ϕm
(1.26)
with respect to adapted local coordinates, where Γkij are components of ∇. We call this connection a pure connection with respect to ϕ [36]. We denote the curvature tensor of the pure connection ∇ by R, which belongs to ℑ13 (M) : R(X,Y)Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
X,Y, Z ∈ ℑ10 (M).
Applying Ricci’s identity to ϕ:
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∇X ((∇Y ϕ)Z) − ∇Y ((∇X ϕ)Z)
= R(X,Y)ϕZ − ϕ (R(X,Y )Z) + ∇[X,Y ] ϕ Z + (∇Y ϕ) (∇X Z) − (∇X ϕ)(∇Y Z) , we get ϕR (X,Y ) Z = R (X,Y )ϕZ by virtue of ∇ϕ = 0. Thus, we have Theorem 14. Let ϕ ∈ ℑ11 (M). If ∇ is a pure connection with respect to ϕ, then ϕR (X,Y ) Z = R (X,Y )ϕZ
(1.27)
for any X,Y, Z ∈ ℑ10 (M). Theorem 15. Let ϕ ∈ ℑ11 (M), and ∇ be a pure connection with respect to ϕ. The curvature tensor field R of ∇ is a pure tensor field with respect to ϕ if and only if ψϕX (∇Y Z) = ∇ϕX ∇Y Z − ϕ (∇X ∇Y Z) = 0
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for any X,Y, Z ∈ Kerψϕ . Proof. Since X,Y, Z ∈ Kerψϕ (see(d´)) and ψϕY X = φϕY X = −(LX ϕ)Y = 0 (∇ϕ = 0, T (X,Y ) = ∇X Y − ∇Y X − [X,Y ] = 0) we have R(X, ϕY )Z = ∇X ∇ϕY Z − ∇ ϕY ∇X Z − ∇[X,ϕY ] Z = ∇X ϕ(∇Y Z) − ∇ϕY ∇X Z − ∇(LX ϕ)Y +ϕLX Y Z = ϕ(∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z) +ϕ(∇Y ∇X Z) − ∇ ϕY ∇X Z = ϕR(X,Y)Z − ψϕY (∇X Z). Thus R(X, ϕY )Z = ϕR(X,Y )Z − ψϕY (∇X Z).
(1.28)
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By virtue of R(X,Y )Z = −R(Y, X)Z, from (1.28) we have R(ϕX,Y )Z = −R(Y, ϕX)Z = −ϕR(Y, X)Z + ψϕX (∇Y Z).
(1.29)
From (1.27)-(1.29), we see that the curvature tensor R of a pure connection ∇ is pure in all arguments if and only if ψϕX (∇Y Z) = 0 for any X,Y, Z ∈ Kerψϕ . Thus Theorem 15 is proved. From (1.29) we see that ψϕX (∇Y Z) ∈ ℑ13 (M). Therefore, in the sequel we write ψϕ ∇ (X,Y, Z) for ψϕX (∇Y Z): ψϕ ∇ (X,Y, Z) = ∇ϕX ∇Y Z − ϕ (∇X ∇Y Z) .
Thus ψϕ ∇ is a ψϕ −operator (or a φϕ −operator) applied to a pure connection ∇. We note that, if a pure connection ∇ is a connection of Kahler-Norden manifolds, then automatically it follows that ψϕ ∇ = 0 (see Chapter 3).
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1.5. Tachibana Operators Applied to a Mixed Tensor Field 1.5.1.
Pure Tensor Fields of Mixed Kind on Submanifolds
Let M˜ be an m−dimensional submanifold of an n−dimensional manifold M defined by immersion i : M˜ → M. The submanifold M˜ can be expressed locally by i ∂x i i A x = x u , i = 1, ..., n, A = 1, ..., m, rank = m. ∂uA i
∂x We put BA = BiA ∂i = ∂u A ∂i , A = 1, ..., m, which are the m tangent vectors ˜ with local expression X˜ = X˜ A ∂A , to i M˜ in M. For an element X˜ ∈ ℑ10 (M) we denote by BX˜ ∈ ℑ10 (M) the tangent vector field to i M˜ in M with local expression
BX˜ = BiA X˜ A ∂i .
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Thus the correspondence X˜ → BX˜ determines a mapping ˜ → ℑ10 i M˜ B : ℑ10 (M)
,
which is the differential of the immersion i : M˜ → M, i.e. B = i∗ and so an 1 ˜ 1 ˜ isomorphism of ℑ0 (M) onto ℑ0 i M . By a tensor field of mixed kind (or a tensor field of type (r, s; p, q)) we mean a mapping
˜ × ... × ℑ10 (M)× ˜ t : ℑ10 (M) × ... × ℑ10 (M) × ℑ10 (M) 0 0 0 ˜ 0 ˜ ˜ ×ℑ1 (M) × ... × ℑ1 (M) × ℑ1 (M) × ... × ℑ1 (M) → ℑ00 (M) ˜ q times, ℑ0 (M): ˜ p times) defined ( ℑ10 (M): s times, ℑ01 (M): r times, ℑ10 (M): 1 ˜ ˜ on M that it is a tensor field of type (p, q) on M for fixed arguments X1 , ..., Xs ∈ ℑ10 (M) and ξ1 , ..., ξr ∈ ℑ01 (M), and is a tensor field of type (r, s) on M for fixed 1 p ˜ and e ˜ arguments X˜1 , ..., X˜ q ∈ ℑ10 (M) ξ , ..., e ξ ∈ ℑ0 (M). 1
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˜ If Definition 4. We assume the existence of ϕ ∈ ℑ11 (M) and ϕ˜ ∈ ℑ11 (M). p 1 r e1 e ˜ ˜ t X1 , ..., ϕXλ , ..., Xs, X1 , ..., Xq, ξ , ..., ξ , ξ , ..., ξ 1 p = t X1 , ..., Xs, X˜1 , ..., ϕ˜ X˜µ , ..., X˜ q, ξ1 , ..., ξr , e ξ , ..., e ξ , λ = 1, ..., s, µ = 1, ..., q,
then we say that t is pure with respect to the pair Xλ , X˜ µ . If t is pure with respect to all pairs of its vector and covector arguments, then it is called a pure ˜ tensor field of type (r, s; p, q) with respect to (ϕ, ϕ).
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∼
˜ (or ξ ∈ ℑ0 (M)) ˜ and X ∈ ℑ10 (M) (or ξ ∈ ℑ0 (M)) The vector fields X˜ ∈ ℑ10 (M) 1 1 ˜ restricted to M is considered to be pure, by convention. In particular, the affinor fields ϕ and ϕ˜ are pure tensor fields. Let K and L be pure tensor fields of type (r1 , s1 ; p1 , q1 ) and (r2 , s2 ; p2 , q2 ), respectively. The definition of pure product can be extended to a pure tensors of mixed kind, as follows: C pure product K(X˜ ,ξ˜ ) ⊗L(Y˜ ,η˜ ) on M, 1 p1 C ˜ 1 , ...,Y ˜ q2 ,η e1 , ...,η e p2 for fixed X˜ 1 , ..., X˜ q, e ξ , ..., e ξ ,Y K ⊗L = C ˜ pure product K(X,ξ) ⊗L(Y,η) on M, 1 r1 for fixed X1 , ..., Xs1 , ξ , ..., ξ ,Y1 , ...,Ys2 , η1 , ..., ηr2 ∗
∗
r,p ˜ ˜ = ∑∞ Then as in Section 1.1, we can define an algebra ℑ(M) r,s,p,q=0 ℑs,q (M) C
∗
r,p ˜ over R with respect to the pure product ⊗, where ℑs,q (M) is the module of all ˜ ˜ on M. pure tensor fields of type (r, s; p, q) with respect to (ϕ, ϕ)
1.5.2.
φϕ,ϕ˜ −operators
Let ϕ ∈ ℑ11 (M) and M˜ be a submanifold of M. If ϕTx M˜ ⊂ Tx (M) for each ˜ then M˜ is said to be invariant in M. x ∈ M, Let M˜ be an invariant submanifold. Then as m vectors BA = B∂˜ A = i ˜ their images are linear BA ∂i , A = 1, ..., m in M span the tangent space of M, ˜ such combinations of themselves. Hence there exists an affinor field ϕ˜ ∈ ℑ11 (M) that
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˜ CA BCi . ϕim Bm A =ϕ
(1.30) ˜ satisfying (1.30), then M˜ Conversely if there exists an affinor field ϕ˜ ∈ ℑ11 (M) is invariant. The equation (1.30) means that BiA is pure tensor field of type (1, 0, 0, 1) on ˜ The equation (1.30) may also be written as M˜ with respect to (ϕ, ϕ).
or
˜ 0 ϕξ) B ϕ˜ X˜ , ξ = B(X,
i.e.
˜ 0 ϕξ) = B X, ˜ ξ ◦ ϕ = (ξ ◦ ϕ) BX˜ = ξ ϕ BX˜ , ξB ϕ˜ X˜ = B(X, ϕ BX˜ = B ϕ˜ X˜
(1.31)
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˜ for any X˜ ∈ ℑ10 (M). ˜ and Now we introduce an operator on M˜ which is associated with ϕ and ϕ, applied to a pure tensors of mixed kind. Definition 5. Let ϕ ∈ ℑ11 (M), and M˜ be an invariant submanifold with an ∗
˜ ˜ → ℑ(M) ˜ ( induced affinor field ϕ˜ ∈ ℑ11 (M). A map φϕ,ϕ˜ |r+s+p+q>0 : ℑ(M) ∗
r,p ˜ ˜ = ∑∞ ℑ(M) r,s,p,q=0 ℑs,q (M) is an algebra of tensors of mixed kind ) is called ˜ if a Tachibana operator or φϕ,ϕ˜ −operator on M, (a) φϕ,ϕ˜ is linear with respect to constant coefficients, ∗ r,p
∗ r,p
˜ → ℑs,q+1 (M) ˜ for all r, s, p and q, (b) φϕ,ϕ˜ : ℑs,q (M) C ∗ C C ˜ (c) φϕ,ϕ˜ (K ⊗L) = φϕ,ϕ˜ K ⊗L + K ⊗ φϕ,ϕ˜ L for all K, L ∈ ℑ(M), ˜ ˜ X˜ for any Y˜ ∈ ℑ10 (M), (d1 ) φϕ,ϕ˜ Y˜ X˜ = φϕ˜ Y˜ X˜ = − (LY˜ ϕ) (d2 ) φϕ,ϕ˜ Y X˜ = φϕY BX˜ = − (LY ϕ) BX˜ for any Y ∈ ℑ10 (M), ˜ Ye ) = φϕ˜ ω ˜ Ye = d ((iY˜ ω)) ˜ ˜ (X, ˜ X, ϕ˜ X˜ (e1 ) φϕ,ϕ˜ ω ˜ for any ω ˜ ˜ X) ˜ ∈ ℑ01 (M), ˜ ◦ ϕ)) ˜ X˜ + ω((L ˜ ϕ) ( ω −d (i ˜ Y Y˜ ˜ ) = φϕ ω BX,Y ˜ (e2 ) φϕ,ϕ˜ ω (X,Y = d ((iY ω)) ϕ BX˜ ˜ for any ω ∈ ℑ0 (M). −d (iY (ω ◦ ϕ)) BX˜ + ω((LY ϕ)(BX)) 1
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φϕ,ϕ˜ −operator Applied to Tensor Fields of Type (1, s, 0, q) and (0, s, 1, q)
1.5.3.
∗ 1,0 e Let t ∈ ℑs,q (M). Then t Y1 , ...,Ys, Ye1 , ..., Yeq ∈ ℑ10 (M) for any Y1 , ...,Ys ∈ ℑ10 (M) e Using Definition 5 we have and Ye1 , ..., Yeq ∈ ℑ1 (M). 0
e 1 , ...,Ys, Ye1 , ..., Yeq) (φϕ,eϕt Y1 , ...,Ys, Ye1 , ..., Yeq )Xe = (φϕ,eϕt)(X,Y s
e ...,Ys, Ye1 , ..., Yeq) + ∑ t(Y1, ..., (φϕ,eϕYλ )X, λ=1 q
e ..., Yeq). + ∑ t(Y1 , ...,Ys, Ye1 , ..., (φϕ,eϕYeµ )X, µ=1
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From this we have following
e 1 , ...,Ys, Ye1 , ..., Yeq) (φϕ,eϕt)(X,Y = (φϕ,eϕt Y1 , ...,Ys, Ye1 , ..., Yeq )Xe
(1.32)
s
e ...,Ys, Ye1 , ..., Yeq) − ∑ t(Y1 , ..., (φϕ,eϕYλ )X, λ=1
e = −(Lt (Y1 ,...,Ys ,Ye1 ,...,Yeq ) ϕ)(BX) s
e ...,Ys, Ye1 , ..., Yeq) + ∑ t(Y1 , ..., (LYλ ϕ)(BX), λ=1 q
e ..., Yeq). e)X, + ∑ t(Y1 , ...,Ys, Ye1 , ..., (LYeµ ϕ µ=1
By setting Xe = ∂C ,Yλ = ∂ jλ , Yeµ = ∂Aµ in the equation (1.32), we see that the components (φϕ,eϕ t)Ci j1 ... js A1 ...Aq may be expressed as follows: (φϕ,eϕ t)Ci j1 ... js A1 ...Aq =
(1.33)
i h i m −BCh (t ij1 ... js A1 ...Aq ∂m ϕih − ϕm h ∂m t j1 ... js A1 ...Aq ) − BC (ϕm ∂h t j1 ... js A1 ...Aq )
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s
25 m
i eC + ∑ BCh t ij1 ...m... js A1 ...Aq ∂ jλ ϕm h + ∑ t j1 ... js A1 ...m...Aq ∂Aµ ϕ µ=1
λ=1
∗i j1 ... js A1 ...Aq
eCm ∂mt ij1 ... js A1 ...Aq − ∂C t = ϕ
q
m
eC + ∑ t ij1 ... js A1 ...m...Aq ∂Aµ ϕ µ=1
s
+BCh t ij1 ... js A1 ...Aq (∂h ϕim − ∂m ϕih ) + ∑ BCh t ij1 ...m... js A1 ...Aq ∂ jλ ϕm h, λ=1
∗
where t ij1 ... js A1 ...Aq = ϕimt ij1 ... js A1 ...Aq . The operator (1.33) first introduced by Tachibana and Koto [98] (see also [93]) . ∗
e Then by similar devices we obtain Let now t ∈ ℑ1,0 s,q (M).
e 1 , ...,Ys, Ye1 , ..., Yeq) = −(L e e (φϕ,eϕt)(X,Y t (Y1 ,...,Ys ,Ye1 ,...,Yeq ) ϕ)X s
e ...,Ys, Ye1 , ..., Yeq) + ∑ t(Y1 , ..., (LYλ ϕ)(BX),
(1.34)
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λ=1 q
e ..., Yeq). e)X, + ∑ t(Y1, ...,Ys, Ye1 , ..., (LYeµ ϕ µ=1
e in (1.32), (1.34) respectively, then In particular, if we put t = ϕ and t = ϕ we have e ) = Nϕ (BX,Y e ), (φϕ,eϕ ϕ e Ye ) = Nϕe (X, e Ye), e)(X, (φϕ,eϕ ϕ)(X,Y
e, respectively. where Nϕ and Nϕe are Nijenhuis tensors of ϕ and ϕ ∗
e Since B ∈ ℑ1,0 0,1 (M), we now consider the operator (1.32) acting on B: e Ye) = −(L e ϕ)(BX) e + B((Le ϕ e e (φϕ,eϕ B)(X, Y )X) BY
e + ϕ(L e BX) e + B((Le ϕ e e = −LBYe (ϕ(BX)) Y )X). BY i h i h e Ye [115, p.63] and (1.31), we have e BYe = B X, By virtue of BX,
e e −ϕ e Ye ) = −L e (B(ϕ e + ϕ(L e BX) e + B(Le ϕX e (LYe X)) eX)) (φϕ,eϕ B)(X, Y BY i h i h BY i h e − Bϕ e + B(Le ϕX) e ϕ e Ye , Xe eYe + ϕB Ye , X = −B X, Y
= 0.
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Thus we have e be an invariant submanifold of M with an Theorem 16. Let ϕ ∈ ℑ11 (M), and M 1 e e ∈ ℑ1 (M). Then induced ϕ φϕ,eϕ B = 0 for B ∈
∗ e ℑ1,0 0,1 (M),
e → M. where B is the differential of the immersion i : M C
e and B are pure tensor fields, a tensor fields ϕ ◦ B = ϕ ⊗ B and Since ϕ, ϕ C e = B⊗ϕ e are also pure tensor fields of type (1, 0, 0, 1), where (ϕ ◦ B)Ye = B◦ϕ e If we apply φϕ,eϕ to (1.31), e)Ye = B(ϕ eYe ) for any Ye ∈ ℑ10 (M). ϕ(BYe ) and (B ◦ ϕ then by virtue of Theorem 2, Theorem 16 and (c) of Definition 5 we have e Ye) = (φϕ,eϕ (B ◦ ϕ e Ye ), e))(X, (φϕ,eϕ (ϕ ◦ B))(X,
e BYe) = (B ◦ (φϕ,eϕ ϕ e Ye) = (B ◦ Nϕe )(X, e Ye) = B(Nϕe (X, e Ye)), e ))(X, (φϕ,eϕ ϕ)(X, Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
e BYe ) = B(Nϕe (X, e Ye)), Nϕ (BX,
(1.35)
e, respectively. Since B is where Nϕ and Nϕe are Nijenhuis tensors of ϕ and ϕ injective, from (1.35) we see that if Nϕ = 0, then Nϕe = 0. Thus we have e be the induced affinor field on invariant Theorem 17. Let ϕ ∈ ℑ11 (M), and ϕ e submanifold M. Then if the Nijenhuis tensor Nϕ of ϕ vanishes, the Nijenhuis e vanishes too. tensor of ϕ
1.5.4.
φϕ,eϕ −operator Applied to Tensor Fields of Type (0, s, 0, 0) and (0, 0, 0, q) ∗
e Let now ω ∈ ℑ0,0 s,0 (M). Using (e2 ), by similar devices, as in Section 1.3.2, we have e 1 , ...,Ys) (φϕ,eϕ ω)(X,Y
e e = (ϕ(BX))(ω(Y 1 , ...,Ys)) − (BX)ω(ϕY1 , ...,Ys) s
e ...,Ys) e − ϕ(LY BX), + ∑ ω(Y1 , ..., LYλ ϕ(BX) λ λ=1
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(1.36)
On Operators Applied to Pure Tensor Fields
27
s
e = (ϕ(BX))(ω(Y 1 , ...,Ys)) − ∑ ω(Y1 , ..., Lϕ(BXe )Yλ , ...,Ys) λ=1 s
e −(BX)(ω ◦ ϕ)(Y1 , ...,Ys) + ∑ (ω ◦ ϕ)(Y1, ..., LBXeYλ , ...,Ys) λ=1
= (Lϕ(BXe ) ω − LBXe (ω ◦ ϕ))(Y1, ...,Ys)
e 1 , ...,Ys). = (φϕ ω)(BX,Y ∗
e e ∈ ℑ0,0 Similarly, if ω 0,q (M), then using (e1 ), we have Let now g ∈ e M by
e Ye1 , ..., Yeq) = (φϕe ω e Ye1 , ..., Yeq). e )(X, e )(X, (φϕ,eϕ ω
(1.37)
ge(Ye1 , Ye2 ) = (i∗ g)(Ye1 , Ye2 ) = g(BYe1 , BYe2),
(1.38)
∗ e ℑ0,0 2,0 (M).
We define a pure tensor field ge of type (0, 0, 0, 2) on
e where i∗ is the transpose of i∗ = B. In fact, by definition for any Ye1 , Ye2 ∈ ℑ10 (M), C
C
C
of pure product ⊗, we easily see that a tensor field ge = (g ⊗ B) ⊗ B is a pure
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∗
e tensor field and ge ∈ ℑ0,0 ϕ to (1.38), then using (1.36) and 2,0 (M). If we apply φϕ,e (1.37) we obtain e Ye1 , Ye2 ) = (φϕ,eϕ ge)(X, e Ye1 , Ye2 ) = (φϕ,eϕ g)(X, e BYe1 , BYe2 ) (φeϕ ge)(X, e BYe1 , BYe2 ). = (φϕ g)(BX,
i.e. φϕe ge = i∗ (φϕ g). From here, it follows easily
e be the induced affinor field on an invariant Theorem 18. Let ϕ ∈ ℑ11 (M), and ϕ ∗
e If φϕ g = 0 for g ∈ ℑ0 (M) then φϕe ge = 0, where ge is the induced submanifold M. 2 pure tensor field of type (0, 2) defined by (1.38). The result in Theorem 18 can be specialized to the Norden geometry (see Section 3.5.2).
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1.6. Yano-Ako Operators 1.6.1.
Definitions
Let S be a tensor field of type (1, 2). Definition 6. A tensor field of type (r, s) is called pure tensor field with respect to S if 1
1
r
r
t(SY1 X1 , ..., Xs; ξ, ..., ξ) = ... = t(X1, ..., SY1 Xs; ξ, ..., ξ) = 1
1
r
r
= t(X1 , ..., Xs;´ (SY1 )ξ, ..., ξ) = ... = t(X1 , ..., Xs; ξ, ...,´(SY1 )ξ) 1
1
r
r
t(SY2 X1 , ..., Xs; ξ, ..., ξ) = ... = t(X1, ..., SY2 Xs; ξ, ..., ξ) = 1
1
r
r
= t(X1 , ..., Xs;´ (SY2 )ξ, ..., ξ) = ... = t(X1 , ..., Xs; ξ, ...,´(SY2 )ξ) 1
r
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for any X1 , ..., Xs,Y1 ,Y2 ∈ ℑ10 (M) and ξ, ..., ξ ∈ ℑ01 (M), where SY1 (SY2 ) denotes an affinor field such that SY1 (Y2 ) =S(Y1 ,Y2 ), (SY2 (Y1 ) = S(Y1 ,Y2 )) , for any Y1 ∈ ℑ10 (M) Y2 ∈ ℑ10 (M) ,´ (SY1 ) ´(SY2 ) is the adjoint operator of SY1 , SY2 ∈ ℑ11 (M).
The condition of pure tensor fields with respect to S may be expressed in terms of the local components as follows: j1 ... jr m ... jr m jr j1 ...m js tmi S = ... = tij11...m Sis k = tim... S = ... = tij11...i Smk , 2 ...is i1 k 1 ...is mk s j ... j
j ... j
m... j
j
j ...m j
1 r m r 1 1 s tmi1 2 ...irs Sm li1 = ... = ti1 ...m Slis = ti1 ...is Slm = ... = ti1 ...is Slm .
We consider for convenience’s sake the vector, kovector and scalar fields as pure tensor fields with respect to S. Pure tensor fields with with respect to S have been studied in [40], [86], [113]. Example 3. Let now t ∈ ℑ11 (M) be a pure tensor field with respect to S. Then, S(Y1 ,tY2) = SY1 (tY2 ) = t(SY1 (Y2 )) = tS(Y1 ,Y2 ), S(tY1,Y2 ) = SY2 (tY1 ) = t(SY2 (Y1 )) = tS(Y1 ,Y2 ), which implies S(tY1 ,Y2 ) = S(Y1 ,tY2 ) = tS(Y1 ,Y2 ), i.e S is a pure tensor field with with respect to t ∈ ℑ11 (M) (see Section 1.2.2).
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29
∗
We denote by S ℑrs (M) the module of all pure tensor fields of type (r, s) on M with respect to S. As in Section 1.1.1, by similiar devices, we shall make the ∗
∞
direct sum S ℑ(M) = ∑ r,s=0
∗ S ℑr (M) s
into an algebra over R by defining the pure
C
product ⊗. Definition 7. [71], [113] Let S ∈ ℑ12 (M), and ℑ(M) = ∑∞p,q=0 ℑrs (M) be a ten∗
sor algebra over R. A map φS |r+s>0 : S ℑ(M) → ℑ(M) is called a Yano-Ako operator or φS -operator on M, if (a) φS is a R- lineer, i.e. φS (at1 + bt2 ) = aφS (t1 ) + bφS (t2 ) for any a, b ∈ ∗
R, t1 ,t2 ∈ S ℑrs (M), ∗
(b) φS : S ℑrs (M) → ℑrs+2 (M) for all r, s, C
C
C
∗
(c) φS (K ⊗ L) = (φS K) ⊗ L + K ⊗ (φS L) for all K, L ∈ S ℑ(M), (d) φS(X1 ,X2 )Y = −(LY S)(X1 , X2 ) for all X1 , X2 ,Y ∈ ℑ10 (M), (e) (φS ω)(X1, X2 ,Y ) = (d(ιY ω))(S(X1, X2 )) − 2(d(ιY (ω ◦ S)))(X1, X2 )+ +ω(LY S)(X1 , X2 ) for all X1 , X2 ,Y ∈ ℑ10 (M), ω ∈ ℑ01 (M), where ιY ω = Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
C
ω(Y ) = ω ⊗ Y, S(X1 , X2 ) = −S(X2 , X1 ), d is the exterior differentiation of (ω ◦ S) ∈ ℑ01 (M). In the sequel we write (φSY )(X1 , X2 ) = φS(X1 ,X2 )Y.
1.6.2.
φS −operator Applied to a Tensor Field of Type (1, s) ∗
Let ϕ ∈ S ℑ11 (M). Then using Definition 7 we obtain (φS ϕY )(X1, X2 ) = (φS ϕ)(X1, X2 ,Y ) + ϕ((φSY )(X1, X2 )), which implies (φS ϕ)(X1 , X2 ,Y ) = (φS ϕY )(X1 , X2 ) − ϕ((φSY )(X1, X2 ))
(1.39)
= −(LϕY S)(X1 , X2 ) + ϕ((LY S)(X1, X2 )) = −[ϕY, S(X1 , X2 )] + S([ϕY, X1 ], X2 ) + S(X1 , [ϕY, X2]) +ϕ([Y, S(X1, X2 )]) − S([Y, X1], X2) − S(X1 , [Y, X2]).
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Arif Salimov On the other hand, using Example 3, from (1.7) and (1.39) we have (φS ϕ)(Y, X1, X2 ) = −(LS(X1 ,X2 ) ϕ)Y + S((LX1 ϕ)Y, X2) + S(X1 , (LX2 ϕ)Y ) = −[S(X1 , X2 ), ϕY ] + ϕ([S(X1, X2 ),Y ]) + S([X1 , ϕY ] −ϕ[X1,Y ], X2) + S(X1 , [X2 , ϕY ] − ϕ[X2 ,Y ]) = −(φϕ S)(X1, X2 ,Y ).
Thus we have ∗
Theorem 19. Let ϕ ∈ S ℑ11 (M).Then (φS ϕ)(Y, X1, X2 ) + (φϕ S)(X1, X2 ,Y ) = 0 for any X1 , X2 ,Y ∈ ℑ10 (M), where φS is the Yano-Ako operator applied to ϕ, φϕ is the Tachibana operator applied to S . ∗
Let now t ∈ S ℑ1s (M), s > 1. Then using Definition 7 we have (φS t)(X1, X2 ,Y1 , ...,Ys) = (φS t(Y1 , ...,Ys))(X1, X2 ) Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
s
− ∑ t(Y1 , ..., (φSYλ )(X1 , X2 ), ...,Ys) λ=1
= (−Lt(Y1 ,...,Ys ) S)(X1 , X2 ) s
+ ∑ t(Y1 , ..., (LYλ S)(X1, X2 ), ...,Ys). λ=1
Thus (φS t)(X1, X2 ,Y1 , ...,Ys) = (−Lt(Y1 ,...,Ys ) S)(X1 , X2 ) s
+ ∑ t(Y1 , ..., (LYλ S)(X1, X2 ), ...,Ys). λ=1
1.6.3.
φS −operator Applied to a Tensor Field of Type (0, s)
Let ω ∈ ℑ01 (M) and S be a vector-valued 2−form. Then using (e) of Definition 7 we have
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(φS ω)(X1, X2 ,Y ) = (d(ιY ω))(S(X1, X2 )) − 2(d(ιY (ω ◦ S)))(X1, X2) +ω(LY S)(X1 , X2 ) = (S(X1, X2 ))(ω(Y )) − X1 ω(S(Y, X2)) + X2 ω(S(Y, X1)) +(ω ◦ S)(Y, [X1, X2]) + ω((LY S)(X1 , X2 )) = (S(X1, X2 ))(ω(Y )) − X1 ω(S(Y, X2)) + X2 ω(S(Y, X1)) −(ω ◦ S)([X1, X2 ],Y )ω([Y, S((X1, X2 ) − S(Y, X1 ), X2 ]) = (LS(X1 ,X2 ) ω)Y − (X1 ωS(Y, X2)) − (ω ◦ S)(Y, [X1, X2 ]) (ω ◦ S)([X1, X2 ],Y ) − (X2 ω(S(X1,Y )) −(ω ◦ S)(X1 , [X2,Y ]) − (ω ◦ S)([X2, X1 ],Y ) −(ω ◦ S)([X1, X2 ],Y ) = (LS(X1 ,X2 ) ω)Y − (LX1 (ω ◦ S))(Y, X2) −(LX2 (ω ◦ S))(X1,Y ) + (ω ◦ S)([X1 , X2 ],Y ). Thus
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(φS ω)(X1, X2 ,Y ) = (LS(X1 ,X2 ) ω)Y − (LX1 (ω ◦ S))(Y, X2) −(LX2 (ω ◦ S))(X1,Y ) + (ω ◦ S)([X1 , X2 ],Y ). ∗
Let now ω ∈ S ℑ0s (M). Similarly we get (see Section 1.2.4) (φS ω)(X1, X2 ,Y1 , ...,Ys) = = S(X1 , X2)(ω(Y1 , ...,Ys) − X1 (ω(S(Y1 , X2 ),Y2 , ...,Ys)) −X2 (ω(S(X1 ,Y1 ),Y2, ...,Ys)) − (ω ◦ S)([X1 , X2 ],Y1,Y2 , ...,Ys) s
+ ∑ ω(Y1 , ..., (LYλ S)(X1, X2 ), ...,Ys) λ=1
= S(X1 , X2)(ω(Y1 , ...,Ys) − X1 (ω(S(Y1 , X2 ),Y2 , ...,Ys)) −X2 (ω(S(X1 ,Y1 ),Y2, ...,Ys)) − (ω ◦ S)([X1 , X2 ],Y1,Y2 , ...,Ys) s
+ ∑ ω(Y1 , ..., LYλ S(X1 , X2 ) − S(LYλ X1 , X2 ) λ=1
−S(X1 , LYλ X2 ), ...,Ys)
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Arif Salimov = (LS(X1 ,X2 ) ω)(Y1, ...,Ys) − X1 ((ω ◦ S)(Y1 , X2 ,Y2 , ...,Ys)) −X2 ((ω ◦ S)(X1 ,Y1 ,Y2 , ...,Ys)) − (ω ◦ S)(LX1 X2 ,Y1 , ...,Ys) s
+ ∑ (ω ◦ S)(Y1 , X2 ,Y2 , ..., LX1Yλ , ...,Ys) λ=1 s
+ ∑ (ω ◦ S)(X1 ,Y1 , ..., LX2Yλ , ...,Ys) λ=1
= (LS(X1 ,X2 ) ω)(Y1, ...,Ys) − (LX1 (ω ◦ S))(Y1, X2 ,Y2 , ...,Ys) −(LX2 (ω ◦ S))(X1 ,Y1 , ...,Ys) + (ω ◦ S)([X1, X2 ],Y1 , ...,Ys). Thus (φS ω)(X1, X2 ,Y1 , ...,Ys) = (LS(X1 ,X2 ) ω)(Y1 , ...,Ys) −(LX1 (ω ◦ S))(Y1, X2 ,Y2 , ...,Ys) −(LX2 (ω ◦ S))(X1,Y1 , ...,Ys) +(ω ◦ S)([X1, X2 ],Y1 , ...,Ys).
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Local expressions of the Yano-Ako operators and their applications to the theory of complete lifts we will consider in Chapter 4.
1.7. ψS −operators Let ∇ be a linear connection on M, and let S ∈ ℑ12 (M). We can replace the condition (d) of Definition 7 by (d ´) ψS(X1 ,X2 )Y = ∇S(X1 ,X2 )Y − S(∇X1 Y, X2 ) − S(X1 , ∇X2 Y ) for any X1 , X2 ,Y ∈ ℑ10 (M). We can consider a new operator as follows. ∗
Definition 8. By a ψS −operator on M, we shall mean a map ψS : S ℑ(M) → ℑ(M), which satisfies conditions (a), (b), (c), (e) of Definition 7 and the condition ( d ´). Remark 8. We easily see that ψS(X1 ,X2 )Y is linear in X1 , and X2 but not Y . In the sequal we write (ψSY )(X1 , X2 ) for ψS(X1 ,X2 )Y .
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ψS −operator Applied to a Tensor Field of Type (1, s), s ≥ 0
1.7.1.
∗
Let t ∈ S ℑ1s (M). Since t(Y1 ,Y2 , ...,Ys) ∈ ℑ10 (M), then using Definition 8, we have (ψS(X1 ,X2 )t)(Y1,Y2 , ...,Ys) = (ψS t)(X1, X2 ,Y1 ,Y2 , ...Ys) s
= ψS(X1 ,X2 )t(Y1 ,Y2 , ...,Ys) − ∑ t(Y1 , ..., (ψS(X1 ,X2 )Yλ ), ...,Ys) λ=1
= ∇S(X1 ,X2 )t(Y1 ,Y2 , ...,Ys) − S(∇X1 (t(Y1,Y2 , ...,Ys), X2)) s
−S(X1 , ∇X2 (t(Y1,Y2 , ...,Ys))) − ∑ t(Y1 , ..., (ψS(X1 ,X2 )Yλ ), ...,Ys), λ=1
(ψS(X1 ,X2 )t)(Y1,Y2 , ...,Ys) s
= ∇S(X1 ,X2 )t(Y1 ,Y2 , ...,Ys) +
∑ t(Y1, ..., ∇S(X ,X )Yλ, ...,Ys) 1
2
λ=1 s
−S((∇X1 t)(Y1 , ...,Ys), X2 ) − S( ∑ t(Y1 , ..., ∇X1 Yλ , ...,Ys), X2 ) Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
λ=1
s
−S(X1 , (∇X2 t)(Y1 , ...,Ys)) − S(X1 , ∑ t(Y1 , ..., ∇X2 Yλ , ...,Ys)) λ=1 s
− ∑ t(Y1 , ..., (ψS(X1 ,X2 )Yλ ), ...,Ys).
(1.40)
λ=1
Since t is a pure tensor field with respect to S, we have
(1.41) s
s
S( ∑ t(Y1, ..., ∇X1 Yλ , ...,Ys), X2) = λ=1
s
S(X1 , ∑ t(Y1, ..., ∇X2 Yλ , ...,Ys)) = λ=1
∑ t(Y1, ..., S(∇X Yλ, X2), ...,Ys), 1
λ=1 s
∑ t(Y1, ..., S(X1, ∇X Yλ), ...,Ys). 2
λ=1
Substituting (1.41) into (1.40), we obtain (ψS t)(X1, X1 ,Y1 , ....,Ys) = (∇S(X1 ,X2 )t − SX1 (∇X2 t) − SX2 (∇X1 t))(Y1, ...,Ys)
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Arif Salimov s
+ ∑ t(Y1 , ..., ∇S(X1 ,X2 )Yλ − S(∇X1 Yλ , X2 ) − S(X1 , ∇X2 Yλ ), ...,Ys) λ=1 s
− ∑ t(Y1 , ..., (ψS(X1 ,X2 )Yλ ), ...,Ys) λ=1
= (∇S(X1 ,X2 )t − SX1 (∇X2 t) − SX2 (∇X1 t))(Y1, ...,Ys). Thus (ψS(X1 ,X2 )t)(Y1 , ...,Ys) = ∇S(X1 ,X2 )t − SX1 (∇X2 t) − SX2 (∇X1 t))(Y1, ...,Ys). (1.42) From (1.42) we have ∗
Theorem 20. Let t ∈ S ℑ1s (M). If ∇t = 0, then t ∈ KerψS .
1.7.2.
ψS −operator Applied to a Tensor Field of Type (0, s)
Let ω ∈ ℑ01 (M). Using Definition 8, we have (ψS ω)(X1, X2 ,Y ) = (ψS(X1 ,X2 ) ω)(Y )
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= S(X1 , X2 )(ιY ω) − 2(d(ιY (ω ◦ S)))(X1, X2 ) −ω(∇S(X1 ,X2 )Y − S(∇X1 Y, X2) − S(X1 , ∇X2 Y )) = (S(X1, X2 ))(ω(Y )) − X1 (ω(S(Y, X2))) +X2 (ω(S(Y, X1))) + (ω ◦ S)(Y, [X1, X2 ]) −ω(∇S(X1 ,X2 )Y − S(∇X1 Y, X2) − S(X1 , ∇X2 Y )) = (∇S(X1 ,X2 ) ω − ∇X1 (ω ◦ SX2 ) − ∇X2 (ω ◦ SX1 ))Y −(ω ◦ S)([X1, X2 ],Y ) for any X,Y ∈ ℑ10 (M), where (ω ◦ SX1 )Y = ω(S(X1,Y )), (ω ◦ SX2 )Y = ω(S(Y, X2)). ∗
Let now ω ∈ S ℑ0s (M), s > 1. As in Section 1.6.3, by similar devices we have (ψS ω)(X1, X2 ,Y1 , ...,Ys) = (ψS(X1 ,X2 ) ω)(Y1 , ...,Ys) = S(X1 , X2 )(ω(Y1 , ...,Ys)) − X1 (ω(S(Y1 , X2 ),Y2 , ...,Ys)) −X2 (ω(S(X1 ,Y1 ),Y2 , ...,Ys)) − (ω ◦ S)([X1, X2 ],Y1 , ...,Ys) s
− ∑ ω(Y1 , ..., ∇S(X1 ,X2 )Yλ − S(∇X1 Yλ , X2 ) − S(X1 , ∇X2 Yλ ), ...,Ys) λ=1
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= (∇S(X1 ,X2 ) ω)(Y1 , ...,Ys) − X1 (ω(S(Y1 , X2 ),Y2 , ...,Ys)) −X2 (ω(S(X1 ,Y1 ),Y2 , ...,Ys)) − (ω ◦ S)([X1, X2 ],Y1 , ...,Ys) s
+ ∑ ω(Y1 , ..., S(∇X1 Yλ , X2 ) + S(X1 , ∇X2 Yλ ), ...,Ys) λ=1
= (∇S(X1 ,X2 ) ω − ∇X1 (ω ◦ SX2 ) − ∇X2 (ω ◦ SX1 ))(Y1, ...,Ys) −(ω ◦ S)([X1 , X2 ],Y1, ...,Ys). Thus (ψS ω)(X1, X2 ,Y1 , ...,Ys) = (∇S(X1 ,X2 ) ω − ∇X1 (ω ◦ SX2 ) − −∇X2 (ω ◦ SX1 ))(Y1, ...,Ys) −(ω ◦ S)([X1, X2 ],Y1, ...,Ys). The connection ∇ on M is a called a S−connection, if ∇S = 0. Theorem 21. If ∇ is a torsion-free S−connection, then φS(X1 ,X2 )t = ψS(X1 ,X2 )t ∗
∗
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for any t ∈ S ℑ1s (M) and t ∈ S ℑ0s (M). Proof. Let T (X,Y ) = ∇X Y − ∇Y X − [X,Y ] = 0. Then [X,Y ] = ∇X Y − ∇Y X. On the other hand, from (d´´) of Definition 8 and (d´) of Definition 3, we obtain φS(X1 ,X2 )Y
= −(LY S(X1 , X2 )) = [S(X1 , X2 ),Y ] − S([X1 ,Y ], X2 ) − S(X1 , [X2,Y ]) = ∇S(X1 ,X2 )Y − ∇Y S(X1 , X2 ) − S(∇X1 Y − ∇Y X1 , X2 ) −S(X1 , ∇X2 Y − ∇Y X2 ) = ∇S(X1 ,X2 )Y − S(∇X1 Y, X2 ) − S(X1 , ∇X2 Y ) = ψS(X1 ,X2 )Y.
1.8. Generalizations The definition of the Yano-Ako operator ( φS −operator) can be extended to S of type (1, q) as follows:
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Arif Salimov Let S be a tensor field of type (1, q), q > 0. As in Section 1.6.1, we denote ∗
∞
by (S ℑ(M) = ∑ r,s=0
∗ C S ℑr (M), ⊗) s
an algebra over R of all pure tensor fields of type
(r, s) with respect to S. ∗
Definition 9. A map φS |r+s>0 : S ℑ(M) → ℑ(M) is called a generalized YanoAko operator or a generalized φS -operator, if (a) φS is a R- lineer, i.e. φS (at1 + bt2 ) = aφS (t1 ) + bφS (t2 ) for any a, b ∈ R ∗
and t1 ,t2 ∈ S ℑrs (M), ∗
(b) φS : S ℑrs (M) → ℑrs+q (M) for all r and s, C
C
C
∗
(c) φS (K ⊗ L) = (φS K) ⊗ L + K ⊗ (φS L) for all K, L ∈ S ℑ(M), (d) φS(X1 ,...,Xq )Y = −(LY S)(X1 , ..., Xq) for all Y, X1, ..., Xq ∈ ℑ10 (M), (e) (φS(X1 ,...,Xq ) ω)Y = (d(ιY ω))(S(X1, ..., Xq)) − q(d(ιY (ω ◦ S)))(X1, ..., Xq) +ω((LY S)(X1, ..., Xq)) for all X1 , ..., Xq,Y ∈ ℑ10 (M), ω ∈ ℑ01 (M), where C
ιY ω = ω(Y ) = ω ⊗ Y, S is a vector-valued q−form, d is the exterior differentiation of ιY (ω ◦ S) ∈ Λq−1 (M). Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
∗
∗
In particular, if t ∈ S ℑ1s (M) and ω ∈ S ℑ0s (M), then as in Section 1.6.3, by similar devices we have (φS t)(X1, ..., Xq,Y1 , ...,Ys) = (−Lt(Y1 ,...,Ys ) S)(X1, ..., Xq) s
+ ∑ t(Y1, ..., (LYλ S)(X1 , ..., Xq), ...,Ys) λ=1
and (φS ω)(X1 , ..., Xq,Y1 , ...,Ys) = (S(X1 , ..., Xq))(ω(Y1 , ...,Ys)) −q(d(ιY1 (ωY2 ,...,Ys ◦ S)))(X1, ..., Xq) s
+ ∑ ω(Y1 , ..., (LYλ S)(X1 , ..., Xq), ...,Ys), λ=1
where ωY2 ,...,Ys ∈ ℑ01 (M) for fixed Y2 , ...,Ys ∈ ℑ10 (M) and ωY2 ,...,Ys ◦ S ∈ Λq (M). If we replace the condition (d) of Definition 9 by q
ψS(X1 ,...,Xq )Y = ∇S(X1 ,...,Xq )Y − ∑ S(X1 , ..., ∇Xλ Y, ..., Xq), λ=1
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we have a generalized ψS −operator. Let now q = 0, i.e. S = X ∈ ℑ10 (M). Then from (d) of Definition 9 we have φX Y = −LY X = LX Y for all X,Y ∈ ℑ10 (M), i.e. φX is the Lie derivative with respect to X, i.e. φX = LX . Analogously, we easily see that ψX is the covariant derivative with respect to X, i.e. ψX = ∇X . Thus the operators φS and ψS , S ∈ ℑ1q (M), q ≥ 0 are generalizations of LX and ∇X , respectively.
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Chapter 2
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Algebraic Structures on Manifolds In this chapter we give the fundamental results and some theorems concerning geometry of holomorphic hypercomplex manifolds which will be needed for the later treatment of special types of hypercomplex manifolds. In Section 2.1, we give the basic definitions and theorems on hypercomplex algebras. Section 2.2 is devoted to the study of hypercomplex structures on manifolds. We introduce the regular hypercomplex structure on manifolds. Section 2.3 treats manifolds with integrable regular hypercomplex structures. We show that such a manifold is a real model of a holomorphic manifold over hypercomplex algebras. Section 2.4 is devoted to the study of pure tensor fields. We compute the components for the pure tensor field with respect to the regular hupercomplex structures and we show that the pure tensor fields are a real models of hypercomplex tensors. In Section 2.5, we discuss holomorphic hypercomplex tensor fields. Using the Tachibana operator we give the condition of holomorphic hypercomplex tensors fields in real coordinate systems. In Section 2.6, we consider pure connections which are real model of the hypercomplex connections. Section 2.7 is devoted to the study of pure hypercomplex torsion tensors. In Section 2.8, we give a real model of holomorphic hypercomplex connections by using the tensor operator applied to pure torsion-free connection. In the last Section 2.9, we consider some properties of pure curvature tensors. The main theorem of this section is that the curvature tensor of holomorphic connection is holomorphic. Finally, we also consider a holomorphic manifold of hypercomplex dimension 1 and we show that the hypercomplex connection on such manifold is holomorphic if and
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Arif Salimov
only if the real manifold is locally flat.
2.1. Algebraic Theory In this section, we present some basic algebraic notions and theorems.
2.1.1.
Associative Algebras
We consider an m−dimensional associative algebra Am over R ( hypercomplex γ algebra [36]) with basis {eα }, α = 1, ..., m and structural constants Cαβ : γ
eα · eβ = Cαβ eγ . γ
We note that Cαβ are components of the (1, 2)−tensor · : Am × Am → Am . In this work we assume Am is a unitial algebra, i.e., it admits the principal unit e1 = 1. We introduce the matrices
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γ γ Cα = (Cαβ ), Ceα = (Cβα ),
(2.1)
where γ and β are the number of the row and column, respectively. Then the asτ Cσ = Cσ Cτ ) can be written sociativity condition eα .(eβ .eγ) = (eα .eβ ).eγ (Cασ βγ αβ σγ in one of the following three equivalent forms: σ CαCβ = Cαβ Cσ ,
eαC´ eβ = Cγ C´ e C´ αβ γ ,
(2.2) (2.3)
CαCeβ = CeβCα ,
(2.4)
εσCσ = εσCeσ = I = (δβα ),
(2.5)
eα is the transpose of Ceα . From e1 = 1 = εσ eσ it follows that where C´
where δβα is the Kronecker delta. e eα , respectively Let C(A) and C´(A) be the algebras are generated by Cα and C´ (see (2.2) and (2.3)). A mapping ρ1 : Am → C(A) : Am 3 a = aσ eσ → aσCσ = C(a) ∈ C(A), a1 , ..., am ∈ R
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is called a regular representation of type I. The representation ρ1 is an isomore phism. In a similar way, we introduce a regular representation ρ2 : Am → C´(A) of type II: eσ = C´(a) e ∈ C´(A), e Am 3 a = aσ eσ → aσC´ a1 , ..., am ∈ R
which is also isomorphism. In this paper the representations appearing in the discussion will be supposed to be representations of type I. Notice that a regular representation of type I is usually called simply a regular representation. From (2.4) we see that all A = aσCeσ , a1 , ..., am ∈ R belongs to the commutator algebra (center) of the algebra C(A). Conversely, using (2.5) we easily see that if CA = AC for any C ∈ C(A), then A = aσCeσ . In fact, we put C = Cα , α = 1, ..., m , then γ σ γ we have AσγCαβ = Cαγ Aβ . Contracting this equation with εβ , we find γ
γ
σ AσγCαβ εβ = Cαγ A β εβ , σ γ a, Aσγδγα = Cαγ
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σ A = (Aσα ) = aγ(Cαγ ) = aγCeγ ,
γ e where aγ = Aβ εβ , A ∈ C(A).Thus we have
Theorem 22. Let A be a matrix of (m × m)-matrices. Then ACα = Cα A for all α if and only if A = aαCeα. By similar devices, we have
eα = C´ eα A for all Theorem 23. Let A be a matrix of (m × m)-matrices. Then AC´ α eα . α if and only if A = a C´
2.1.2.
Commutative Algebras
We restrict ourselves to the consideration of commutative hypercomplex algebras. In this and next sections, we always assume that the Am is a commutative hypercomplex algebra. The commutativity condition eα · eβ = eβ · eα can be written in one of the following equivalent forms: γ
γ
γ
Cαβ = Cβα (Cα = Ceα ), γ
σ σ (CαCβ = CβCα ). = CβσCαδ CασCβδ
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(2.6) (2.7)
42
Arif Salimov
e Since Cα = Ceα , then we have C´(A) = C´(A). Therefore, C´(A) is called a transpose regular representations of commutative algebras. From Theorems 22 and 23 we have (see [36]) Theorem 24. Let Am be a commutative hypercomplex algebra. Then C(A) and C´(A) are maximal commutative subalgebras of the algebra of (m×m)-matrices. Among of commutative algebras, a special role is played by Frobenius algebras for which there exist constants λγ such that γ
ϕαβ = Cαβ λγ
(2.8)
form a nonsingular symmetric matrix and can be taken as components of the metric tensor of the Frobenius metric. Then, along with the basis {eα }, we can introduce the dual basis {eα }, where eα = ϕαβ eβ . For the Frobenius metric, we have relations β σ σ α ϕασCγβ = ϕβσCγα , ϕασCγσ = ϕβσCγσ , (2.9) β
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α γ α β eα · eβ = Cβγ e , e · e = ϕασCγσ eγ , ϕαβ εβ = λα ,
(2.10)
where εβ are components of e1 = 1. An automorphism ψ : Am −→ Am is called a conjugation of Am, if ψ is a involution, i.e ψ2 = id. From definition of the conjugation ψ : x −→ x ∈ Am we have β γ eα = ψ(eα ) = ψα eα , ψαγ ψβ = δαβ , (2.11) γ
β
γ
τ Cαβ ψασ ψτ = Cστ ψτ , β
ψα εα = εβ ,
(2.12) (2.13)
where e1 = 1 = εβ eβ . A conjugate to a ∈ Am is given by a = aα eα −→ a = aα eα = aα ψσα eσ . From (2.11) follows that, there exists an adapted basis of Am such that ψαβ = ±δαβ . We set ( β ψαβ 1 , i f ψα = δαβ , α 1 ψβ = β ψαβ 2 , i f ψα = −δαβ , 2
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where 1 ≤ αi < α, 1 ≤ βi < β, i = 1, 2 . Then from (2.11), we have eα1 = eα1 , eα2 = −eα2 . Since ψ(e1 ) = e1 (see (2.13)), we have e1 ∈ {eα1 }, where {eα1 } denotes the plane spaned by eα1 , 1 ≤ α1 < α. In particular, if A2 = C(m = 2) is the complex 1 0 and e1 = e1 = 1, e2 = −e2 = −i, i2 = −1. Takalgebra then, ψαβ = 0 −1 ing account of (2.14) and writing expression (2.12), for the different indices, we find γ γ γ γ Cα21 β = 0, Cα22 β = 0,Cα11 β = Cβ1 α1 = 0 2
1
2
2
with respect to the adapted basis.
2.1.3.
Holomorphic Functions
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Let z = xαeα be a variable in Am , where xα (α = 1, ..., m) are real variables.Using a real-valued C∞ −functions yβ (x) = yβ (x1 , ..., xm), β = 1, ..., m, we introduce a hypercomplex function w = yβ (x)eβ of variable z ∈ Am. Let dz = dxα eα and dw = dyα eα be respectively the differentials of z and w(z). We shall say that the function w = w(z) is a holomorphic function if there exists a functions w´(z) such that dw = w´(z)dz.
(2.14)
We shall call w´(z) the derivative of w(z). Theorem 25. [89], [36], [107, p.87] The hypercomplex function w = w(z) is holomorphic if and only if the Scheffers conditions hold: Cα D = DCα .
(2.15)
α
where D = ( ∂y ) is the Jacobian matrix of yα (x), Cα is the matrix defined by ∂xβ (2.1). e α eα . Then Proof. Let w = w(z) be a holomorphic function. We put w´(z) = w from (2.11) we have dw =dyα eα =
∂yα β γ e α eα dxβ eβ = w e α dxβCαβ dx eα = w eγ . ∂xβ
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Arif Salimov
From here, we obtain
∂yα γ e αCαβ . (2.16) =w ∂xβ Thus, the hypercomplex function α w = w(z) is a holomorphic function if and only if the Jacobian matrix ∂y have the form (2.16). Contracting (2.16) with ∂xβ σ σ ε (1 = ε eσ ) and using (2.5), we find
i.e.
e γ = εβ w
∂yγ , ∂xβ
∂yγ eγ . (2.17) ∂xβ Now applying Theorem 22 to (2.16), we see that the condition (2.16) is equivalent to the Scheffers conditions. From (2.17) follows that the Jacobian matrix γ of ∂y ∂yγ β β ∂2 yγ = ε ∂ D, where D = , εβ ∂x has components of the form D´= ε β β ∂xβ ∂xα ∂xβ
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w´(z) =εβ
and the Jacobian matrix D´ = εβ ∂β D also satisfying the Scheffers conditions. From here we easily see that exists the successive derivatives w´´(z), w´´(z), ... of w(z). We note that, in particular, if A2 = C , where C is the complex algebra, then the Scheffers conditions (2.15) reduces to the Cauchy-Riemann conditions. In fact, by virtue of 1 1 1 1 C11 C12 1 0 C21 C22 0 −1 C1 = = , C2 = = 2 2 2 2 C11 C12 C21 0 1 C22 1 0 from (2.15) we have
∂y1 ∂y2 ∂y2 ∂y1 = , = − , ∂x1 ∂x2 ∂x1 ∂x2 where z = x1 + ix2 , w = y1 (x1 , x2 ) + iy2 (x1 , x2 ), i2 = −1. Remark 9. It is well known that the concepts of holomorphic and analytic complex functions are equivalent. The hypercomplex function w = w(z) is said to be analytic if w(z) admits a convergent power series. In general, for hypercomplex functions the concepts of holomorphic and analytic functions are not equivalent (see [107, p.88], [11]). The concept the holomorphic hypercomplex functions can be immediately extended to the case of several algebraic variables. Let zu = x(u−1)m+αeα , (u = 1, ..., r)
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be an variables in Am. In fact the function w(z1 , ..., zr) = yβ (x1 , ..., xrm)eβ is a holomorphic function variables z1 , ..., zr if and only if Cα Du = DuCα for of the α ∂y , u = 1, ..., r. any u, where Du = ∂x(u−1)m+β The holomorphic hypercomplex functions possess the following properties: (i) (w1 + w2 )´= w1´+ w2´if w1 and w2 are holomorphic functions, (ii) (w1 w2 )´= w1´w2 + w1 w2´if w1 and w2 are holomorphic functions, (iii) If F and w are holomorphic functions of w and z respectively then F = dF dw F(w(z)) is a holomorphic function of z and Fz´= dF dw w´, where Fz´= dz , w´= dz . γ
γ
β
Proof. (ii) We put w1 w2 = wα1 w2 eα eγ = wα1 w2Cαγ eβ = w = wβ eβ . Then from (2.17) we have β
γ
(w1 w2 )´ = εσ (∂σ wβ )eβ = εσCαγ (∂σ (wα1 w2 ))eβ β
γ
γ
= εσCαγ ((∂σ wα1 )w2 + wα1 (∂σ w2 ))eβ γ
γ
= ((εσ ∂σ wα1 )w2 + wα1 (εσ ∂σ w2 ))eα eγ = w1´w2 + w1 w2´
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proof of (i) is analogous to (ii). (iii) We put F = Fα eα , w = yβ (x)eβ , z = xγ eγ. If by virtue of (2.17), we have
dF dw
= Feα eα , w´= w eα eα , then
dF ∂F β = εα α eβ dz ∂x β γ ∂F ∂y β θ γ = εα γ α eβ = εα FeσCσγ w e Cθα eβ ∂y ∂x β θ γ e δθ eβ = Feσ w eγ eσ eγ = FeσCσγ w dF dw dF = = w´. dw dz dw
Fz´ =
Example 4. Let A2 = R(ε) be a dual algebra with canonical bases {1, ε}, ε2 = 0. Using 1 1 C21 C22 0 0 C2 = = 2 2 1 0 C21 C22 we see that, the condition (2.12) reduces to the following equations: ∂y1 ∂y2 ∂y1 = , = 0, ∂x1 ∂x2 ∂x2
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Arif Salimov
where z = x1 +εx2 , w(z) = y1 (x1 , x2 )+εy2 (x1 , x2 ), ε2 = 0. From last equations we have w(z) = f(x1 ) + ε(x2 f´(x1 ) + g(x1 )). We call w = w(z) in this form the synectic function [99], [16]. If g(x1 ) = 0, then the function w(z) = f(x1 ) + εx2 f´(x1 ) is said to be natural extension of the real C∞ - functions f(x1 ) to R(ε). Example 5. Let A2 = A(e), e2 = 1 be an algebra of paracomplex numbers. From 0 1 C2 = 1 0 we easily see that, the Scheffers conditions reduces to the para-Cauchy-Riemann ∂y1 ∂y2 ∂y2 ∂y1 conditions (see [11]) ∂x 1 = ∂x2 , ∂x1 = ∂x2 ,where z = x1 + ex2 , w(z) = y1 (x1 , x2 ) + ey2 (x1 , x2 ), e2 = 1.
2.2. Algebraic Π−structures on Manifolds If a collection of (1, 1)-tensor (affinor or endomorphism) fields ϕ, ϕ, ... are given Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
1 2
on a C∞ -manifold M, then one says that a polyaffinor structure (or Π−structure) is given on M : Π = {ϕ}. This structure is said to be rigid [36], if every ϕ ∈ Π is reduced to the constant form in a certain (in general, nonholonomic) frame {Xi }, i = 1, ..., n in a neigborhood Ux of every point x ∈ M. In this case, we shall say that {Xi }, i = 1, ..., n is adapted frame with respect to the Π−structure. If every ϕ ∈ Π is constant on a certain set of holonomic(natural) adapted frames {Xi } = {∂i }, i = 1, ..., n of a smooth atlas, then the Π−structure is said to be integrable. Obviously, the integrable Π−structure is always rigid: the converse holds only under certain additional requirements on the Π−structure. For example, if Π = ϕ, i.e. Π−structure consists of one affinor, and if there exists a torsion-free ϕ-connection ∇ on M preserving the rigid ϕ−structure, i.e. the realiton ∇ϕ = 0 holds, then the ϕ−structure is integrable [92], [36]. It is well known that, for simplest rigid ϕ−structures (almost complex and almost paracomplex structures et al), the integrability is equivalent to the vanishing of the Nijenhuis tensor (see Chapter 1). Definition 10. Let ∇ be a linear connection on M. ∇ is called Π−connection with respect to the Π−structure, if ∇ϕ = 0 for any ϕ ∈ Π.
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Definition 11. Π−structure on M is called almost integrable, if there exists a torsion-free Π−connection. We note that for some simplest Π−structures (ϕ−structures, regular Π−structures at al) the concepts of integrability and almost integrability are equivalent. Let Am be a hypercomplex algebra. An almost hypercomplex structure on M is a polyaffinor Π−structure such that γ
ϕ · ϕ = Cαβ ϕ,
(2.18)
γ
α β
i.e. if there exists an isomorphism Am ↔ Π, where ϕ are structural affinors α
corresponding to the bases elements eα ∈ Am , α = 1, ..., m. Definition 12. [36] An almost hypercomplex structure on M is said to be a regular (or r-regular) Π−structure if matrices of ϕ of order n, α = 1, ..., m si-
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α
multaneously reduced to the form Cα 0 · · · ! 0 Cα · · · ϕij = . .. .. . ··· α 0 0 ···
0 0 .. . Cα
, α = 1, ..., m; i, j = 1, ..., n
(2.19)
γ
with respect to the adapted frame {Xi }, where Cα = (Cαβ ) is the regular representation of Am, r is a number of Cα −blocks. Remark 10. Putting C´α in (2.19), we have a transpose regular Π−structure on M. From definition immediately follows that the regular Π−structures are rigid structures. In particular, for almost complex and paracomplex structures the condition (2.19) immediately follows from (2.18), i.e. almost complex and paracomplex structures on M (dimM = 2r), automatically are regular structures. For example, a Π−structure Π = {I, ϕ}, I = idM , ϕ2 = 0 on M is an isomorphic representation of dual algebra R(ε), ε2 = 0 but it is nonregular Π−structure on M, in general (see [102], [103], [108]). Let now Π = ϕ , α = 1, ..., m be a regular Π−structure on M. Then from α
(2.12), we have
n = mr
(n = dim M , m = dim Am),
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(2.20)
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Arif Salimov
where r is a number of the bloks Cα . Thus, the condition (2.20) is a necessary condition for exists of regular Π−structures on M and in this case we putting i = (u − 1)m + α (i = 1, ..., n; u = 1, ..., r; α = 1, ..., m) or i = uα, j = vβ, k = wγ, ... In other words, the structural affinors ϕ have the coordinates α
γ
uα ϕij = ϕvβ = δuvCαβ (δuv − Kronecker delta). α
(2.21)
α
Let Xi´ = Sii´Xi (Det(Sii´) 6= 0) be a transformations of adapted frame {Xi } with respect to the regular Π−structure. Then we have ! ! j
ϕi´j´ = (Sii´) ϕij (S j´),
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α
(2.22)
α
where (Sii´) = (Sii´)−1 . If {Xi } is an adapted frame, then we call transformation Xi → Xi´ the admissible transformation Xi → Xi´. Since in this case {Xi´} is adapted ! ! frame then
ϕi´j´ =
ϕij , and we have from (2.22) α
α
Sϕ = ϕS, α j
(2.23)
α
!
ϕij . Thus we have
where S = (S j´) and ϕ = α
α
Theorem 26. Let Π = ϕ be a regular Π−structures on M. A transformation α
S : Xi → Xi´ of adapted frames is an admissible if and only if it satisfies the condition (2.23).
Using Theorem 22 and Theorem 23 we see that the matrix S has the special structure α Sii´ = ∆uσ (i = uα, i´= u´α´). (2.24) u´ Cσα´
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For inverse matrix, by similar devices, we have α´ Sii´ = ∆u´σ u Cσα
(i = uα, i´= u´α´).
(2.25)
We can associate matrices from the algebra Am : ∗
∗
∗
∗
u´ u´σ S = (Suu´) = (∆uσ u´ eσ ), S = (Su ) = (∆u eσ ). ∗ ∗ −1
∗
= I, where 1 0 = .. .
From here, we easily see that SS e1 0 · · · 0 0 e1 · · · 0 ∗ I = . .. . . . .. . .. . 0
· · · e1
0
(2.26)
In fact, from (2.24) and (2.25) we obtain
0 ··· 0 1 ··· 0 i .. . . .. = (δ j ). . . . 0 0 ··· 1
α u´ε α´ δij = Sii´Si´j = ∆uσ u´ Cσα´∆v Cεβ γ
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u´ε uσ u´ε = ∆uσ u´ ∆v Cσε eγ = ∆u´ eσ ∆v eε ∗ ∗
= Suu´Su´v. With each vector field ξ = ξi Xi = ξuα Xuα, where {Xi } is the adapted frame on ∗
M, we can associate r coordinates ξu (u = 1, ..., r) from the algebra Am : ∗
ξu = ξuαeα . i´
We easily see that, if ξ =
Sii´ξi ,
∗ u´
∗ ∗
then ξ = Su´uξu . In fact, from (2.21) we obtain
uα u´σ α´ uα ξu´α´ = Su´α´ uα ξ = ∆u Cσα ξ
or ∗ u´
ξ
α´ uα = ξu´α´eα´ = ∆u´σ u Cσα ξ eα´ ∗ ∗
uα u´ u = ∆u´σ u eσ ξ eα = Su ξ .
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Arif Salimov
Let now ϕ ∈ Π. Then ϕ = aα ϕ. The action of ϕ on a vector field ξ = ξi Xi , i.e the α
equation ηi = ϕij ξ j reduces to ∗
α vβ ηu = ηuαeα = aσ ϕij ξ j eα = aσ δuvCσβ ξ eα σ ∗
= aσ δuv ξvβ eσ eβ = aσ eσ ξuβ eβ = aξu , where i = uα, j = vβ, a ∈ Am. Thus we have Theorem 27. If Π is a regular structure on M then, each tangent space Tx (M), x ∈ M serves as a real model of the module Tr (Am) over algebra Am. In particular, in the case ϕ = ϕ we have α ∗
∗
ηu = eα ξu .
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2.3. Integrable Regular Π−structure Let Π = ϕ be an integrable regular Π−structures on Mmr and let xi = xuα α
and xi´ = xu´α´- adapted local coordinates in Ux (x ∈ Mmr ). It is well known that the structural affinors ϕ have the constant form (2.19) with respect to the adapted αn n o o frames ∂x∂ i and ∂x∂ i´ . In this case the admissible transformation has the ∂ ∂xi´
= Sii´ ∂x∂ i , where Sii´ =
∂xi , ∂xi´
α i´ u´σ α´ Sii´ = ∆uσ C , i = uα, i´= u´α´(see u´ Cσα´, Si = ∆ uα u uασα ∂x ∂x for (2.21) and (2.24)). Then from Theorem 26 we have ∂x u´α´ Cσ = Cσ ∂xu´α´
form
fixed u and u´, i.e. zu´ = xu´α´eα´ is holomorphic function of zu = xuα eα . Using (2.17) and (2.26), we have ∗ u´α´ ∂zu´ α ∂x α u´σ α´ u´σ α´ u´σ u´ = ε e = ε ∆ C e = ∆ δ e = ∆ e = S α´ α´ α´ σ u ασ u σ u u. ∂zu ∂xuα
By similar devices, we have
∗ ∂zu = Suu´. u´ ∂z
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Thus, on the intersection of two coordinate neigbourhoods with local adapted coordinates xuα and xu´α´, the transition functions zu´ = zu´(zu ) (zu = xuαeα , zu´ = xu´α´eα´) are holomorphic, i.e. Mmr is holomorphic A-manifold of dimension r : Xr (A). Conversely let Xr (A) be a holomorphic A-manifold. Then on the intersection of two A-coordinate neigborhoods the transition functions zu´ = zu´(zu ) are holomorphic. Then by virtue of Theorem 25 we have uα uα ∂x ∂x Cσ = Cσ u´α´ ∂x ∂xu´α´ for fixed u and u´. From here, we have uα uα ∂x ∂x ϕ=ϕ u´α´ u´α´ ∂x σ σ ∂x
u γ for any u and u´, where ϕij = ϕuα C , from which, it follows that Π = ϕ = δ v σβ vβ σ
α
σ
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is integrable and regular Π−structure.Thus we have
Theorem 28. A real model of a A-holomorphic manifold Xr (A) is a real manifold Mmr with integrable regular Π−structure Π = ϕ . α
Remark 11. Let Mmr be a real manifold with regular Π−structure. Then the following properties are equivalent [35]: (i) Regular Π−structure is integrable, (ii) Regular Π−structure is almost integrable, (iii) NΠ = 0, where NΠ denotes the Nijenhuis-Shirokov tensor determined by regular Π−structure. Example 6. Let Xr (C) be a complex analytic (C-holomorphic) manifold [112, p.50]. To show that every complex analytic manifold carries a natural almost complex structure on the real model M2r of Xr (C), we consider the space Cr of r−tuples of complex numbers (z1 , z2 , ..., zr) with zi = xi + iyi , i2 = −1, i =1, ..., r . With respect to the real coordinate system (x1 , ..., xr, y1 , ..., yr) we define an almost complex structure ϕ on R2r (real model of Cr ) ∂ ∂ ∂ ∂ ϕ (2.27) = i, ϕ = − i , i = 1, ..., r. i i ∂x ∂y ∂y ∂x
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Since the transition functions of two charts of Xr (C) are C-holomorphic, therefore to define an almost complex structures on M2r , we transfer the almost complex structure of R2r in the form (2.27) to M2r by means of such charts. From the construction of the almost complex structure ϕ given by (2.27), it is clear that the components of ϕ with respect to the local (holonomic) coordinate x1 , ..., xr, y1 , ..., yr system are constant and hence ϕ is integrable. On the other hand, every almost complex structure is regular. In fact, for ϕ there exist o elen ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ r ments ∂x1 , ..., ∂xr of C such that ∂x1 , ∂x2 , ..., ∂xr , ϕ( ∂x1 ), ϕ( ∂x2 ), ..., ϕ( ∂xr ) is a basic for R2r (real model of Cr ) (see [34, p.141]). If we put ∂ ∂ ∂ ∂ ∂ ∂ , ϕ( 1 ), 2 , ϕ( 2 ), ..., r , ϕ( r ) , ∂x1 ∂x ∂x ∂x ∂x ∂x
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then we see that ϕ is given by the matrix 0 −1 ... 0 1 0 , (ϕ) = 0 −1 0... 1 0
i.e. ϕ is regular structure. Thus a real model of complex analytic manifold is a real manifold with integrable regular natural almost complex structure given by (2.27). Example 7. By similar devices, we easily see that a real model of paracomplex (paraholomorphic) manifold Xr (A(e)) (see [11] ) is a real manifold M2r with integrable regular natural almost para-complex structure ψ given by ∂ ∂ ∂ ∂ ψ = i , ψ = i , i = 1, ..., r, i i ∂x ∂y ∂y ∂x where A(e) = {zi = xi + eyi , e2 = 1}. Example 8. Let T (Mn ) be a tangent bundle of Mn (see [114] for details). The tangent bundle of Mn consist of pair (x, y), where x ∈ Mn and y ∈ Tx (Mn ). Let π : T (Mn ) → Mn defined by π(x, y) = x be the natural projection of T (Mn ) onto Mn . Let (U, x = (x1 , ..., xn)) be a coordinate chart on Mn . Then it induces local coordinates (x1 , ..., xn, x1 , ..., xn) on π−1 (U), where x1 , ..., xn represent n the o components of vector fields on Mn with respect to local frame {∂i } = ∂xii .
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In the following we use the notation i = i + n for all i = 1, ..., n. If (U´, x´ = (x1´, ..., xn´)) is another coordinate chart on Mn , then the induced coordinates (x1´, ..., xn´, x1´, ..., xn´) with respect π−1 (U´) will be given by ( xi´ = xi´(xi ), i = 1, ..., n, (2.28) ∂xi´ i i´ x = ∂xi x , i = n + 1, ...2n. The Jacobian of (2.28) is given by matrix S=
∂xα´ = ∂xα
∂xi´ ∂xi 2 xi´ xs ∂x∂ i ∂x s
0 ∂xi´ ∂xi
!
, α = 1, ..., 2n.
It is clear that there exist a natural affinor field 0 0 ϕ = ϕαβ = (I − identity matrix of degree n) I 0 such that
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Sϕ = ϕS, i.e. the transformation S : {∂α } → {∂α´} is an admissible with respect to the structure ϕ. It is clear that ϕ is integrable and ϕ2 = 0. Also ϕ is a regular persentation of the algebra of dual numbers R(ε), ε2 = 0. In fact, if we put {∂1 , ∂1 , ∂2 , ∂2 , ..., ∂n, ∂n } instead of {∂1 , ∂2 , ..., ∂n, ∂1 , ∂2 , ..., ∂n}, then the affinor field ϕ has the matrix 0 0 ... 0 1 0 ϕ= 0 0 0... 1 0
with respect to the frame {∂1 , ∂1 , ∂2 , ∂2 , ..., ∂n , ∂n }, i.e. ϕ is a regular. Thus, on the tangent bundle T (Mn ) there exist a natural integrable affinor ϕ-structure [100], [94] which is a regular presentation of algebra of dual numbers R(ε) (we note that an isomorphic representation of R(ε) is nonregular (see [106]), in general). Therefore with each induced coordinates (xi , xi) in π−1 (U) ⊂ T (Mn ), we associate the local dual coordinates X i = xi + εxi , ε2 = 0. Using (2.28) we see that the local dual coordinates X i = xi + εxi transformed by X i´ = xi´(xi) + εxs ∂s (xi´(xi)).
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(2.29)
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Arif Salimov
The equation (2.29) show that the quantities X i are R(ε)-holomorphic functions of X i = xi + εxi (see, Example 9). Thus the tangent bundle T (Mn ) with a natural integrable regular ϕ-structure is real model of R(ε)-holomorphic manifold Xn (R(ε)).
2.4. Pure Tensors with Respect to the Regular Structure In particular, being applied to a (1, 1)-tensor field t, the purity with respect to the regular Π−structure means that in the local coordinates the following conditions should hold: t mj ϕim = tmi ϕmj for any ϕ ∈ Π. (2.30) α We putting i = uα, j = vβ, m = wσ and ϕ = (δuvCγβ ) instead of ϕ ∈ Π. Then γ
γ
from (2.30), we have wσ u α uα w σ tvβ δwCγσ = twσ δv Cγβ , uσ α tvβ Cγσ
(2.31)
uα σ tvσ Cγβ .
=
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Using contraction with εβ (1 = εβ eβ ) and (2.5), from (2.31) we have uσ β α uα σ β uα uα σ tvβ ε Cγσ = tvσ Cγβ ε = tvσ δγ = tvγ
or
σ
uα α t ij = tvσ = ℑuvCσβ ,
(2.32) ∗
σ
uσ β where ℑuv = tvβ ε . Thus, a pure tensor field t ∈ ℑ11 (M) has the form (2.32). Conversely, from (2.32) it follows that the tensor field t of type (1, 1) is pure if ε
ℑ are an arbitrary functions. In fact, substituting (2.32) into (2.30), we find ε
ε
u α w σ σ u α ℑw v Cεβ δwCγσ = ℑwCεσ δv Cγβ , ε σ α α σ ℑuv (Cεβ Cγσ −Cεσ Cγβ )
(2.33)
= 0.
σ α α σ Since Am is a commutative algebra Cεβ Cγσ = Cεσ Cγβ , we see that the equation ε
(2.30) is satisfies for arbitrary functions ℑ . Thus, the tensor field t of type (0, 2) is pure with respect to the regular Π− structure if and only if t has form (2.32)
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∗
ε
for arbitrary functions ℑ . In the case g ∈ ℑ02 (M), we by similar devices see that the tensor field of type (0, 2) is pure if and only if has the form σ gi j = guαvβ = ℑuvσCαβ ∗
for arbitrary functions ℑ..σ . In the case G ∈ ℑ20 (M) the situation is more difficult. The purity condition of G is given by Gm j ϕim = Gim ϕmj for any ϕ ∈ Π. By similar way, we have β
α GuσvβCγσ = GuαvσCγσ .
(2.34)
After contraction of (2.34) with λα , we have σ
σ
β
ϕγσ Guσvβ = ℑuvCγσ (ℑuv = Guαvσ λα ),
(2.35)
γ
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where ϕαβ = λγCαβ . If Det(ϕαβ ) 6= 0, i.e. if Am is a Frobenius algebra, then from (2.35) we have following solition σ
α µβ Gi j = Guαvβ = ℑuvCσµ ϕ .
(2.36)
Conversely, from (2.36) by virtue of (2.7) it follows that the tensor field G ∈ ∗
∗
σ
ℑ20 (M) is pure if ℑ are an arbitrary functions. Thus, in the case G ∈ ℑ20 (M), algebra Am must be Frobenius algebra. In general, the case for t ∈ ℑrs (M), in the space of Am we consider the Kruchkovich tensors [36]: Bαβ1 β2 ...βs Bαβ1 β2
α
= Cβα1 α1 Cβα1α2 ...Cβ s−2β 2
=
Cβα1 β2
,
Bαβ
s−1 s
=
(s > 2),
δαβ .
If Am is the Frobenius algebra with metric ϕαβ , then we have ...α r Bβα1...β
= Bαβ r ...β λ1 ...λr−1 ϕλ1 α1 ...ϕλr−1 αr−1 ,
Bα1 ...αr
= Bαλ1r ...λr−1 ϕλ1 α1 ...ϕλr−1 αr−1 ,
1
s
1
Bβ1 ...βs = Bαβ β
s
1 2 ...β s−1
ϕαβs .
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Arif Salimov
...α r Now we state some properties of Kruchkovich tensors Bαβ 1...β : 1 s α1 ...αr (B1 ) Bβ ...β is a symmetric tensor with respect to the indices α1 , ..., αr and 1 s β1 , ..., βs , α1 ...α r σ α1 ...αr λ σα1 ...αr 1 ...αr Bβ ...β = Bλα (B2 ) Cσµ µβ1 ...βs , Cλµ Bσβ1 ...βs = Bλµβ1 ...βs , 1 s α1 ...αr σ α1 ...αr 1 ...αr (B3 ) Bσα β1 ...βs λσ = Bσβ1 ...βs ε = Bβ1 ...βs . Proof of (B1 ), (B2 ) , (B3 ) immediately follows from (2.5), (2.7) and (2.9). ∗
Similarly, pure tensor field t ∈ ℑrs (M) of type (r, s) with respect to the regular Π− structures has components of the form σ
...ur α1 ...αr = ℑ vu11 ...v B s β ...β
r t ij11...i ... js
1
(2.37)
s
(ia = ua αa, jb = ub αb , a = 1, ..., r, b = 1, ..., s), σ
where ℑ is an arbitrary functions in the adapted coordinate chart U. We note that in the cases r = 0 and r = 1, the formulae (2.37) is true, if even the algebra Am is non-Frobenius algebra.With each pure tensor field (2.37) we can associate a hypercomplex values from Am :
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∗ u ...u t v11 ...vrs ∗
We easily see that t (r, s), i.e.
σ
u1 ...ur βs r β1 = t ij11...i ... js ε ...ε λα1 ...λαr−1 eαr = ℑ v1 ...vs eσ .
u1 ...ur v1 ...vs
(2.38)
are components of hypercomplex tensor field of type
∗ u´ ...u´ t v´11 ...v´rs
∗
∗
∗
∗
∗ u ...u 1 r v1 ...vs .
= S u´u11 ...S u´urr S vv´11 ...S vv´ss t
In fact, for simplicity we take r = s = 1 , then from (2.24) and (2.25) we have β
vβ
u´α´ uα u´σ α´ vε uα t ij´´ = tv´β´ = Su´α´ uα Sv´β´tvβ = ∆u Cσα ∆v´ Cεβ´tvβ
which implies ∗ u´ t v´
u´α´ β´ = t ij´´εβ´eα´ = tv´β ´ ε eα´ β
α´ vε uα β´ = ∆u´σ u Cσα ∆v´ Cεβ´tvβ ε eα´ vβ
uα = ∆u´σ u ∆v´ eσ eα tvβ ε
vβ
u α = ∆u´σ u ∆v´ eσ eα ℑv Bεβ vβ
ε
u = ∆u´σ u ∆v´ ℑv eσ eβ eε ∗
∗
∗u v
= S u´uS vv´ t
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by virtue of (2.26). Thus, we have Theorem 29. Let Π be an regular integrable Π−structure on Mmr . Then pure ∗ tensor fields t of type (r, s) on Mmr are a real models of hypercomplex tensors t in the A−holomorphic manifold Xr (Am). In the next section, we will study a real model of the A−holomorphic con∗ ditions of t.
2.5. A-holomorphic Tensors in Real Coordinate Systems ∗
Let Am be a Frobenius hypercomplex algebra and t ∈ ℑrs (Xr (Am)) be an hypercomplex tensor field on Xr (Am). Then the real model of such a tensor field t is ∗
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a pure tensor field of type (r, s) on Mmr . In general, t is not A−holomorphic. Using the Tachibana operator we give the condition of A−holomorphic tensors in real coordinate systems, i.e. the following theorem is true [36], [83]: Theorem 30. Let on Mmr be given the integrable regular Π−structure. The ∗ hypercomplex tensor field t ∈ ℑrs (Xr (Am)) is A−holomorphic tensor field if and ∗
∗
only if the pure tensor field t ∈ ℑrs (Mmr ) ( the real model of t ) satisfies the equation φϕ t = 0, α = 1, ..., m, α
where φϕ t is the Tachibana operator defined by (1.16). α r Proof. It is well known that the components (φϕt)ij11...i ... js of φϕt with respect to α
α
local coordinate system x1 , ..., xmr may be expressed as (1.19). In the adapted charts (∂k ϕij = 0), by virtue of (2.37), from (1.19) we have (ia = uaαa , jb = vb βb , k = wγ, a = 1, ..., r, b = 1, ..., s) r (φϕt)ij11...i ... js α
i1 ...ir i1 ...ir = ϕm k ∂m t j1 ... js − ∂k (t ◦ ϕ) j1 ... js α
α
λ
σ
α1 ...α r ...ur ...ur −Cαγ ∂wµ ℑ vu11 ...v )Bλβ = (Cαγ ∂wµ ℑ vu11 ...v s s ...β = 0. µ
µ
1
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Arif Salimov
From here and property B3 (see Section 2.4), it follows that the condition φϕt = 0 α
is equivalent to the condition µ
λ
µ
σ
...ur ...ur Cαγ ∂wµ ℑ vu11 ...v = Cαγ ∂wµ ℑ vu11 ...v , s s ∗
σ
r which is the Scheffers condition (see (2.15)) of A−holomorphity of t uv11 ...u ...vs = ℑ u1 ...ur e with respect to the local coordinates zu = xuα e from X (A ). Thus the α r m v1 ...vs σ proof is complete. An infinitesimal automorphism of a regular Π− structure on Mmr is a vector field X such that LX ϕ = 0, α = 1, ..., m, where LX denotes the Lie differentiation
α
with respect to X ∈ ℑ10 (Mmr ). From Theorem 30 and (d) of Definition 2, we have Corollary 31. Let on Mrm be given the integrable regular Π−structure. A vector field X ∈ ℑ10 (Mmr ) is an infinitesimal automorphism of Π−structure if and only if X is A−holomorphic. Remark 12. Let Mrm on be given the non-integrable regular Π−structure. Then if t ∈ Kerφϕ , we say that t is an almost A−holomorphic tensor field. Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
α
2.6. Pure Connections In this section, we always assume that the regular Π−structure is an integrable. By a local coordinates we shall mean an adapted coordinates with respect to the Π−structure. Let ∇ be an Π−connection on Mmr , i.e. ∇ϕ = 0 for any ϕ ∈ Π. Since the components of ϕ with respect to the local adapted coordinates x1 , ..., xmr are constant, we have i ∇ϕ = 0 ⇔ Γikm ϕmj = Γm (2.39) k j ϕm . By the same arguments as developed in Section 2.5, we see that the Π− connection has components of the form σ
u α Γik j = Γuα wγvβ = τ wγvCσβ (i = uα, j = vβ, k = wγ),
(2.40)
σ
where τ− arbitrary functions in the adapted chart U. In fact, if ϕ = ϕ, then from σ
(2.39), using contraction with εβ , we obtain
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tε tε uα Γuα wγtε ϕ vβ = Γwγvβ ϕ tε , σ
σ
t ε u α tε Γuα wγtε δvCσβ = Γwγvβ δt Cσε ,
.
uε β α Γuα wγvσ = Γwγvβ ε Cσε ε
α = τ uwγvCεσ , ε
β where τ uwγv = Γuε wγvβε , m = tε. With each Π− connection (2.40) we can associate a hypercomplex values from Am: ∗
σ
γ β u γ Γ uwv = Γuα wγvβ ε ε eα = τ wγv ε eσ .
(2.41)
Definition 13. If the hypercomplex values (2.41) satisfies the connection condition ∗ u´ ∂zu´ ∂zw ∂zv ∗ u ∂2 zu ∂zu´ Γw´v´ = u w´ v´ Γwv + v´ w´ u , ∂z ∂z ∂z ∂z ∂z ∂z ∗
∗
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i.e. if Γ uwv are components of hypercomplex connection ∇ in Xr (Am), we say that the Π−connection ∇ is a pure. Theorem 32. Let Π be a regular integrable Π−structure on Mmr . The α Π−connection on Mmr is pure if and only if τ uwγv in (2.40) satisfies the condition αu σ α τ wγv = τ uwvCσγ . (2.42) Proof. Let Γik j = Γuα wγvβ be the components of the Π−connection ∇. Then, taking account of the {∂i } → {∂i´} of adapted frames admissible u´α´transformation ∂xi´ ∂x i´ with matrix (Si ) = ∂xi = ∂xuα , we have Γu´α´ w´γ´v´β´ =
∂2 zuα ∂zu´α´ ∂zu´α´ ∂zwγ ∂zvβ uα Γ + . ∂zuα ∂zw´γ´ ∂zv´β´ wγvβ ∂zw´γ´∂zv´β´ ∂zuα
After contraction with εγ´εβ´eα´, by virtue of (2.24), (2.25) and (2.26), we obtain ∗
γ´ β´ Γ u´w´v´ = Γu´α´ w´γ´v´β´ε ε eα´ σ
ε
γ
θ
β
α´ v uα γ´ β´ = ∆ u´uCσα ∆w w´Cεγ´∆ v´Cθβ´Γwγvβ ε ε eα´ σ
α´ +∆ u´uCσα (
∂ ε u α γ´ β´ (∆ C ))ε ε eα´ ∂zw´γ´ v´ εβ´
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Arif Salimov
or ∗
σ
ε
∗
∗
θ
ω
σ
∗
∂ αu (∆ )eα , ∂zw´γ´ v´
γ β u v α u´ γ´ Γ u´w´v´ = ∆ u´u∆ w w´∆ v´eσ eα δε δv τ wγvCωβ + ∆ u eσ ε ( γ
θ
u u´ γ´ = S u´uS vv´∆ w w´τ wγv eθ + S u ε (
∂ αu (∆ )eα ∂zw´γ´ v´
where
∂zu´ u ∂zu u , Su´ = u´ , z = xuα eα (see Section 2.3). ∂zu ∂z Using (2.17), we see that ∗
S u´u =
εγ´(
∂ αu ∂2 zu ( ∆ )e = . α ∂zw´γ´ v´ ∂zw´∂zv´
Thus, we have ∗
Γ u´w´v´ =
∂zu´ γ w ∂zv θ u ∂2 zu ∂zu´ ∆ τ e + . θ wγv w´ ∂zu ∂zv´ ∂zv´∂zw´ ∂zu ∗
(2.43)
∗
From (2.43) we easily see that ∇ with components Γ uwv is hypercomplex conCopyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
θ
nection on Xr (Am ) if and only if τ uwγv satisfies the condition (2.42). The proof is completed. If we put (2.42), we get from (2.40) and (2.41), respectively σ
µ
σ
u α u α Γik j = Γuα wγvβ = τ wvCσγCµβ = τ wv Bσγβ
and
∗
σ
Γ uwv = τ uwv eσ ,
(2.44)
(2.45)
where Bασγβ is the Kruchkovich tensor. Thus, we have Corollary 33. A pure Π−connection ∇ has components (2.44) with respect to the adapted coordinates. Corollary 34. A pure Π−connection ∇ is a real model of the hypercomplex ∗
connection ∇ with components (2.45).
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From Corollary 33 and (2.37) follows that the pure Π−connection as pure tensor fields of type (1, 2) is defined by i i m Γikm ϕ mj = Γm k j ϕ m = Γm j ϕ k , α = 1, ..., m α
α
α
with respect to the adapted charts (for pure torsion-free connection, see (1.26)). But the pure tensor fields of type (1, 2) (see Section 2.4 ) is defined by similiar equation with respect to the arbitrary charts.
2.7. Torsion Tensors of Pure Π−connections Let S be a torsion tensor field of pure Π−connection ∇. Since Bασγβ = Bασβγ (see Section 2.4), we have from (2.44): Sik j = Γik j − Γijk σ
(2.46)
σ
= (τ uwv − τ uvw )Bασγβ ,
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i.e. S is a pure tensor (see (2.37)). Conversely, we now assume that S is a pure tensor of Π−connection. Then, by virtue of (2.37), we have λ
u α Sik j = Suα wγvβ = σ wv Bλγβ .
(2.47)
On the other hand, from (2.40) we have Sik j = Γik j − Γijk λ
λ
α α = τ uwγvCλβ − τ uvβwCλγ .
(2.48)
From (2.47) and (2.48), by virtue of contraction with εγ , we have λ u λu α γ α , τvβw = τ wγv ε − σ vβw Cλβ which shows the condition of type (2.42) is true, i.e. the Π−connection is a pure. Thus, we have Theorem 35. The Π−connection ∇ is pure if and only if the torsion tensor of ∇ is pure.
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Arif Salimov For pure Π−connection by virtue of (2.38), (2.45) and (2.46), we obtain ∗
σ
σ
S uwv = (τ uwv − τ uvw )eσ ∗
∗
= Γ uwv − Γ uvw . Thus, we have Corollary 36. The pure torsion tensor field of the Π−connection ∇ is a real ∗
model of the hypercomplex torsion tensor of hypercomplex connection ∇. Of course, a zero tensor field is pure, therefore we have Corollary 37. A torsion-free Π−connection ∇ is always pure. Also, from Corollary 36 and Corollary 37 we have ∗
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Corollary 38. If ∇ is a torsion-free Π−connection, then ∇ with components (2.45) is a torsion-free connection.
2.8. A−holomorphic Hypercomplex Connection In this and next sections, we give a real model of A−holomorphic hypercomplex connections by using the ψϕ −operator applied to pure torsion-free connection (see Section 1.4). Let R be a curvature tensor of pure torsion-free connection ∇ with respect to the integrable regular Π−structure. Using (2.44) and properties of Kruchkovich tensors, we have Rijkl = Ruα vβwγtδ uα uα xε uα xε = ∂vβ Γuα wγtδ − ∂wγ Γvβtδ + Γvβxε Γwγtδ − Γwγxε Γvβtδ σ
σ
= ∂vβ (τ uwt Bασγδ ) − ∂wγ (τ uvt Bασβδ ) σ
θ
σ
(2.49)
θ
+τ uvx Bασβε τ xwt Bεθγδ − τ uwx Bασγε τ xvt Bεθβδ σ
σ
σ
θ
σ
θ
= (∂vβ τ uwt )Bασγδ − (∂wγ τ uvt )Bασβδ + (τ uvx τ xwt − τ uwx τ xvt )Bασγθβδ .
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We now assume that R is a pure tensor. Then by virtue of (2.37) we have λ
u α Rijkl = Ruα vβwγtδ = ρ vwt Bλβγδ
(2.50)
From (2.49) and (2.50), we obtain σ
λ
σ
ρ uvwt Bαλβγδ = (∂vβ τ uwt )Bασγδ − (∂wγ τ uvt )Bασβδ σ
θ
σ
(2.51)
θ
+(τ uvx τ xwt − τ uwx τ xvt )Bασγθβδ . After contraction with εβ εδ of (2.51) and by virtue of properties of Kruchkovich tensors, we have λ
σ
σ
σ
θ
σ
θ
σ
θ
α α λ α ρ uvwt Cλγ = εβ (∂vβ τ uwt )Cσγ − ∂wγ τ uvt + (τ uvx τ xwt − τ uwx τ xvt )Cσθ Cλγ α
λ
σ
θ
λ λ α = −∂wγ τ uvt + (εβ ∂vβ τ uwt + τ uvx τ xwt Cσθ − τ uwx τ xvt Cσθ )Cλγ ,
from which
λ
α
α ∂wγ τ uvt = P uvwt Cλγ ,
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where
λ
λ
σ
θ
σ
(2.52) θ
λ
λ λ P uvwt = εβ ∂vβ τ uwt + τ uvx τ xwt Cσθ − τ uwx τ xvt Cσθ − ρ uvwt .
By virtue of (2.15), for fixed u, v, w and t (2.52) is the Scheffers condition ∗
of A−holomorphity of Γ zu = xuα eα from Xr (Am ). ∗
u wt
α u vt eα
=τ
with respect to the local coordinates
α
Conversely, if Γ uwt = τ uvt eα is a A−holomorphic connection, then from Theorem 22 (Ceα = Cα ) and (2.15) we obtain the condition (2.52). Using (2.52), from (2.49) we have Rijkl = Ruα vβwγtδ λ
λ
σ
θ
σ
θ
σ σ = (P uwvt Cλβ )Bασγδ − (P uvwt Cλγ )Bασβδ + (τ uvx τ xwt − τ uwx τ xvt )Bασγθβδ λ
λ
σ
θ
σ
θ
λ α Bλγβδ = P uwvt Bαλβγδ − P uvwt Bαλγβδ + (τ uvx τ xwt − τ uwx τ xvt )Cσθ λ
= ρ uvwt Bαλβγδ , where λ
λ
λ
σ
θ
σ
θ
λ ρ uvwt = P uwvt − P uvwt + (τ uvx τ xwt − τ uwx τ xvt )Cσθ .
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(2.53)
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Arif Salimov
Thus, from (2.37) and (2.53), we see that R is pure curvature tensor. Summing up, we have ∗
Theorem 39. [36], [104] Let ∇ be a hypercomplex connection on Xr (Am) and ∇ its a real model (a pure connection) on Mmr . The curvature tensor R of ∇ is ∗
pure if and only if ∇ is A− holomorphic connection. On the other hand, from Theorem 15 and Theorem 39 we have Theorem 40. Let ∇ be a pure connection which satisfies the condition (ψϕ ∇)(X,Y.Z) = ∇ϕX ∇Y Z − ϕ(∇X ∇Y Z) = 0 α
α
α
for any A−holomorphic vector fields X,Y, Z and α = 1, ..., m.Then such con∗
nection is a real model of A−holomorphic connection ∇ .
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2.9. Some Properties of Pure Curvature Tensors Let now R be a pure tensor field of pure connection ∇ which satisfies the condition ∇ ∈ Kerψϕ . Using (2.38), from (2.49) we have α
∗
R uvwt
β γ δ = Ruα vβwγtδ ε ε ε eα σ
σ
σ
θ
σ
θ
α = εβ (∂vβ τ uwt )eα − εγ (∂wγ τ uvt )eα + (τ uvx τ xwt − τ uwx τ xvt )Cσθ eα . α Since, Cσθ eα = eσ eθ , by virtue of (2.17) and (2.45), we have ∗u
∗u
∗u
∗u ∗x
∗u ∗x
Rvwt = ∂v Γwt − ∂w Γvt + Γvx Γwt − Γwx Γvt , ∗
∗
i.e. R is a curvature tensor of Γ. Thus, we have ∗
Theorem 41. Let ∇ be a real model of A−holomorphic connection ∇. The ∗
hypercomplex components R uvwt of the pure curvature tensor R are components ∗
of curvature tensor of ∇.
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∗
Let ∇ be a A−holomorphic hypercomplex connection on Xr (Am) and a torsion-free ∇ its real model on Mmr . From Theorem 39 we see that the curvature tensor R of ∇ is pure. Since the curvature tensor R is pure, we can apply the Tachibana φϕ − operator (see Section 1.2) to R. Using ∇ϕ = 0, α = 1, ..., m, (1.20) and Theorem 13, α
we have (φϕ R)(X,Y1,Y2 ,Y3 ) = (φϕX R)(Y1,Y2 ,Y3 ) − ϕ(∇X R)(Y1,Y2 ,Y3 ), (2.54) α
α
α
ℑ10 (Mmr ).
X,Y1 ,Y2 ,Y3 ∈
Using the purity of R and applying the Bianchi’s 2nd identity to (2.54), we get (φϕ R)(X,Y1,Y2 ,Y3 ) = (φϕX R)(Y1,Y2 ,Y3 ) − ϕ(∇X R)(Y1 ,Y2 ,Y3 ) α
α
α
= −(∇Y1 R)(Y2 , ϕX,Y3 ) − (∇Y2 R)(ϕX,Y1 ,Y3 ) α
α
−ϕ(∇X R)(Y1 ,Y2 ,Y3 ). α
On the other hand, using ∇ϕ = 0, we find Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
α
(∇Y2 R)(ϕX,Y1 ,Y3 ) = ∇Y2 (R(ϕX,Y1 ,Y3 )) − R(∇Y2 (ϕX),Y1 ,Y3 ) α
α
α
−R(ϕX, ∇Y2 Y1 ,Y3 ) − R(ϕX,Y1 , ∇Y2 Y3 ) α
α
= (∇Y2 ϕ)(R(X,Y1,Y3 )) + ϕ(∇Y2 (R(X,Y1,Y3 )) α
α
−R((∇Y2 ϕ)X + ϕ((∇Y2 X),Y1 ,Y3 )) α
α
−R(ϕX, ∇Y2 Y1 ,Y3 ) − R(ϕX,Y1 , ∇Y2 Y3 ) α
α
= ϕ(∇Y2 (R(X,Y1,Y3 )) − ϕ(R(∇Y2 X,Y1 ,Y3 )) (2.55) α
α
−ϕ(R(X, ∇Y2 Y1 ,Y3 )) − ϕ(R(X,Y1 , ∇Y2 Y3 )) α
α
= ϕ((∇Y2 R)(X,Y1,Y3 )). α
Similarly (∇Y1 R)(Y2 , ϕX,Y3 ) = ϕ((∇Y1 R)(Y2, X,Y3 )). α
α
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Arif Salimov
Substituting (2.55) and (2.56) into (2.54), and using again the Bianchi’s 2nd identity, we obtain (φϕ R)(X,Y1,Y2 ,Y3 ) = −ϕ((∇Y1 R)(Y2, X,Y3 )) − ϕ((∇Y2 R)(X,Y1,Y3 )) α
α
α
−ϕ(∇X R)((Y1,Y2 ,Y3 )) α
= −ϕ(σ{(∇X R)(Y1,Y2 },Y3 ) = 0, α
where σ denotes the cyclic sum with respect to X,Y1 and Y2 . Therefore, by virtue of Theorem 30 and Theorem 39, we have ∗
∗
Theorem 42. The curvature tensor R of A− holomorphic connection ∇ is A− holomorphic tensor. Example 9. Let now r = 1, i.e. we consider a A−holomorphic manifold X1 (Am ) of hypercomplex dimension 1. Since u = v = w = t = 1, we have from (2.49)
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Rijkl = R1α 1β1γ1δ σ
σ
σ
σ
σ
θ
σ
θ
= (∂1β τ 111 )Bασγδ − (∂1γ τ 111 )Bασβδ + (τ 111 τ 111 − τ 111 τ 111 )Bασγθβδ = (∂1β τ 111 )Bασγδ − (∂1γ τ 111 )Bασβδ σ
(2.57)
σ
α ε ε α = (∂β τ)Cδε Cσγ − (∂γ τ)Cσβ Cεδ σ
σ
ε ε α = (Cσγ ∂β τ −Cσβ ∂γ τ)Cδε , σ
σ
where τ = τ 111 . We now assume that
σ
σ
ε ε ∂γ τ. Cσγ ∂β τ = Cσβ
(2.58)
After contraction with εγ , we have from (2.58) ε
ε
ε , ∂β τ = εγ(∂γ τ)Cσβ
i.e. by virtue of (2.17), τ is A−holomorphic function of x = xαeα ∈ Am .
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(2.59)
Algebraic Structures on Manifolds
67
Conversely, let τ = τ(x) be a holomorphic function. Then, from (2.59) we have ε
σ
θ ε θ Cετ ∂β τ = εγ (∂γ τ)Cσβ Cετ σ
θ ε = εγ (∂γ τ)Cβε Cστ σ
ε θ = εγ (∂γ τ)Cστ Cβε ε
σ θ = εγ (∂σ τ)Cγτ Cβε ε
θ = (∂τ τ)Cβε ,
i.e. the condition (2.58) is true and the condition (2.58) is equivalent to the Scheffers condition (2.15). Thus, if τ = τ(x) is A−holomorphic, then Rijkl = 0. Conversely, if R = 0, then from (2.57) we have δ 0 = R1α 1β1γ1δ ε σ
σ
ε ε α δ = (Cσγ ∂β τ −Cσβ ∂γ τ)Cδε ε σ
σ
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α α = Cσγ ∂β τ −Cσβ ∂γ τ,
i.e. the function τ = τ(x) is A−holomorphic. Obviously, we have ∗
Theorem 43. Let Mm be a real model of X1 (Am). The connection ∇ with com∗ σ ponents τ = τeσ on X1 (Am ) is A−holomorphic if and only if the real manifold Mm is locally flat. σ
σ
Remark 13. In particular, if τ = τ 111 = εσ (1 = εσ eσ ∈ Am), then from (2.44) α we have Γαγβ = Cγβ , i.e. Mm is the Vranceanu space [109], [63].
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Chapter 3
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Applications to the Norden Geometry In this chapter, we study the various hypercomplex Norden manifolds. In Section 3.1 we give the fundamental properties of almost hypercomplex Norden manifolds. We give the condition for an almost hypercomplex Norden manifold to be holomorphic (K¨ahlerian). In Section 3.2 we discuss complex Norden manifolds. We define the twin Norden metric, the main theorem of this section is that the Levi-Civita connection of K¨ahler-Norden metric coincides with the Levi-Civita connection of twin Norden metric. Section 3.3 devoted to the study of almost product Riemannian manifolds. We give the definitions of decomposable Riemannian manifolds and para-K¨ahler-Norden manifolds. The final theorem of this section is that there does not exist para-K¨ahler-Norden warped metric. In Section 3.4, we give some examples of dual-K¨ahler-Norden manifolds. In Section 3.5, we consider Norden-Hessian structures. We give the condition for a K¨ahler (para-K¨ahler-Norden) manifold to be Norden-Hessian. In Section 3.6, we shall focus our attention to Norden manifolds of dimension four. The main purpose of the present section is to study complex Norden metrics on 4-dimensional Walker manifolds. We discuss the integrability and Kahler (holomorphic) conditions for these structures. The curvature properties for Norden-Walker metrics is also investigated. Examples of Norden-Walker metrics are constructed from an arbitrary harmonic function of two variables. We define the isotropic K¨ahler structures and moreover, show that a proper almost complex structure on almost Norden-Walker manifold is isotropic K¨ahler. We also consider the quasi-K¨ahler-Norden metric and give the condition for an
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Arif Salimov
almost Norden manifold to be quasi-K¨ahler-Nordenian. In this section we also give progress to the conjecture of Goldberg under the additional restriction on Norden-Walker metric. Section 3.7 is devoted to the analysis of opposite almost complex structures on Norden-Walker 4-manifolds. In Section 3.8, we shall focus our attention to para-Norden manifolds of dimension four. Using a Walker metric we construct para-Norden-Walker metrics together with a proper almost paracomplex structures. In Section 3.9, we give some examples of NordenWalker metrics on 8-dimensional manifolds.
3.1. Hyper-K¨ahler-Norden Manifolds Let Mmr be a Riemannian manifold with metric g , which is not necessarily positive definite, and let on Mmr be given the regular hypercomplex Π−structure
u α Π = ϕ , ϕ ij = ϕ uα vβ = δv Cσβ , i = uα, j = vβ; i, j = 1, ..., mr; α, β, σ = 1, ...., m; σ
σ
σ
u, v = 1, ..., r (see Section 2.2). A Norden metric with respect to the hypercomplex structure is a Riemannian metric g such that g(ϕX,Y ) = g(X, ϕY ), α = 1, ..., m
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α
(3.1)
α
for any X,Y ∈ ℑ10 (Mmr ), i.e. g is pure with respect to the regular hypercomplex Π−structure. Such Riemannian metrics were studied in [104], where they were said to be B-metrics, since the metric tensor g with respect to the Π−structure is B-tensor according to the terminology accepted by Norden [58]. If (Mmr , Π) is an almost hypercomplex manifold with Norden metric, we say that (Mmr , Π, g) is an almost hypercomplex Norden manifold. If Π−structure is integrable, we say that (Mmr , Π, g) is a hypercomplex Norden manifold. β
Remark 14. Let ψ be a conjugation of Am (see Section 2.1.2). We put ϕ = ψα ϕ. α
β
A Riemannian metric g which satisfies g(ϕX,Y ) = g(X, ϕY ), α = 1, ..., m α
α
for any X,Y ∈ ℑ10 (Mmr ) is called a hybrid [37] with respect to the conjugation ψ. In particular, if ψ = idAm , then the hybrid tensor g is pure. If Am = C (m = 2) is complex algebra, then for the conjugation ψ 6= id , we have
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1 0 β ϕ = ψ(ϕ) = −ϕ, ψα = , 0 −1
i.e. a hybrid tensor g with respect to the conjugation is characterized by g (ϕX,Y ) = −g (X, ϕY ) for any X,Y ∈ ℑ10 (M2r ). It is well known that, if g hybrid with respect to the conjugation ψ 6= id and positive-definite, then the triple (Mmr , Π, g) is an almost Hermitian manifold. A Norden metric g is called a A−holomorphic(see Theorem 30) if (φϕ g) (X,Y, Z) = (ϕX) (g (Y, Z)) − X(g(ϕY, Z)) α
α
α
+g((LY ϕ)X, Z) + g(Y, (LZ ϕ)X) α
α
= (LϕX g − LX (g ◦ ϕ))(Y, Z) = 0, α = 1, ..., m α
α
for any X,Y, Z ∈ ℑ10 (Mmr ), where φϕ g, α = 1, ..., m are Tachibana operators Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
α
applied to a Norden metric. If (Mmr , Π, g) is a hypercomplex Norden manifold with A−holomorphic Norden metric g, we say that (Mmr , Π, g) is a A−holomorphic Norden manifold. Since the case where dimM = m, such a manifold is flat (see Theorem 43 and Theorem 47), we assume in the sequel that dim M ≥ 2m, i.e. r ≥ 2. In some aspects, A−holomorphic Norden manifolds are similar to K¨ahler manifolds. The following theorem is analogous to the next known result: An almost Hermitian manifold is K¨ahler if and only if the almost complex structure is parallel with respect to the Levi-Civita connection [34, Chapter 9]. The following theorem plays very important role in Norden geometry: Theorem 44. An almost hypercomplex Norden manifold is A−holomorphic Norden manifold if and only if structure affinors are parallel with respect to the Levi-Civita connection ∇g . Proof. Using (3.1) and [X,Y ] = ∇X Y − ∇Y X, we get from (1.13) (φϕ g)(X, Z1 , Z2 ) = (LϕX g − LX (g ◦ ϕ))(Z1 , Z2 ) α
α
α
+g(Z1, ϕLX Z2 ) − g(ϕZ1 , LX Z2 ) α
α
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Arif Salimov = (ϕX)g(Z1 , Z2 ) − Xg(ϕZ1 , Z2 ) − g(∇ϕX Z1 , Z2 ) α
α
α
+g(∇Z1 ϕX, Z2 ) − g(Z1 , ∇ϕX Z2 ) + g(Z1 , ∇Z2 ϕX) α
α
α
+g(ϕ(∇X Z1 ), Z2 ) − g(ϕ(∇Z1 X), Z2 ) α
α
+g(ϕZ1 , ∇X Z2 ) − g(Z1 , ϕ(∇Z2 X)). α
α
We find g(∇Z1 ϕX, Z2 ) − g(ϕ(∇Z1 X), Z2 ) + g(Z1 , ∇Z2 ϕX) − g(Z1 , ϕ(∇Z2 X)) α
α
α
α
= g((∇ϕ)(X, Z1 ), Z2 ) + g(Z1 , (∇ϕ)(X, Z2 )).
(3.2)
Substituting (3.2) into (3.2), we have (φϕ g)(X, Z1 , Z2 ) = (ϕX)g(Z1 , Z2 ) − Xg(ϕZ1 , Z2 ) + g((∇ϕ)(X, Z1 ), Z2 ) α
α
α
α
+g(Z1 , (∇ϕ)(X, Z2 )) − g(∇ϕX Z1 , Z2 ) − g(Z1 , ∇ϕX Z2 ) α
α
α
+g(ϕ(∇X Z1 ), Z2 ) + g(ϕZ1 , ∇X Z2 ). Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
α
(3.3)
α
On the other hand, with respect to the Levi-Civita connection ∇, we have (ϕX)g(Z1 , Z2 ) − g(∇ϕX Z1 , Z2 ) − g(Z1 , ∇ϕX Z2 ) = (∇ϕX g)(Z1 , Z2 ) = 0 α
α
(3.4)
α
α
and −Xg(ϕZ1 , Z2 ) + g(ϕ(∇X Z1 ), Z2 ) + g(ϕZ1 , ∇X Z2 ) = −g((∇X ϕ)Z1 ), Z2 ). (3.5) α
α
α
α
By virtue of (3.4) and (3.5), (3.3) reduces to (φϕ g)(X, Z1, Z2 ) = −g((∇X ϕ)Z1 ), Z2 ) + g((∇Z1 ϕ)X, Z2 ) + g(Z1 , (∇X ϕ)Z2 ). α
α
α
α
(3.6) Similarly, we have (φϕ g)(Z2 , Z1 , X) = −g((∇Z2 ϕ)Z1 ), X) + g((∇Z1 ϕ)Z2 , X) + g(Z1 , (∇Z2 ϕ)X). α
α
α
α
(3.7)
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73
The sufficiency follows easily from (3.6) or (3.7). By virtue of g(Z, (∇Y ϕ)X) = α
g((∇Y ϕ)Z, X) we find α
(φϕ g)(X, Z1, Z2 ) + (φϕ g)(Z2 , Z1 , X) = 2g(X, (∇Z1 ϕ)Z2 ). α
(3.8)
α
α
Now, putting φϕ g = 0 in (3.8), we find ∇ϕ = 0, α = 1, ..., m from which the α
α
necessity follows. Thus Theorem 44 is proved. Using Remark 11, from Theorem 44 we have Corollary 45. The Π−structure on almost hypercomplex Norden manifold is integrable if φϕ g = 0, α = 1, ..., m. α
Also, from Theorem 44 we have
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Corollary 46. The Levi-Civita connection ∇g on A−holomorphic Norden manifolds is pure connection. Remark 15. Recall that a hyper-K¨ahler-Norden manifold can be defined as a triple (M2n , Π, g) which consists of a manifold Mmr endowed with a hypercomplex structure Π and a pseudo-Riemannian metric g such that ∇ϕ = 0, α = α
1, ..., m , where ∇ is the Levi-Civita connection of g and the metric is assumed to be Nordenian: g(ϕX,Y ) = g(X, ϕY ), α = 1, ..., m. Therefore, there exists a α
α
one-to-one correspondence between hyper-K¨ahler-Norden manifolds and complex Riemannian manifolds with a A−holomorphic metric. From (3.1) we have σ gi j = guαvβ = GuvσCαβ
(3.9)
for arbitrary functions Guvσ (see Section 2.4). The corresponding hypercomplex ∗ tensor guv is defined by ∗
guv = guαvβ εβ εα = Guvα eα ,
(3.10) ∗
where eα = ϕαβ eβ , ϕαβ is the Frobenius metric. Since is guv symmetric, non∗
∗
∗
singular (Det(guv ) 6= 0) and A−holomorphic, it follows that ds2 = guvdzu dzv can
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Arif Salimov
be regarded as hyper-K¨ahler-Norden metric in the A−holomorphic manifold Xr (A). Since the Levi-Civita connection ∇g of A−holomorphic Norden manifolds is pure connection, by virtue of (2.44), we obtain σ
σ
µ
σ
σ
uα α Kki j = Kwγvβ = k uwvCσγCµβ = k uwv Bασγβ , k uwv = k uvw ,
(3.11)
where are Kki j components of ∇g and i = uα, j = vβ, k = wγ. Substituting (3.9) and (3.11) into ∇k gi j = 0, we find σ
σ
µ
µ
σ v τ v τ ∂wγ (GuvσCαβ ) = k twuCσγ Cvα GtvτCµβ + k twvCσγ Cvβ GtuτCµα ,
After contraction with εβ εγεα , by virtue of (2.5) and (2.10), we obtain σ
σ
εγ ∂wγ Guvσeα = (k twu Gtvµ + k twv Gtuµ )eσ eµ . By virtue of (2.17), (2.45) and (3.10), we have ∗
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∗ ∂guv ∗ t ∗ ∗ = k wu gtv + k twv gut , w ∂z
(3.12) ∗
from which we easily see that the Christoffel symbols formed with g has components ∗ ∗ ∗ ∗ 1 ∗ ut ∂g ∂gwt ∂guv k uwv = g ( tv + − ). (3.13) 2 ∂zw ∂zv ∂zw ∗
Since the hypercomplex Norden metric g is A−holomorphic, exists the suc∗
∗
cessive derivatives of g (see Section 2.1.3), i.e. from (3.13) it follows that k is A−holomorphic. Thus we have
u wv
Theorem 47. The Levi-Civita connection of hyper-K¨ahler-Norden manifold is A− holomorphic. Using Theorem 39 we have Theorem 48. The Riemannian curvature tensor field of hyper-K¨ahler-Norden manifold is pure.
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From Theorem 42 we have Theorem 49. The Riemannian curvature tensor field of hyper-K¨ahler-Norden manifold is A−holomorphic. In particular, if Am = C (m = 2), an alternative proof of Theorem 48 will be made in Section 3.2 by using the real model of hyper-K¨ahler-Norden manifolds. From Theorem 6 we have: A necessary and sufficient condition for an exact 1-form d f , f ∈ ℑ00 (M2m) to be A−holomorphic is that associated 1-forms d f ◦ ϕ, α = 1, ..., m be closed 1-forms, i.e. α
d(d f ◦ ϕ) = 0, α = 1, ..., m.
(3.14)
α
If there exists functions g, α = 1, ..., m in a hyper-K¨ahler-Norden manifold α
such that d f ◦ ϕ = dg, α = 1, ..., m for a function f , then we shall call f a α
α
A−holomorphic function and g, α = 1, ..., m associated functions. If such a α
function f is defined locally, then we call it a locally A−holomorphic function. We notice that the equation (3.14) is equivalent to d f ◦ ϕ = dg, α = 1, ..., m Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
α
α
only locally. Hence the condition for f to be locally A−holomorphic (ϕm i ∂m f = α
∂i g) also is given by α
m m (φϕ d f ) = ϕm i ∂m ∂i f − ∂i (ϕi ∂m f ) + (∂i ϕi )∂m f = 0 , α = 1, ..., m. α
α
α
α
Let (Mmr , Π, g) be a A−holomorphic manifold with hyper-K¨ahler-Norden metric g. Then using Theorem 13 and Theorem 47, by virtue of (1.23) we find that in hyper-K¨ahler-Norden manifolds the covariant derivative of the curvature tensor field ∇R is also pure. Therefore, the covariant derivative of the Ricci tensor R ji = Rss ji = gts Rt jis is pure in all its indices and hence ϕts ∇s R ji = ϕsj ∇t Rsi , α = 1, ..., m. α
α
Transvecting this equation with contravariant hyper-K¨ahler-Norden metric g ji , we find ∗
ϕts ∇s R = g ji ϕsj ∇t Rsi = ∇t (g ji ϕsj Rsi ) = ∇t R , α = 1, ..., m, α
α
α
α
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(3.15)
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Arif Salimov
where R = gi j Ri j are curvature scalars of hyper-K¨ahler-Norden metric g and ∗
R = g ji ϕsj Rsi . From (3.15) we have α
α
Theorem 50. In a hyper-K¨ahler-Norden manifold, the curvature scalar R is a locally A−holomorphic function. Remark 16. In the next sections, we shall easily see that if Am(m = 2) is com∗
plex algebra (or paracomplex algebra), then the associated function R is a curvature scalar of twin-Norden metric.
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3.2. Complex K¨ahler-Norden Manifolds Let Am = C (m = 2) be a complex algebra. In this section we shall only consider C−holomorphic Norden (i.e. K¨ahler-Norden) manifolds. Let M2n be a pseudo-Riemannian manifold with a neutral metric, i.e. with a pseudo-Riemannian metric g of signature (n, n). We say (M2n , ϕ) is an almost complex manifold if M2n can be endowed with an affinor field ϕ ∈ ℑ11 (M2n) such that ϕ2 = −I, where I is a field of identity endomorphisms. If the Nijenhuis tensor field Nϕ ∈ ℑ12 (M2n ) vanishes, then ϕ is a complex structure and moreover M2n is a C−holomorphic manifold Xn (C) whose transition functions are C−holomorphic mappings. A metric g is a Norden metric [19] if g(ϕX,Y ) = g(X, ϕY ),
(3.16)
for any X,Y ∈ ℑ10 (M2n), i.e. g is pure with respect to ϕ. Metrics of this kind have been also studied under the names: anti-Hermitian and B-metrics (see [5], [7], [14], [15], [25], [26], [31], [42], [50], [51], [52], [60], [61], [64], [70], [71], [72], [80], [81], [82], [83], [101], [104], [105]). If (M2n , ϕ) is an almost complex manifold with Norden metric, we say that (M2n , ϕ, g) is an almost Norden manifold. If ϕ is integrable, we say that (M2n , ϕ, g) is a Norden manifold. Let (M2n, ϕ, g) be an almost Norden manifold. The twin Norden metric of almost Norden manifold is defined by G(X,Y ) = (g ◦ ϕ)(X,Y),
(3.17)
for all vector fields X and Y on M2n . One can easily prove that G is a metric, which is also called the associated (or dual) metric of g and it plays a role similar
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to the Kahler form in Hermitian Geometry. We shall now apply the Tachibana operator to the pure Riemannian metric G : (φϕ G)(X,Y, Z) = (LϕX G − LX (G ◦ ϕ))(Y, Z) +G(Y, ϕLX Z) − G(ϕY, LX Z) = (φϕ g)(X, ϕY, Z) + g(Nϕ (X,Y ), Z).
(3.18)
Thus (3.18) implies the following Theorem 51. In an almost Norden manifold, we have φϕ G = (φϕ g) ◦ ϕ + g ◦ (Nϕ ). Corollary 52. In a Norden manifold the following conditions are equivalent: a) φϕ g = 0, b) φϕ G = 0.
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From Theorem 44 and Theorem 51 we have Theorem 53. Almost Norden manifold with conditions φϕ G = 0 and Nϕ 6= 0, i.e. analogues of the almost K¨ahler manifolds with closed K¨ahler form, do not exist. We denote ∇g by the covariant differentiation of Levi-Civita connection of Norden metric g. Then, we have φϕ G = (φϕ g) ◦ ϕ + g ◦ (φg ϕ) = g ◦ (∇g ϕ), which implies φϕ G = 0 by virtue of Theorem 44. Therefore we have Theorem 54. Let (M2n , ϕ, g) be a K¨ahler-Norden manifold. Then the LeviCivita connection of Norden metric g coincides with the Levi-Civita connection of twin Norden metric G. We note that, in the case where Am = C (m = 2), we can prove the Theorem 48 by using Theorem 54. In fact, let R and S be the curvature tensors formed by g and G respectively, then for the K¨ahler-Norden manifold we have R = S by means of the Theorem 54. Applying the Ricci’s identity to ϕ, we get ϕ(R(X,Y)Z) = R(X,Y )ϕZ
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(3.19)
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Arif Salimov
by virtue of ∇ϕ = 0. Hence R(X1, X2 , X3 , X4 ) = g(R(X1, X2 )X3 , X4 ) is pure with respect to X3 and X4 and also pure with respect to X1 and X2 : R(X1 , X2 , ϕX3 , X4 ) = g(R(X1, X2 )ϕX3 , X4 ) = g(ϕ(R(X1, X2 )X3 ), X4 ) = g(R(X1, X2 )X3 , ϕX4 ) = R(X1 , X2 , X3 , ϕX4 ). On the other hand, S being the curvature tensor formed by twin metric G, if we put S(X1 , X2 , X3 , X4 ) = G(S(X1, X2 )X3 , X4 ) , then we have S(X1 , X2 , X3 , X4 ) = S(X3 , X4, X1 , X2 ).
(3.20)
Taking account of (3.16), (3.17), (3.19) and R = S , we find that S(X1 , X2 , X3 , X4 ) = G(S(X1, X2 )X3 , X4 ) = g(ϕ(S(X1 , X2 )X3 ), X4 ) = g(S(X1, X2 )X3 , ϕX4 ) Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
= g(R(X1, X2 )X3 , ϕX4 ) = R(X1, X2 , X3 , ϕX4 ) and S(X3 , X4 , X1 , X2 ) = G(S(X1, X2 )X3 , X4 ) = g(ϕ(S(X3 , X4 )X1 ), X2 ) = g(S(X3, X4 )X1 , ϕX2 ) = g(R(X3, X4 )X1 , ϕX2 ) = R(X3, X4 , X1 , ϕX2 ) = R(X1, ϕX2 , X3 , X4 ). Thus the equation (3.20) becomes R(X1 , X2 , X3 , ϕX4 ) = R(X1 , ϕX2 , X3 , X4), which shows that R(X1 , X2, X3 , X4 ) is pure with respect to X2 and X4 . Therefore R(X1 , X2 , X3 , X4 ) is pure.
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We now consider a K¨ahler-Norden manifold and its invariant submanifold. Let (M2n , ϕ, g) be a K¨ahler-Norden manifold . From (1.31) we have e = ϕB(ϕ e = B(ϕ e eX) e2 X) −BXe = ϕ2 (BX)
or
e + Xe ) = 0 e2 X B(ϕ
e2m).Since B is injective, from here we can see ϕ e 2 = −I , so an for any Xe ∈ ℑ10 (M e2m becomes an almost complex manifold by virtue of invariant submanifold M e. the induced affinor ϕ If we take account of Theorem 16, then we get
Theorem 55. An invariant submanifold in a K¨ahler-Norden manifold is itself K¨ahler-Norden manifold with respect to the induced structure.
3.3. Almost Product Riemannian Manifolds
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3.3.1.
Decomposable Riemannian Manifolds
Let Mn be an almost product manifold with almost product structure F . If Mn admits a Riemannian metric g such that g(FX,Y ) = g(X, FY ) for X,Y ∈ ℑ10 (Mn ), i.e. if g is pure with respect to F, then Mn is called an almost product Riemannian manifold [56], [115, p.423]. We define the operator φF : ℑ02 (Mn ) → ℑ03 (Mn ) associated with F and applied to the pure metric g : (φF g)(X,Y1,Y2 ) = (FX)(g(Y1,Y2 )) − X(g(FY1 ,Y2 ) +g((LY1 F)X,Y2 ) + g(Y1 , (LY2 F)X). By similar devices (see Section 3.1), we have (φF g)(X, Z1 , Z2 ) = −g((∇X F)Z1 , Z2 ) + g((∇Z1 F)X, Z2 ) + g(Z1 , (∇Z2 F)X) and (φF g)(X, Z1, Z2 ) + (φF g)(Z2 , Z1 , X) = 2g(X, (∇Z1 F)Z2 ).
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Putting φF g in the last equation, we find ∇F = 0. On the other hand, we know that the integrability of the almost product structure F is equivalent to the existing a torsion-free affine connection with respect to which the equation ∇F = 0 holds. Since the Levi-Civita connection ∇ of g is a torsion-free affine connection, we have Theorem 56. Let (Mn , F) be an almost product Riemannian manifold with pure metric g. Then F is integrable if φF g = 0 . We note that, if ∇F = 0, then the condition φF g = 0 follows from the above expression of (φF g)(X, Z1, Z2 ). Thus we have Theorem 57. [76] For an almost product Riemannian manifold with pure metric g, the condition φF g = 0 is equivalent to ∇F = 0, where ∇ is the Levi-Civita connection of g.
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An integrable almost product Riemannian manifold with structure tensor F is usually called a locally product Riemannian manifold. If the metric of a locally product Riemannian manifold Mn has the form ds2 = gab (xc )dxa dxb + gab (xc )dxa dxb , a, b, c, ... = 1, ..., m, a, b, c, ... = m + 1, ..., n that gab is are functions of xc only, gab = 0 and gab are functions xc only, then we call the manifold Mn a locally decomposable Riemannian manifold . On the other hand, we know that the locally product Riemannian manifold with structure tensor F is locally decomposable if and only if F is covariantly constant with respect to the Levi-Civita connection ∇ [115, p.420]. Thus, by Theorem 56 and Theorem 57 we have Theorem 58. Let (Mn , F) be an almost product Riemannian manifold with pure metric g. A necessary and sufficient condition for (Mn , F) to be a locally decomposable Riemannian manifold is that φF g = 0.
3.3.2.
Para-K¨ahler-Norden Manifolds
Let Am = A(e),(m = 2) be a paracomplex algebra. We shall only consider A(e)holomorphic Norden (i.e. para-K¨ahler-Norden) manifolds.
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An almost paracomplex manifold is an almost product manifold (Mn , ϕ), ϕ2 = id, such that the two eigenbundles T + Mn and T − Mn associated to the two eigenvalues +1 and −1 of ϕ, respectively, have the same rank. Note that the dimension of an almost paracomplex manifold is necessarily even. Considering the paracomplex structure ϕ, we obtain the following set of affinors on Mn :{id, ϕ}, ϕ2 = id, which form a bases of a representation of the algebra of order 2 over the field of real numbers R, which is called the paracomplex (or double) numbers and is denoted by A(e) = algebra of 2 a0 + a1 e | e = 1; a0, a1 ∈ R . Obviously, it is associative, commutative and unitial, i.e., it admits principal unit 1. The canonical bases of this algebra has the form {1, e}. Structural constants of an algebra are defined by the multiplication law of the base units of this algebra: ei e j = Cikj ek . The components of Cikj 1 = C2 = C2 = C1 = 1, all the others being zero with respect are given by C11 12 21 22 to the canonical bases of A(e). Consider A(e) endowed with the usual topology of and a domain U of A(e). Let X = x1 + ex2 be a variable in A(e), where xi are real coordinates of a point of a certain domain U for i = 1, 2. Using two real-valued functions f i (x1 , x2 ), i = 1, 2, we introduce a paracomplex function F = f1 +ef2 of variable X. It follows that F is paraholomorphic if and only if f 1 and f 2 satisfy the para-Cauchy-Riemann equations(see Example 5): ∂f1 ∂f2 ∂f1 ∂f2 = , = . ∂x1 ∂x2 ∂x2 ∂x1 By similar devices (see Theorem 54), we can prove a following: In a paraK¨ahler-Norden case, the Levi-Civita connection of is also the Levi-Civita connection of the twin metric G given by G(X,Y) = g(ϕX,Y ), moreover, in such a manifold, the Riemannian curvature tensor is paraholomorphic. Remark 17. The leaves of the foliations defined by the paracomplex structure of a para-K¨ahler-Norden manifold are totally geoedesic submanifolds (see [56] or the book Yano and Kon [115, p.420]). In the paper of Naveira [56] Riemannian almost product manifolds were classified.
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Example 10. Let now Mn be the locally product Riemannian manifold with integrable almost product structure ! δij 0 , i, j = 1, ..., k, i, j = k + 1, ...., n ϕ= 0 −δij and let n = 2k. Then the paracomplex manifold M2k , admit a metric of paraNorden manifold: gi j 0 g= , gi j = gi j (xt , xt ), gi j = gi j (xt , xt ). 0 gi j Suppose that the metric of the locally product Riemannian manifold M2k has the form
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ds2 = gi j (xt )dxi dx j + gi j (xt )dxi dx j ,
i, j,t = 1, ..., k ; i, j,t = k + 1, ..., 2k,
that is gi j (x) are functions of xt only, gi j = 0, and gi j (x) are functions of xt only, i.e. the manifold M2k is a locally decomposable Riemannian manifold. Since the necessary and sufficient condition for a locally product Riemannian manifold to be a locally decomposable Riemannian manifold is that ∇g ϕ = 0, we have from Theorem 44 Theorem 59. A locally decomposable Riemannian manifold M2k is a paraK¨ahler-Norden manifold. Example 11. [85] Let (M2k, ω) be a symplectic manifold and let D be a Lagrangian distribution, which is a k−dimensional distribution having ω/D = 0. Then, M may be endowed with an almost para-Norden structure.First of all, we shall prove that there exist a transversal Lagrangian distribution. Taking into account that (M, ω) is an almost symplectic manifold one can find (see [1] or [62]) an almost Hermitian structure (J, G) on M such that ω(X,Y ) = G(JX,Y ). Let D⊥ the G−orthogonal distribution to D. Then one has: (1) If X,Y ∈ D, then G(JX,Y ) = ω(X,Y ), thus proving that J(D) = D⊥ . (2) D⊥ is a Lagrangian distribution, because ω(JX, JY ) = ω(X,Y ), for all X,Y ∈ ℑ10 (Mn ). Let F be the almost product structure defined by D and D⊥ , i.e.,F + = D and F − = D⊥ . Then, one easily check that J ◦ F = −F ◦ J. Moreover, one can
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prove that (M, F, G) is a Riemannian almost product manifold: If X ∈ ℑ10 (Mn ), then X = X1 + X2 ,where X1 ∈ F + = D and X2 ∈ F − = D⊥ = J(D) one can write X2 = J(X3 ), with X3 ∈ F + . Using this notation we obtain: G(X,Y ) = G(X1 + JX3 ,Y1 + JY3 ) = G(X1 ,Y1 ) + G(JX3 , JY3 ) = G(X1 ,Y1 ) + G(X3 ,Y3 ) and G(FX, FY ) = G(X1 − JX3 ,Y1 − JY3 ) = G(X1 ,Y1 ) + G(JX3 , JY3 ) = G(X1,Y1 ) + G(X3 ,Y3 ) thus proving G(X,Y ) = G(FX, FY ) .
3.3.3.
Nonexistence of Para-K¨ahler-Norden Warped Metrics
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Bishop and O’Neill [3] introduced warped product manifolds as follows: Suppose (B, gB) and (F, gF ) are semi-Riemannian manifolds, and let f : B → (0, ∞) be a smooth function. The warped product M = B × f F is the product manifold B × F furnished with metric tensor w
g = π∗ (gB ) + ( f ◦ π)2 σ∗ (gF ),
where π and σ are the projections of B × F onto B and F, respectively, and ∗ denotes the pull-back operator. Here, B is called the base of M, and F the fiber, and f the warping function. Explicitly, if X and Y are tangent to B × F at (p, q), then w g(X,Y ) = gB (π∗ X, π∗Y ) + f 2 (p)gF (σ∗ X, σ∗ X), where π∗ and σ∗ are differentials of the projections π and σ respectively. If the warping function f is constant , then the manifold B × f F is said to be trivial, i.e. B × f F reduces to a semi-Riemannian product manifold with decomposable metric g = π∗ (gB ) + σ∗ ( f 2 gF ). On the other hand, the warped product manifold B × f F has a canonical product structure ϕ ∈ ℑ11 (Mn ), ϕ2 = id. A Norden metric on B × f F is a pure semi-Riemannian metric on B × f F with respect to the almost product structure ϕ. The case, where ϕ is an almost paracomplex structure, we discuss the paraK¨ahler-Norden conditions of warped metrics with respect to the structure ϕ, and we prove the following theorem:
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Theorem 60. Let dimB = dim F = k , and M2k = Bk × f Fk be a warped paracomplex manifold with metric w g. Then (Bk × f Fk ,w g, ϕ) is a para-Norden manifold and there do not exist para-K¨ahler-Norden warped metric on this manifold. Proof. Let dim B = dim F = k , and M2k = Bk × f Fk be a warped paracomplex manifold with metric w g = π∗ (gB) + ( f ◦ π)2 σ∗ (gF ). The horizontal (vertical) lift of X ∈ ℑ10 (Bk ) (of U ∈ ℑ10 (Fk )) to M2k = Bk × f Fk is the unique element H X ∈ ℑ1 (B × F ) (V U ∈ ℑ1 (B × F ) ) that is π−related ( σ−related) to X (to 0 k f k 0 k f k U ) and σ−related ( π−related) to the zero vector field on Fk (Bk ), i.e. π∗ H X = X, σ∗ H X = 0 (π∗ V U = 0, σ∗ V U = U)
(3.21)
Let X,Y ∈ ℑ10 (Bk) and U,W ∈ ℑ10 (Fk ). Then for Lie bracket of these vector fields we have [59, p.25] [H X,H Y ] =H [X,Y ], [V U,V W ] =V [U,W], [H X,V U] = 0.
(3.22)
Putting, now ϕ(H X) =H X, ϕ(V U) = −V U
(3.23)
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we have a canonical integrable paracomplex structure ϕ on Bk × f Fk .We put e Ye ) =W g(ϕX, e Ye) −W g(X, e ϕYe ) G(X,
e Ye ∈ ℑ1 (Bk × f Fk ). If G(X, e Ye ) = 0 for all vector fields Xe and Ye which for any X, 0 H H V V are of the form X, Y or U, W then G = 0. By virtue of (3.21) and (3.23), we have G(H X,H W ) =
W
g(ϕH X,H Y ) −W g(H X, ϕH Y ) = 0,
G(H X,V W ) =
W
g(ϕH X,V W ) −W g(H X, ϕV W ) = 2W g(H X,V W )
V 2 H V = gB (πH ∗ X, π∗ W ) + f gF (σ∗ X, σ∗ W ) = 0,
G(V U,H Y ) =
w
g(ϕV U,H Y ) −w g(V U, ϕH Y ) = 2w g(V U,H Y )
2 V H = gB (πV U, πH ∗ Y ) + f gF (σ∗ U, σ∗ Y ) = 0,
G(V U,V W ) =
w
g(ϕV U,V W ) −w g(V U, ϕV W )
= −w g(V U,V W ) +w g(V U,V W ) = 0, i.e. (Bk × f Fk ,w g, ϕ) is a para-Norden warped manifold.
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We now define φϕ −operator applied to the para-Norden Walker metric w g by e Ye1 , Ye2 ) = (φX)( e W g(Ye1 , Ye2 )) − X eW g(ϕYe1 , Ye2 ) (φϕ W g)(X, e Ye2 ) +W g(Ye1 , (Le ϕ)X) e +W g((Le ϕ)X, Y1
Y2
e Ye1 , Ye2 ∈ ℑ1 (Bk × f Fk ). Using (3.21), (3.22) and (3.23) we have for any X, 0 (φϕ W g)(V U,H Y1 ,H Y2 )
= (ϕV U)W g(H Y1 ,H Y2 ) −V U W g(ϕH Y1 ,H Y2 ) + +W g((LHY1 ϕ)V U,H Y2 ) +W g(H Y1 , (LHY2 V U) = −2V U W g(HY1 ,H Y2 ) +W g(LHY1 (ϕV U) −ϕ(LH Y1 V U),H Y2 ) +W g(HY1 , LH Y2 (ϕV U) − ϕ(LH Y2 V U)) = −2V U W g(HY1 ,H Y2 ) +W g(−LH Y1 V U − ϕ(LH Y1 V U),H Y2 ) + +W g(H Y1 , −LH Y2 V U − ϕ(LH Y2 V U)) H = −2V U W g(HY1 ,H Y2 ) = −2V UgB (πH ∗ Y1 , π∗ Y2 ) = 0
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for any U ∈ ℑ10 (Fk ) and Y1 , Y2 ∈ ℑ10 (Bk ). By similar devices, we have (φϕ W g)(H X,H Y1 ,H Y2 ) = (φϕ W g)(H X,V U,H Y ) = (φϕ W g)(H X,H Y,V U) = (φϕ W g)(V U,V W,H Y ) = (φϕ W g)(V U,H Y,V W ) = (φϕ W g)(V U,V W1 ,V W2 ) = 0. In the case, where Xe =H X, Ye1 =V W1 , Ye2 =V W2 we obtain (φϕW g)(H X,V W1 ,V W2 )
= 2H X W g(V W1 ,V W2 ) +W g((LV W1 ϕ)H X,V W2 ) +W g((V W1 , LV W2 ϕ)H X) = 2H X W g(V W1 ,V W2 ) +W g(LV W1 H X − ϕ(LV W1 H X),V W2 ) +W g(V W2 , LV W2 H X − ϕ(LV W2 H X)) = 2H X(( f ◦ π)2 gF (σ∗V W1 , σ∗V W2 )) = 2(H X( f ◦ π)2 )gF = 4 f (X f )gF , from which we see that the equations φϕW g = 0 and f =constant ( f > 0) are equivalent. On the other hand, as stated in Theorem 44, a para-Norden metric
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W
g on a warped manifold is a paraholomorphic ( φϕW g = 0 ), if and only if it satisfies the para-K¨ahler-Norden condition. Thus proof of Theorem 60 is complete.
3.4. Dual-K¨ahler-Norden Manifolds
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Let Am − C(ε)(m = 2) be a dual algebra. In this section we shall only consider C(ε)-holomorphic Norden (i.e. dual-K¨ahler-Norden) manifolds. We suppose that the manifold M2n is the tangent bundle π : T (Vn ) → Vn of a Riemannian manifold Vn . If (u1 , u2 , ..., un) are local coordinates on Vn , then xi = ui ◦ π together with the fibre coordinates xi = yi , i = n + 1, ..., 2n form local coordinates on T (Vn). It is well known that there exists a tensor field of type (1, 1) which has components of the form 0 0 γ= (3.24) I 0 with respect to the induced coordinates (xi , xi ) in T (Vn), I being unit matrix in Mn and γ satisfies γ2 = 0.Thus T (Vn ) has a natural integrable regular dual γ−structure (see Example 8). If X is a vector field on Vn , its vertical lift V X on T (Vn ) is the vector field defined by V X(ιω) = ω(X) ◦ π =V (ω(X)), where ω is a 1-form on Vn, ιω is regarded as a function on T (Vn), which has local expression ιω = ωi xi . For a vector field X on Vn , the complete lift C X of X is defined by C X C f =C (X f ), where C f = ι(d f ) = xi ∂i f , f ∈ ℑ00 (Vn). V X and C X have, respectively local expression of the form V
X = Xi
∂ , ∂xi
C
X = Xi
∂ ∂ + xS ∂S X i i ∂x ∂xi
(3.25)
with respect to the induced coordinates (xi , xi ) in T (Vn). From (3.24) and (3.25) we have γV X = 0, γC X =V X. (3.26) Since a tensor field of type (0, q) (or (1, q) ) on T (Vn) completely determined by its action on complete lifts of vector field [114, p.33], the complete lift C g of
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the Riemennian metric g is defined by C
g(C X,C Y ) =C (g(X,Y )),
which is a Riemannian metric on T (Vn ) too. C g satisfies the equation C
g(V X,C Y ) =C g(C X,V Y ) =V (g(X,Y)).
(3.27)
From (3.26) and (3.27), we have C
g(γC X,C Y ) =C g(V X,C Y ) =V (g(X,Y )),
C
g(C X, γCY ) =C g(C X,V Y ) =V (g(X,Y )),
i.e. C g is a pue metric with respect to γ. Thus T (Vn ), γ,C g is a plural(dual)Norden manifold. The complete lift C ∇ of the Levi – Civita connection ∇ is defined by ∇C X C Y =C (∇X Y ),
(3.28)
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which is a Levi Civita connection of C g and satisfies the equation C
∇V X C Y =C ∇C X V Y =V (∇X Y ), ∀X,Y ∈ ℑ10 (Vn ).
(3.29)
Using (3.26), (3.28) and (3.29), we have for any X,Y ∈ ℑ10 (Vn) (C ∇γ)(CY,C X) =
C
∇C X (γCY ) − γ(C ∇C X C Y )
=
C
∇C X V Y − γ C (∇X Y )
=
V
(∇X Y ) −V (∇X Y ) = 0,
from which we see that C ∇γ = 0. Then φCγ g = 0 by virtue of Theorem 44. Thus we have Theorem 61. (T (Vn ), γ,C g) is a dual-K¨ahler-Norden manifold. Remark 18. By virtue of Detγ = 0 and G(X,Y ) = g(γX,Y ) = (g ◦ γ)(X,Y), it follows that a twin dual-Norden metric G is non-Riemannian, but it is symmetric, i.e. G is a degenerate metric.
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Let now G be a Riemannian metric I + II in the tangent bundle (see [114, p. 139]), i.e. G =C g +V g , where V g is a vertical lift of the Riemannian metric g. By similar devices, we have from (3.26) and (3.27) G(γC X,C Y ) = =
C
=
V
G(C X, γC Y ) = =
C
=
g(γC X,C Y ) +V g(γC X,C Y ) C V C g( X, Y ) +V g(V X,C Y ) (g(X,Y )),
g(C X, γCY ) +V g(C X, γCY ) C C V g( X, Y ) +V g(C X,V Y ) V
(g(X,Y )),
i.e. (T (Mn ), γ, G) is also a dual-Norden manifold. Since the Levi-Civita connection of the metric C g and the metric G = C g +V g coincide [14, p.149], from the equation C ∇γ = 0 it follows that φγ G = 0. From this, we have
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Theorem 62. (T (Mn ), γ, G) is a dual-K¨ahler-Norden manifold. By similar devices, we can prove that is also (T (Mn ), γ,C g) is a dual-K¨ahlerNorden manifold, where C g =S g +V a (V a− vertical lift of a symmetric tensor field a ∈ T20 (Mn )) is a synectic lift of g [16], [99]. Remark 19. Let now T 2 (Mn ) be a tangent bundle of order 2 over Mn . It is also well know there exists a affinor field bγ ∈ ℑ11 (T 2 (Mn )) which has components of the form 0 0 0 bγ = I 0 0 , bγ3 = 0 0 I 0 2 with respect to the induced coordinates (xi , xn+i , x2n+i) in T 2 (M n n ), i.e.oT (Mn ) 2 has a natural integrable regular plural Π−structure: Π = I, bγ, bγ , which
is a isomorphic repersentation of the algebra of plural numbers R(1, ε, ε2 ), ε3 = 0 [106], [108]. The 2−nd lift of g, i.e. CC g =H g (see [114, p.332]) is a plural-Norden metric with respect to bγ and CC ∇C g = 0, where CC ∇ denote the 2−nd lift of the Levi-Civita connection ∇ which is necessarily the Levi-Civita connection determined by CC g. Thus, (T 2 (Mn ), Π,CC g) is a pluralK¨ahler-Norden manifold.
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3.5. Norden–Hessian Structures Let (M2n , g) now be a Riemannian manifold with a metric tensor g. The gradient grad f of a function f ∈ ℑ00 (M2n ) is the vector field metrically equivalent to the differential d f ∈ ℑ01 (M2n). In terms of a coordinate system grad f = (gi j ∂i f )∂ j . Thus g(grad f , X) = gi j (∂i f )X k g jk = X f = (d f )(X). The Hessian of a function f ∈ ℑ00 (M2n ) is its second covariant differential h = ∇(∇ f ) = ∇2 f with respect to the Levi-Civita connection of g, i.e. since ∇Y f = Y f = (d f )(Y ), h(Y, X) = (∇(∇ f ))(Y, X) = (∇(d f ))(Y, X) = X((d f )(Y )) − (d f )(∇X Y ) = XY f − (∇X Y ) f . We easily see that h is symmetric tensor field. Also, we have
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g(∇X (grad f ),Y ) = h(Y, X). For the natural coordinates in Euclidean space, the components of h are just the 2 second partials ∂x∂i ∂xf j . If the differentiable function f : M2n → R on a Riemannian manifold (M2n , g) is convex (strictly), then the Hessian h = ∇2 f is indefinite (positive definite) [24], [32]. Hence, h = ∇2 f defines a new metric on M2n if h is nondegenerate, and is called a pseudo-Riemannian Hessian metric. p Let Γi j be the Christoffel symbols and Rm i jk be the components of the curvature tensor fields produced by the Riemannian metric g. If h pk are the contravariant components of the pseudo-Riemannian Hessian metric h, then the components of Levi-Civita connection h ∇ of h are given by the following formula e p = Γ p + 1 h pk [(∇i ∇ j ∇k f ) + (Rm + Rm )∇m f ]. Γ jki ik j ij ij 2 Let (M2n, g, ϕ) be a K¨ahler (para-K¨ahler-Norden) manifold. If there exist a func∗
∗
tion f on a K¨ahler (para-K¨ahler-Norden) manifold such that d f ◦ ϕ = d f for a function f , then we shall call f a holomorphic (para-holomorphic) function and ∗
f its associated function (see Section 3.1). If such a function f is defined locally, then we call it a locally holomorphic (para-holomorphic) function.
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Remark 20. If (M2n, ϕ) is a complex manifold, then in terms of a real coor∗
dinates (xi , xi ), i = 1, ..., n; i = n + 1, ..., 2n the equation d f ◦ ϕ = d f reduces to ∗ ∂i f = ∂i f , ∂ f = −∂ ∗f , i
i
∗
which is the Cauchy-Riemann equations for the complex function F = f + i f (see [34, p. 122]).
We notice that the condition for f to be locally holomorphic (paraholomorphic) also is given by m m (φϕ d f )i j = ϕm i ∂m ∂ j f − ∂i (ϕ j ∂m f ) + (∂ j ϕi )∂m f = 0.
If we assume that f is holomorphic (para-holomorphic), then, from (1.13), we have (φϕ (d f ))(X,Y ) = (ϕX)((d f )(Y)) − X((d f )(ϕY )) + (d f )((LY ϕ)(X))
(3.30)
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= ϕ(X)((d f )(Y)) − X((d f )(ϕY )) + (d f )([Y, ϕX] − ϕ([Y, X])) = ϕ(X)((d f )(Y)) − X((d f )(ϕY )) +(d f )(∇Y ϕX − ∇ϕX Y − ϕ(∇Y X − ∇X Y )) = (∇ϕX d f )(Y ) − (∇X d f )(ϕY ) + (d f )(∇ϕX Y ) −(d f )(∇X ϕY ) + (d f )((∇ϕ)(X,Y ) − ∇ϕX Y + ϕ(∇X Y ) = (∇ϕX d f )(Y ) − (∇X d f )(ϕY ) − (∇ϕ)(Y, X) = 0. We now consider a holomorphic (para-holomorphic) function f on a K¨ahler (para-K¨ahler-Norden) manifold (M2n , g, ϕ) . On a K¨ahler (para-K¨ahler-Norden) manifold (M2n , g, ϕ) (∇ϕ = 0), the equation (3.30) equivalent to the equation (∇2 f )(Y, ϕX) = (∇2 f )(ϕY, X), i.e. h = ∇2 f is pure and a manifold (M2n , ϕ, h = ∇2 f ) is a Norden manifold. Thus, h naturally defines a Norden metric on the K¨ahler (para-K¨ahler-Norden) manifold (M2n , g, ϕ) . We call it Norden-Hessian metric. Thus we have Theorem 63. Let (M2n , g, ϕ) be a K¨ahler (para-K¨ahler-Norden) manifold. Then, M2n admits a Norden-Hessian structure (ϕ, h = ∇2 f ), if f ∈ ℑ00 (M2n) is holomorphic (para-holomorphic).
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Let (M2n , g, J) be a locally decomposable Riemannian manifold with integrable paracomplex structure E 0 J= , 0 −E being (n × n)-unit matrix . In such manifolds, g is pure with respect to J, moreover ∇J = 0, i.e. a triple (M2n , g, J) is a para-K¨ahler-Norden manifold. Also g is para-holomorphic and the curvature tensor field R of g is pure with respect to the structure J. Let (M2n , h = ∇2 f , J) be a Hessian-Norden structure, which exists on a paracomplex decomposable Riemannian manifold. Then (∇2 f )(JX,Y ) = (∇2 f )(X, JY ), from which we have (∇3 f )(JX,Y, Z) = (∇3 f )(X, JY, Z).
(3.31)
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Using the Ricci equation, from (3.31) we obtain (∇3 f )(X, JY, Z) = ∇Z (∇JY (∇X f )
(3.32)
= ∇JY (∇Z (∇X f )) − (d f )(R(Z, JY )X) = (∇3 f )(X, Z, JY ) − (d f )(R(Z, JY )X) and (∇3 f )(JX,Y, Z) = ∇Z (∇Y (∇JX f )
(3.33)
= ∇Y (∇Z (∇JX f )) − (d f )(R(Z,Y )JX) = (∇3 f )(JX, Z,Y ) − (d f )(R(Z,Y )JX). Since h is symmetric and the curvature tensor R of g is pure with respect to J, from (3.32) and (3.33) we have (∇3 f )(Z, JX,Y ) = (∇3 f )(Z, X, JY ), i.e. a tensor field ∇3 f is pure in all arguments. On the other hand
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(φJ h)(X, Z1 , Z2 )
(3.35)
= J(X)(h(Z1 , Z2 )) − X(h(JZ1 , Z2 )) −h(∇JX Z1 , Z2 ) + h((∇J)(X, Z1 ), Z2 ) + h(Z1 , (∇J)(X, Z2 )) −h(Z1 , ∇JX Z2 ) + h(J(∇X Z1 ), Z2 ) + h(JZ1 , ∇X Z2 ) = (∇JX h)(Z1 , Z2 ) − (∇X h)(JZ1 , Z2 ) + h((∇J)(X, Z1 ), Z2 ) +h(Z1 , (∇J)(X, Z2 )). Substituting h(Z1 , Z2 ) = ∇Z1 ∇Z2 f and ∇J = 0 in (3.35), by virtue of (3.34) we have (φJ h)(X, Z1, Z2 ) = (φJ h)(X, Z2 , Z1 ) = (∇JX (∇2 f ))(Z2 , Z1 ) − (∇X (∇2 f ))(Z2 , JZ1 ) = (∇3 f )(Z2 , Z1 , JX) − (∇3 f )(Z2 , JZ1 , X) = 0,
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i.e. h is para-holomorphic. Then, using Theorem 44, we see that h ∇J = 0. Thus, we have Theorem 64. Let (M2n , g, J) be a paracomplex decomposable Riemannian manifold. If f is paraholomorphic, then a triple (M2n, h = ∇2 f , J) is a paraK¨ahler-Norden-Hessian manifold. Remark 21. For K¨ahler manifold (M2n, g, J), the curvature tensor R of the Hermitian metric g is not pure in all arguments. Therefore, K¨ahler manifolds may not always locally admit any K¨ahler-Norden-Hessian metric.
3.6. Norden-Walker Manifolds with Proper Structures In the present section, we shall focus our attention to Norden manifolds of dimension four. The main purpose of the present section is to study complex Norden metrics on 4-dimensional Walker manifolds. A neutral metric g on a 4-manifold M4 is said to be a Walker metric if there exists a 2-dimensional null distribution D on M4 , which is parallel with respect
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to g. From Walker’s theorem [110], there is a system of coordinates (x, y, z,t) with respect to g which takes the following local canonical form 0 0 1 0 0 0 0 1 g = (gi j ) = (3.36) 1 0 a c , 0 1 c b
where a, b, c are smooth functions of the coordinates (x, y, z,t). The paralel null ∂ 2-plane D is spanned locally by {∂x , ∂y }, where ∂x = ∂x , ∂y = ∂∂y .
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3.6.1.
Almost Norden-Walker Metrics
In [47], a proper almost complex structure with respect to g is defined as a gorthogonal almost complex structure J so that J is a standard generator of a positive π2 rotation on D, i.e., J∂x = ∂y and J∂y = −∂x . Then for the Walker metric g, such a proper almost complex structure J is determined uniquely as 1 0 −1 −c 2 (a − b) 1 0 1 (a − b) c 2 . (3.37) 0 0 0 −1 0 0 1 0
In [5], for such a proper almost complex structure J on Walker 4-manifold M, an almost Norden structure (gN+, J) is constructed, where gN+ is a metric on M, with properties gN+ (JX, JY ) = −gN+ (X,Y ). In fact, as one of these examples, such a metric takes the form (see Proposition 6 in [5], unfortunately, the calculations of the component gN+ 44 in [5] are erroneous):
0 −2 −2 0 gN+ = 0 −a −b −2c
0 −a 0 1 2 (1 − ab)
−b −2c . 1 2 (1 − ab) −2bc
We may call this an almost Norden-Walker metric. The construction of such a structure in [5] is to find a Norden metric for a given almost complex structure, which is different form the Walker metric. The purpose of the present section is to find also an almost Norden-Walker structure (g, F), where the metric is nothing but the Walker metric g, with an
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appropriate almost complex structure F, to be determined. That is, for a fixed metric g, we will find an almost complex structure F which satisfy g(FX, FY ) = −g(X,Y ). In [5], for a given almost complex structure, a metric is constructed. Our method is, however, for a given metric, an almost complex structure is constructed. Let F be an almost complex structure on a Walker manifold M4 , which satisfies (i) F 2 = −I, (ii) g(FX,Y ) = g(X, FY ) (Nordenian property), (iii) F∂x = ∂y , F∂y = −∂x ( F induces a positive π2 −rotation on D). We easily see that these three properties define F non-uniquely, i.e., F∂x = ∂y , F∂y = −∂x , 1 F∂z = α∂x + 2 (b + a)∂y − ∂t , 1 F∂t = − 2 (b + a)∂x + α∂y + ∂z and has the local components
0 −1 1 0 F = (Fji ) = 0 0 0 0
α − 12 (a + b) 1 α 2 (a + b) 0 1 −1 0
with respect to the natural frame {∂x , ∂y , ∂y , ∂t }, where α = α(x, y, z,t) is an arbitrary function. We must note that the proper almost complex structure J as in (3.37) is determined uniquely. In our case of the almost Norden-Walker structure, the almost complex structure F just obtained contains an arbitrary function α(x, y, z,t). Our purpose is to find a nontrivial almost Norden-Walker structure with the Walker metric g explicitly (see Theorem 66 and Example 12). Therefore, we now put α = c. Then g defines a unique almost complex structure 0 −1 c − 21 (a + b) 1 0 1 (a + b) c 2 ϕ = (ϕij ) = (3.38) 0 0 0 1 0 0 −1 0
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The triple (M4 , ϕ, g) is called almost Norden-Walker manifold. In conformity with the terminology of [12], [13], [43], [44], [45], [46], [47], [48] we call ϕ the proper almost complex structure. Remark 22. From (3.38) we immediately see that in the case a = −b and c = 0, ϕ is integrable.
3.6.2.
Integrability of Proper Almost Complex Structures
We consider the general case for integrability. The almost complex structure ϕ on almost Norden-Walker manifolds is integrable if and only if i i m i m (Nϕ )ijk = ϕmj ∂m ϕik − ϕm k ∂m ϕ j − ϕm ∂ j ϕk + ϕm ∂k ϕ j = 0.
(3.39)
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From (3.38) and (3.39) find the following integrability condition: Theorem 65. The proper almost complex structure on almost Norden-Walker manifolds is integrable if and only if the following PDEs hold: ax + bx + 2cy = 0, (3.40) ay + by − 2cx = 0. From this theorem, we easily see that if a = −b and c = 0, then ϕ is integrable (see Remark 22). Let (M4 , ϕ, g) be a Norden-Walker manifolds (Nϕ = 0) and a = b. Then the equation (3.40) reduces to ax = −cy , (3.41) ay = cx , from which follows axx + ayy = 0,
(3.42)
cxx + cyy = 0, e.g. the functions a and c are harmonic with respect to the arguments x and y. Thus we have Theorem 66. If the triple (M4 , ϕ, g) is Norden-Walker and a = b, then a and c are all harmonic with respect to the arguments x, y .
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Example 12. We now apply the Theorem 66 to establish the existence of special types of Norden-Walker metrics. In our arguments, the harmonic function plays an important part. Let a = b and h(x, y) be a harmonic function of variables x and y, for example h(x, y) = ex cosy. We put a = a(x, y, z,t) = h(x, y) + α(z,t) = ex cos y + α(z,t) where α is an arbitrary smooth function of z and t. Then, a is also hormonic with respect to x and y. We have ax = ex cos y, ay = −ex siny. From (3.41), we have PDEs for c to satisfy as cx = ay = −ex sinx, cy = −ax = −ex cosy. For these PDEs, we have solutions Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
c = −ex siny + β(z,t), where β is arbitrary smooth function of z and t. Thus the Norden-Walker metric has components of the form 0 0 1 0 0 0 0 1 g = (gi j ) = 1 0 ex cos y + α(z,t) −ex siny + β(z,t) . 0 1 −ex siny + β(z,t) ex cos y + α(z,t)
3.6.3.
Holomorphic Norden-Walker (K¨ahler-Norden-Walker) Metrics
Let (M4 , ϕ, g) be an almost Norden-Walker manifold. If m m m m (Φϕ g)ki j = ϕm k ∂m gi j − ϕi ∂k gm j − gm j (∂i ϕk − ∂k ϕi ) + gim ∂ j ϕk = 0,
(3.43)
then by virtue of Theorem 44 ϕ is integrable and the triple (M4 , ϕ, g) is called a holomorphic Norden-Walker or a K¨ahler-Norden-Walker manifold. Taking
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account of Corollary 45, we see that almost K¨ahler-Norden-Walker manifold with condition Φϕ g = 0 and Nϕ 6= 0 does not exist. Substituting (3.36) and (3.38) in (3.43), we see that the non-vanishing components of Φϕ g are 1 (Φϕg)xzz = ay, (Φϕ g)xzt = (Φϕg)xtz = (bx − ax ) + cy , (3.44) 2 (Φϕ g)xtt = by − 2cx , (Φϕ g)yzz = −ax , 1 (Φϕ g)yzt = (Φϕg)ytz = (by − ay ) − cx , (Φϕ g)ytt = −bx − 2cy , 2 (Φϕg)zxz = (Φϕg)zzx = (Φϕg)txt = (Φϕ g)ttx = cx , 1 (Φϕ g)zxt = (Φϕg)ztx = −(Φϕg)txz = −(Φϕg)tzx = (ax + by ), 2 (Φϕg)zyz = (Φϕg)zzy = (Φϕg)tyt = (Φϕ g)tty = cy , 1 (Φϕ g)zyt = (Φϕg)zty = −(Φϕ g)tyz = −(Φϕ g)tzy = (ax + by ), 2 1 (Φϕg)zzz = cax − at + 2cz + (a + b)ay , 2 1 (Φϕg)zzt = (Φϕg)ztz = ccx + bz + (a + b)cy , 2 1 (Φϕ g)ztt = cbx + at − 2cz + (a + b)by , 2 1 (Φϕg)tzz = cay − bz − (a + b)ax, 2 1 (Φϕ g)tzt = (Φϕg)ttz = ccy − at + 2cz − (a + b)cx , 2 1 (Φϕg)ttt = cby + bz − (a + b)bx. 2 From these equations we have Theorem 67. The triple (M4 , ϕ, g) is K¨ahler-Norden-Walker if and only if the following PDEs hold: ax = ay = cx = cy = bx = by = bz = 0, at − 2cz = 0. Corollary 68. The triple (M4 , ϕ, g) with metric
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0 0 g = (gi j ) = 1 0
is always K¨ahler-Norden-Walker.
3.6.4.
0 1 0 0 0 1 0 a(z) 0 1 0 b(t)
Curvature Properties of Norden-Walker Manifolds
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If R and r are respectively the curvature and the scalar curvature of the Walker metric, then the non-vanishing components of R and r have, respectively, expressions (see [47], Appendix A and C) 1 1 1 Rxzxz = − axx, Rxzxt = − cxx , Rxzyz = − axy, (3.46) 2 2 2 1 1 1 1 1 Rxzyt = − cxy , Rxzzt = axt − cxz − ay bx + cx cy , 2 2 2 4 4 1 1 1 Rxtxt = − bxx, Rxtyz = − cxy , Rxtyt = − bxy , 2 2 2 1 1 1 1 1 1 2 Rxtzt = cxt − bxz − (cx ) + ax bx − bxcy + by cx , 2 2 4 4 4 4 1 1 1 1 1 1 Ryzyz = − ayy, Ryzyt = − cyy , Rytzt = cyt − byz − cx cy + ay bx , 2 2 2 2 4 4 1 1 1 1 1 1 1 2 Ryzzt = ayt − cyz − ax cy + ay cx − ayby + (cy ) , Rytyt = − byy, 2 2 4 4 4 4 2 1 1 1 1 1 1 Rztzt = czt − att − bzz − a(cx )2 + aaxbx + caxby − ccx cy 2 2 4 4 4 2 1 1 1 1 1 1 − at cx + ax ct − ax bz + cay bx + bay by − b(cy)2 2 2 4 4 4 4 1 1 1 1 − bz cy + ay bt + by cz − at by 2 4 2 4 and r = axx + 2cxy + byy.
(3.47)
Suppose that the triple (M4 , ϕ, g) is K¨ahler-Norden-Walker. Then from the last equation in (3.45) and (3.46), we see that 1 1 Rztzt = czt − att = − (at − 2cz )t = 0. 2 2
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From (3.45) we easily we see that the another components of in (3.46) directly all vanish. Thus we have Theorem 69. If a Norden-Walker manifold (M4 , ϕ, g) is K¨ahler-Norden-Walker, then M4 is flat. Remark 23. In general, a K¨ahler-Norden manifold is non-flat and in such manifold curvature tensor is pure and holomorphic. In particular, from Theorem 69 we see that a K¨ahler-Norden-Walker manifold is flat, and therefore g is Einstein.
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Let (M4 , ϕ, g) be a Norden-Walker manifold with the integrable proper structure ϕ, i.e. Nϕ = 0. If a = b, then from proof of the Theorem 66 we see that the equation (3.41) hold. If c = c(y, z,t) and c = c(x, z,t) , then cxy = (cx )y = (cy )x = 0. In these cases, by virtue of (3.41) we find a = a(x, z,t) and a = a(y, z,t), respectively. Using of cxy = 0 and axx + byy = 0 (see (3.42)), we from (3.47) obtain r = 0. Thus we have Theorem 70. If (M4 , ϕ, g) is a Norden-Walker non-K¨ahler manifold with metrics 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 g= 1 0 a(x, z,t) c(y, z,t) , ge = 1 0 a(y, z,t) c(x, z,t) 0 1 c(y, z,t) a(x, z,t) 0 1 c(x, z,t) a(y, z,t) then M4 is scalar flat.
3.6.5.
Isotropic K¨ahler-Norden-Walker Structures
It is well known that the inner product in the vector space can be extended to an inner product in the tensor space. In fact, if t and t are tensors of type (r, s) with components t i1 ...ir and t k1 ...kr , then 1 j1 ... js
1
2
2l1 ...ls
g(t , t ) = gi1 k1 ...girkr g j1 l1 ...g jsls t i1 ...ir t k1 ...kr . 1 2
1 j1 ... js 2l1 ...ls
If t = t = ∇ϕ ∈ ℑ12 (M2n ), then the square norm k∇ϕk2 of ∇ϕ is defined by 1
2
s k∇ϕk2 = gi j gkl gms (∇ϕ)m ik (∇ϕ) jl .
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A proper almost complex structure ϕ on Norden-Walker manifold (M4 , ϕ, g) is said to be isotropic K¨ahler if k∇ϕk2 = 0, but ∇ϕ 6= 0. Examples of isotropic K¨ahler structures were given in [4], [20]. Our purpose in this section is to show that a proper almost complex structure on almost Norden-Walker manifold (M4 , ϕ, g) is isotropic K¨ahler as we will see Theorem 71. The inverse of the metric tensor (3.36), g−1 = (gi j ), given by −a −c 1 0 −c −b 0 1 . g−1 = (3.48) 1 0 0 0 0 1 0 0
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For the covariant derivative ∇ϕ of the almost complex structure put (∇ϕ)kij = ∇i ϕkj . Then, after some calculations we see that the non-vanishing components of ∇ϕ are ∇x ϕxz
=
∇z ϕxx
=
∇z ϕyx
=
∇x ϕxz
=
∇z ϕxz
=
∇z ϕtx
=
∇z ϕt
y
=
∇t ϕxx
=
∇t ϕyx
=
∇t ϕxz
=
∇t ϕyz
=
∇t ϕtx
=
∇t ϕty
=
y
y
∇x ϕt = cx , ∇y ϕxz = ∇y ϕt = cy , 1 1 −∇z ϕyy = ∇z ϕzz = −∇z ϕtt = ay + cx , 2 2 1 1 x t z ∇z ϕy = ∇z ϕz = ∇z ϕt = − ax + cy , 2 2 1 1 1 2cz + cax − at − ccy − acx + bcy , 2 2 2 1 1 3 1 az + acy − bcy + cay + aax + bax , 4 4 4 4 1 1 3 1 aax − bax + cay + bcy + ccx + acy , 4 4 4 4 1 1 1 1 2cz + ccy − at + bay + cax − acx , 2 2 2 2 1 1 y z t −∇t ϕy = ∇t ϕz = −∇t ϕt = cy + bx , 2 2 1 1 ∇t ϕxy = ∇t ϕtz = ∇t ϕtz = − cx + by , 2 2 3 1 1 1 ccx + bz − cby − abx + bcy , 2 2 2 2 1 1 1 1 aby − bby − acx + bcx , 4 4 4 4 1 1 1 1 acx − bcx + ccx + bby + cbx − aby , 4 4 4 4 1 1 1 1 cby + bz + bcy + ccx − abx . 2 2 2 2
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Using (3.36), (3.48) and (3.49) we find s k∇ϕk2 = gi j gkl gms (∇ϕ)m ik (∇ϕ) jl = 0.
Thus we have Theorem 71. A proper almost complex structure on almost Norden-Walker manifold (M4 , ϕ, g) is isotropic K¨ahler.
3.6.6.
Quasi-K¨ahler-Norden-Walker Structures
The basis class of non-integrable almost complex manifolds with Norden metric is the class of the quasi-K¨ahler manifolds. An almost Norden manifold (M2n , ϕ, g) is called a quasi- K¨ahler [42], if σ g((∇X ϕ)Y, Z) = 0,
X,Y,Z
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where σ is the cyclic sum by three arguments. If we add (φϕ g)(X,Y, Z) and (φϕ g)(Z,Y, X) (see (3.6) or (3.7)), then by virtue of g(Z, (∇Y ϕ)X) = g((∇Y ϕ)Z, X), we find (φϕ g)(X,Y, Z) + (φϕ g)(Z,Y, X) = 2g((∇Y ϕ)Z, X). Since, from last equation we have (φϕ g)(X,Y, Z) + (φϕ g)(Y, Z, X) + (φϕ g)(Z, X,Y ) = σ g((∇X ϕ)Y, Z). X,Y,Z
Thus we have Theorem 72. Let (M2n , ϕ, g) be an almost Norden manifold. Then the Norden metric g is a quasi-K¨ahler-Norden if and only if (Φϕg)(X,Y, Z) + (Φϕg)(Y, Z, X) + (Φϕg)(Z, X,Y ) = 0 for any X,Y, Z ∈ ℑ10 (M2n ). The equation in Theorem 72 can also be written in the form (Φϕ g)(X,Y, Z) + 2g((∇X ϕ)Y, Z) = 0.
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From here we see that, if we take a local coordinate system, then a NordenWalker manifold (M4 , ϕ, g) satisfying the condition Φk gi j + 2∇k Gi j to be zero is called a quasi-K¨ahler manifold, where G is defined by Gi j = ϕm i gm j . For the covariant derivative ∇G of the twin metric G put (∇G)i jk = ∇i G jk . After some calculations we see that the non-vanishing components of ∇G are ∇x Gzz = ∇x Gtt = cx , ∇y Gzz = ∇y Gtt = cy , 1 1 ∇z Gxz = ∇z Gzx = −∇z Gyt = −∇z Gty = ay + cx , 2 2 1 1 ∇z Gxt = ∇z Gtx = ∇z Gyz = ∇z Gzy = cy − ax , 2 2 1 ∇z Gzz = 2cz − at + ay(a + b) + cax , 2 1 1 1 1 ∇z Gzt = ∇z Gtz = cay + ccx − ax (a + b) + cy (a + b), 2 2 4 4 1 ∇z Gtt = 2cz − at + ccy − cx (a + b), 2 1 1 ∇t Gxz = ∇t Gzx = −∇t Gyt = −∇t Gty = bx + cy , 2 2 1 1 ∇t Gxt = ∇t Gtx = ∇t Gyz = ∇t Gzy = by − cx , 2 2 1 1 1 1 ∇t Gzt = ∇t Gtz = ccy + cbx − cx (a + b) + by (a + b), 2 2 4 4 1 ∇t Gzz = bz + ccx + cy (a + b), 2 1 ∇t Gtt = bz + cby − bx (a + b). 2
(3.50)
From (3.44) and (3.50) we have Theorem 73. A triple (M4 , ϕ, g) is a quasi-K¨ahler-Norden-Walker manifold if and only if the following PDEs hold: bx = by = bz = 0, ay − 2cx = 0, ax − 2cy = 0, cax − at + 2cz − (a + b)cx = 0.
3.6.7.
On the Goldberg Conjecture
Let now (M4 , ϕ, g) be an almost Hermitian manifold. The Goldberg conjecture (see [48]) states that an almost Hermitian manifold (M4 , ϕ, g) must be K¨ahler
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(or ϕ must be integrable) if the following three conditions are imposed: (G1 ) if M is compact and (G2 ) g is Einstein, and (G3 ) if the fundamental 2-form is closed. It should be noted that no progress has been made on the Goldberg conjecture, and the orginal conjecture is still an open problem. Despite many papers by various authors concerning the Goldberg conjecture, there are only Sekigawa papers (see for example [90]) which obtained substantial results to the orginal Goldberg conjecture: Let be (M4 , ϕ, g) be an almost Hermitian manifold, which satisfies the three conditions (G1 ),(G2 ) and (G3 ). If the scalar curvature of M is nonnegative, then ϕ must be integrable. Let (M4 , ϕ,w g) be an indefinite almost K¨ahler-Walker-Einstein compact manifold with the proper almost complex structure (3.37). As noted before, many examples of Norden-Walker metrics can be obtained by gN+ (JX, JY ) = −gN+ (X,Y ) (see Section 3.7.1), and as one of these examples, such a metric has components 0 −2 0 −b −2 0 −a −2c . gN+ = 1 0 −a 0 (1 − ab) 2 1 −b −2c 2 (1 − ab) −2bc with respect to the Walker coordinates. Using Corollary 45, we have
Theorem 74. The proper almost complex structure ϕ on indefinite almost K¨ahler-Walker-Einstein compact manifold (M4 , ϕ,w g) is integrable if φϕ gN+ = 0, where gN+ is the induced Norden-Walker metric on M4 . This resolves a conjecture of Goldberg under the additional restriction on Norden-Walker metric( gN+ ∈ Kerφϕ ). Example 13. (Counterexample of noncompact and neutral type to the Goldberg conjecture). Let (M4 , ϕ, g) be an almost Norden-Walker manifold. Consider the metric 0 0 1 0 0 0 0 1 . g = (gi j ) = 1 0 a(x, y, z,t) 0 0 1 0 a(x, y, z,t)
That is the metric is defined by putting a = b, c = 0 in the generic canonical form (3.36).
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Let Ri j and S denote the Ricci curvature and the scalar curvature of the metric g in (3.36). The Einstein tensor is defined by Gi j = Ri j − 41 Sgi j and has non zero components as follows (see [47], Appendix D): 1 1 1 1 axx − byy, Gxt = cxx + bxy, (3.51) 4 4 2 2 1 1 1 1 Gyz = axy + cyy, Gyt = byy − axx , 2 2 4 4 1 1 1 1 Gzz = aaxx + caxy + bayy − ayt + cyz − aycx + ax cy 4 2 2 2 1 1 1 1 + ayby − (cy)2 − acxy − abyy, 2 2 2 4 1 1 1 1 1 1 1 Gzz = acxx + ccxy + axt − cxz − ay bx + cx cy + bcyy 2 2 2 2 2 2 2 1 1 1 1 − cyt + byz − caxx − cbyy, 2 2 4 4 1 1 1 1 Gtt = abxx + cbxy + cxt − bxz − (cx )2 + axbx − bxcy 2 2 2 2 1 1 1 1 + bycx + bbyy − baxx − bcxy. 2 4 4 2 The metric g in (3.36) is almost Norden-Walker-Einstein if all the above components Gi j vanish ( Gi j = 0 ). In this case, we see from (3.51) that the Einstein condition consist of the following PDE’s :
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Gxz =
aaxx − 2ayt + (ay )2 = 0, axx − ayy = 0, axy = 0, axt − ax ay + azy = 0, aaxx − 2axz + (ax )2 = 0. If a is independent of y and t, and if a contains x only linearly, the first four PDE’s hold trivially, and the last one reduces to: 2axz − (ax)2 = 0. We see that a = − 2xz is a solution to the PDE, and therefore the metric 0 0 1 0 0 0 0 1 g = (gi j ) = (3.52) 1 0 − 2x 0 z 0 1 0 − 20x z is Einstein on the coordinate patch z > 0 (or z < 0). Thus, the second condition (G2 ) of Goldberg conjecture holds. We know that this metric admits a proper almost complex structure as follows:
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ϕ∂x = ∂y , ϕ∂y = −∂x , ϕ∂z = a∂y − ∂t , ϕ∂t = −a∂x + ∂z .
105
(3.53)
For the Einstein metric (3.52), the proper almost complex structure ϕ in (3.53) becomes 2x 2x ϕ∂x = ∂y , ϕ∂y = −∂x , ϕ∂z = − ∂y − ∂t , ϕ∂t = ∂x + ∂z . z z Then, the integrability of ϕ, given in Theorem 65, becomes 4 ax + bx + 2cy = 2ax = − = 6 0, ay + by − 2cx = 2ay = 0. z Thus, ϕ cannot be integrable.
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3.7. Opposite Almost Complex Structure It is known that an oriented 4-manifold with a field of 2-planes, or equivalently endowed with a neutral indefinite metric, admits a pair of almost comlex structure ϕ and an opposite almost complex structure ϕ´, which satisfy the following properties ( [47]): i) ϕ2 = ϕ´2 = −1, ii) g(ϕX, ϕY ) = g(ϕ´X, ϕ´Y ) = g(X,Y ), iii) ϕϕ´= ϕ´ϕ, iv) the preferred orientation of ϕ coincides with that of M4 , v) the preferred orientation of ϕ´is opposite to that of M4 . We devote this section to the analysis of opposite almost complex structures on Norden-Walker 4-manifolds. Let (M4 , ϕ, g) be an almost Norden-Walker manifolds. For a Walker manifold M4 , with the proper almost complex structure ϕ, the g-orthogonal opposite almost complex structure ϕ´takes the form θ2 θ1 a)∂1 − b∂2 + θ2 ∂3 + θ1 ∂4 , 2 2 θ1 θ2 = (− a + θ2 c)∂1 + b∂2 + θ1 ∂3 − θ2 ∂4 , 2 2 θ1 θ1 θ2 2 θ2 θ1 = −( ac + a + 2 )∂1 − ( ab + 2 )∂2 2 2 4 4 θ1 + θ2 θ1 + θ22 θ2 θ1 + a∂3 + a∂4 , 2 2
ϕ´∂1 = −(θ1 c + ϕ´∂2 ϕ´∂3
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Arif Salimov θ1 θ2 θ1 ab + (ac − bc) + 2 )∂1 4 2 θ1 + θ22 θ1 θ2 θ2 θ1 θ2 +(− bc + b2 + 2 )∂2 + ( b + θ2 c)∂3 + (θ1 c − b)∂4 , 2 2 4 2 2 θ1 + θ2
ϕ´∂4 = −(θ1 c2 +
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where θ1 and θ2 are two parameters. We shall focus our attention to one of explicit forms of ϕ´, obtained by fixing two parameters as θ1 = 1 and θ2 = 0 (only for simplicity), as follows: 1 1 ϕ´∂1 = −c∂1 − b∂2 + ∂4 , ϕ´∂2 = − a∂1 + ∂3 , 2 2 1 1 1 ϕ´∂3 = − ac∂1 − ( ab + 1)∂2 + a∂4 , 2 4 2 1 1 1 2 ϕ´∂4 = −(c + ab + 1)∂1 − bc∂2 + b∂3 + c∂4 4 2 2 and ϕ´has the local components −c − 21 a − 12 ac −(c2 + 14 ab + 1) 1 1 − b 0 −( 4 ab + 1) − 12 bc 2 ϕ´= (ϕ´ij ) = 1 0 1 0 2b 1 1 0 a c 2
.
By similar devices (as in the previous section for proper almost complex structure ϕ ), we can prove the following theorems for opposite almost complex structure ϕ´(see [81]): Theorem 75. The opposite almost complex structure of an almost NordenWalker manifold is integrable if and only if the following PDEs hold: abx − 2bz = 0, by = 0, ax − 2cy = 0, bay − 2at − 2acx + 4ccy + 4cz = 0.
Theorem 76. A triple (M4 , ϕ´, g) is a K¨ahler-Norden-Walker manifold if and only if the following PDEs hold: ax = ay = bx = by = bz = cx = cy = 0, at − 2cz = 0. Theorem 77. The opposite almost complex structure of an almost NordenWalker manifold (M4 , ϕ´, g) is isotropic K¨ahler if and only if the following PDEs hold: cx (2bay − 2acx + 4cz − 2at + 2cax ) + cy (2bz − 2abx) = 0
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Theorem 78. A triple (M4 , ϕ´, g) is a quasi-K¨ahler-Norden-Walker manifold if and only if the following PDEs hold: ax = ay = bx = by = bz = cx = cy = 0, at − 2cz = 0.
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3.8. Para-Norden-Walker Metrics In the present section, we shall focus our attention to para-Norden manifolds of dimension four. Using a Walker metric we construct new para-Norden-Walker metrics together with a proper almost paracomplex structures. Let F be an almost paracomplex structure on a Walker manifold M4 , which satisfies i) F 2 = I, ii) g(FX,Y ) = g(X, FY ) (Nordenian property), iii) F∂x = ∂y , F∂y = ∂x . We easily see that these three properties define F non-uniquely, i.e., F∂x = ∂y , F∂y = ∂x , 1 F∂ = −α∂ z x − 2 (b − a)∂y + ∂t , F∂t = 21 (b − a)∂x + α∂y + ∂z and has the local components
0 1 F = (Fji ) = 0 0
1 −α 0 − 12 (b − a) 0 0 0 1
1 2 (b − a)
α 1 0
with respect to the natural frame {∂x , ∂y , ∂z , ∂t }, where α = α(x, y, z,t) is an arbitrary function. Our propose is to find a nontrivial para-Norden structure with Walker metric g explicitly. If α = c, then we have explicit examples of Walker metrics with hyperbolic functions a, b and c (Theorem 80). Therefore, we put α = c. Then g defines a unique almost paracomplex structure 1 0 1 −c 2 (b − a) 1 0 − 1 (b − a) c 2 . ϕ = (ϕij ) = (3.54) 0 0 0 1 0 0 1 0
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The triple (M4 , ϕ, g) is called almost para-Norden-Walker manifold. Using the Nijenhuis tensor for paracomplex structure (3.54), we have Theorem 79. The proper almost paracomplex structure ϕ of an almost paraNorden-Walker manifolds is integrable if and only if the following PDEs hold: ax − bx + 2cy = 0, (3.55) ay − by + 2cx = 0. From this theorem, we easily see that if a = b and c = 0, then ϕ is integrable. Let (M4 , ϕ, g) be a para-Norden-Walker manifolds (Nϕ = 0) and a = −b. Then the equation (3.55) reduces to ax = −cy , ay = −cx , from which follows axx − ayy = 0,
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cxx − cyy = 0, e.g., the functions a and c are hyperbolic with respect to the arguments x and y. Thus we have Theorem 80. If the triple (M4 , ϕ, g) is para-Norden-Walker and a = −b, then a, b and c are all hyperbolic with respect to the arguments x,y . Let now (M4 , ϕ, g) be an almost para-Norden-Walker manifold. Next analysis of para-Norden-Walker manifolds is very similar to the analysis of NordenWalker Manifolds (see, Section 3.6), i.e the situtaion is as folows [84]: Theorem 81. A triple(M4, ϕ, g) is a para-K¨ahler-Norden-Walker manifold if and only if the following PDEs hold: ax = ay = bx = by = bz = cx = cy = 0, at − 2cz = 0. Theorem 82. A triple (M4 , ϕ, g) is a quasi-para-K¨ahler-Norden-Walker manifold if and only if the following PDEs hold: bx = by = bz = 0, ay − 2cx = 0, ax − 2cy = 0, cax − at + 2cz − (a − b)cx = 0.
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Theorem 83. If (M4 , ϕ, g) is an almost para-Norden-Walker manifold, then it is isotropic para-K¨ahlerian. Theorem 84. If a para-Norden-Walker manifold (M4 , ϕ, g) is para-K¨ahlerNorden-Walker, then M4 is flat. Theorem 85. Let (M4 , ϕ, g) be a para-Norden-Walker manifold. If ax = ay = bx = by = 0, then g is para-Norden-Walker-Einstein.
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We note that the almost para-Norden-Walker structure is a specialized almost product metric structure on a pseudo-Riemannian manifold (There is an extensive literature on almost product metric structures, see, for examples [11], [17], [18], [22], [23], [53], [56], [96]). In this context, we consider some properties of almost para-Norden-Walker manifolds. An integrable almost product manifold is usually called a locally product manifold. Thus a para-Norden-Walker manifold (M4 , ϕ, g) is a locally product pseudo-Riemannian manifold such that dimT + M4 = dim T − M4 = 2. From Theorem 79 we have Corollary 86. An almost para-Norden-Walker manifold (M4 , ϕ, g) is a locally product pseudo-Riemannian manifold if and only if ax − bx + 2cy = 0 and ay − by + 2cx = 0. Let now (M4 , ϕ, g) be a paraholomorphic Norden-Walker manifold (∇ϕ = 0). It is well known that a locally product (pseudo-)Riemannian manifold (M4 , ϕ, g) is a decomposable if and only if ∇ϕ = 0, where ∇ is a Levi-Civita connection of g. Then, from Theorem 44 and Theorem 81, we have Corollary 87. A para-Norden-Walker manifold (M4 , ϕ, g) is a locally decomposable if and only if ax = ay = bx = by = bz = cx = cy = 0 and at − 2cz = 0. Gil-Medrano and Naveira established in [24] that both distributions of the almost product structures on the (pseudo-)Riemannian manifold (M, ϕ, g) are totally geodesic if and only if g((∇X P)Y, Z) + g((∇Y P)Z, X) + g((∇Z P)X,Y) = σ g((∇X P)Y, Z) = 0 X,Y,Z
(3.56) ℑ10 (M).
for any X,Y, Z ∈ From Theorem 82 and (3.56), we have
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Corollary 88. Let (M4 , ϕ, g) be an almost para-Norden-Walker manifold. Both distributions of the manifold (M4 , ϕ, g) are a totally geodesic if and only if bx = by = bz = 0, ay − 2cx = 0, ax − 2cy = 0, cax − at + 2cz − (a − b)cx = 0. The minimal almost product structure on the (pseudo-)Riemannian manifold (M, P, g) is characterized by the equality of trg J = (g jk J ijk ) to zero on M [23], [56], [96], where is the Jordan tensor field, which in coordinate form can be written as j
J ijk = Psj (∇s Pk + ∇ k Psi ) + Pks (∇s Pij + ∇ j Psi ). We can rewrite the condition of minimality as [96] j
∇ j Pk = 0.
(3.57)
Let now (M4 , ϕ, g) be an almost para-Norden-Walker manifold. Then using (3.57), we see that ∇1 ϕ1k + ∇2 ϕ2k + ∇3 ϕ3k + ∇4 ϕ4k = 0, k = 1, 2, 3, 4 if and only if ax − bx − 2cy = 0, (3.58) ay − by − 2cx = 0. Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
Thus, we have Corollary 89. An almost para-Norden-Walker manifold (M4 , ϕ, g) is a minimal, if and only if g satisfies the condition (3.58). Let ∗ T (Mn ) denote the cotangent bundle of a manifold Mn and let π :∗ T (Mn ) → Mn be the natural projection. A point ξ of the cotangent bundle is represented by an ordered pair (p, ω), where p = π(ξ) is a point on Mn and ω is a 1-form on cotangent spaces ∗ Tp (Mn ). For each coordinate neighborhood i (U, xi ) on Mn with p∈ U, of ω in denote by x , i = n + 1, ..., 2n the components i the natural coframe dx . Then, for any local coordinates (U, xi ) on Mn , (xi , xi ) are natural induced coordinates in π−1 (U) ⊂∗ T (Mn ). For a given symmetric connection ∇ on Mn , the cotangent bundle ∗ T (Mn ) may be equipped with a pseudo-Riemann metric ∇ g of signature (n, n) [115, p.268]: The Riemann extension of ∇, given by ∇ V C g( ω, X) =V (ω(X)), ∇ g(V ω,V θ) = 0, ∇ C C g( X, Y ) = −γ(∇X Y, ∇Y X),
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where V ω,V θ are the vertical lifts to ∗ T (Mn ) of 1-forms ω, θ and C X,C Y denote the complete lifts to ∗ T (Mn ) of vector fields X,Y on Mn . Moreover, for any vector field Z on Mn (Z = Z i ∂i ) γZ is the function on ∗ T (Mn ) defined by γZ = ∑xi Z i . In a system of induced coordinates (xi , xi ) on ∗ T (Mn ), the Riemann i
extension is expressed by ∇
g=
−2∑xk Γkij δij k
j
δi
0
!
with respect to {∂i , ∂i }, where Γkıj are the components of ∇. Since inverse (∇g)−1 of the matrix ∇ g is given by ! j 0 δi ∇ −1 , ( g) = δi 2 xk Γk ∑ ij j k
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the Walker metrics (3.36) can be viewed as Riemann extensions, i.e., they are locally isometric to the cotangent bundle (∗ T (∑α ),∇ g) of a surface (∑2 , ∇) equipped with the metric ∇g. Then we have Corollary 90. Let ∗ T (∑2 ) a cotangent bundle of a surface (∑2 , g). Then (∗ T (∑2 ),∇ g) is an almost para-Norden-Walker manifold with an almost paracomplex structure (3.54), where a = 2(x3 Γ111 + x4 Γ211 ), b = 2(x3 Γ122 + x4 Γ222 ), c = 2(x3 Γ112 + x4 Γ212 ) .
3.9. Some Notes Concerning Norden-Walker 8-manifolds In the present section, we shall focus our attention to the Norden manifolds in dimension eight. Using the Walker metric we constructive a new Norden-Walker metrics together with so called proper almost complex structures. Note that an indefinite K¨ahler-Einstein metric on an eight-dimensional Walker manifolds has been recently investigated in [49]. A neutral metric g on a 8-manifold M8 is said to be Walker metric if there exists a 4-dimensional null distribution D on M8 , which is parallel with respect to g. From Walker theorem [110], there is a system of coordinates (x1 , ..., x8)
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Arif Salimov
with respect to which g takes the local canonical form 0 I4 g = (gi j ) = , I4 B
(3.59)
where I4 is the unit 4 × 4 matrix and B is a 4 × 4 symmetric matrix whose entries are functions of the coordinates (x1 , ..., x8). Note that g is of neutral signature (+ + + + − − −−), and that the paralel null 4-plane D is spanned locally by {∂1 , ∂2 , ∂3 , ∂4 }, where ∂i = ∂x∂ i , i = 1, ..., 4 . In this section, we consider the specific Walker metrics on M8 with B of the form a 0 0 0 0 0 0 0 (3.60) B= 0 0 b 0 , 0 0 0 0
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where a, b are smooth functions of the coordinates (x1 , ..., x8). We can construct various almost complex structures ϕ on a Walker 8manifold M8 with the metric g as in (3.59), (3.60) so that (M8 , ϕ, g) is almost Nordenian. The following ϕ is one of the simplest examples of such an almost complex structure: ϕ∂1 = ∂3 , ϕ∂2 = ∂4 , ϕ∂3 = −∂1 , ϕ∂4 = −∂2 , 1 ϕ∂5 = (a + b)∂3 − ∂7 , ϕ∂6 = −∂8 , 2 1 ϕ∂7 = − (a + b)∂1 + ∂5 , ϕ∂8 = ∂6 . 2 In conformity with the terminology of Matsushita (see, [47]- [49]) we call ϕ the proper almost complex structure. The proper almost complex structure ϕ has the local components 0 0 −1 0 0 0 − 21 (a + b) 0 0 0 0 −1 0 0 0 0 1 1 0 0 0 2 (a + b) 0 0 0 0 1 0 0 0 0 0 0 i (3.61) ϕ = (ϕ j ) = 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 with respect to the natural frame {∂i } , i = 1, ..., 8 .
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Remark 24. From (3.61) we see that in the case a = −b, ϕ is integrable.
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The proper almost complex structure ϕ on almost Norden-Walker manifolds is integrable if and only if (Nϕ )ijk = 0.Since N ijk = −Nki j , we need only consider N ijk ( j < k). By explicit calculation, the nonzero components of the Nijenhuis tensor are as follows: 1 1 1 3 3 N15 = N37 = N17 = −N35 = (a1 + b1 ), 2 1 3 (a + b)(a1 + b1 ), N57 = 4 1 1 1 3 3 N25 = N47 = N27 = −N45 = (a2 + b2 ), 2 1 1 1 3 3 N17 = −N35 = −N15 = −N37 = − (a3 + b3 ), 2 1 1 N57 = − (a + b)(a3 + b3 ), 4 1 1 1 3 3 N27 = N45 = N25 = −N47 = (a4 + b4 ), 2 1 1 1 3 3 N56 = −N78 = N58 = −N67 = − (a6 + b6 ), 2 1 1 1 3 3 N58 = −N67 = −N56 = N78 = − (a8 + b8 ). 2
(3.62)
From (3.62) we have Theorem 91. The proper almost complex structure ϕ on almost Norden-Walker 8-manifolds is integrable if and only if the following PDEs hold: a1 + b1 = 0, a2 + b2 = 0, a3 + b3 = 0, a4 + b4 = 0, a6 + b6 = 0, a8 + b8 = 0 Corollary 92. i) The proper almost complex structure on almost NordenWalker 8-manifolds is integrable if and only if a = −b + ξ where ξ is any function of x5 and x7 alone,
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Arif Salimov ii) The metric (3.59) with matrix −b(x1 , ..., x8) + ξ(x5 , x7 ) 0 B= 0 0
0 0 0 0 0 b(x1 , ..., x8) 0 0
0 0 0 0
is always Norden-Walker with respect to the proper complex structure ϕ. Let (M8 , ϕ, g) be an almost Norden-Walker manifold. If m m m m (Φϕ g)ki j = ϕm k ∂m gi j − ϕi ∂k gm j + gm j (∂i ϕk − ∂k ϕi ) + gim ∂ j ϕk = 0,
(3.63)
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then by virtue of Theorem 44, ϕ is integrable and the triple (M8 , ϕ, g) is called a holomorphic Norden-Walker or a K¨ahler-Norden-Walker manifold. We will write (3.59), (3.60) and (3.31) in (3.63). Since (Φϕg)i jk = (Φϕg)ik j , we need only consider (Φϕ g)i jk ( j < k). By explicit calculation, the nonzero components of Φϕ g the tensor are as follows: (Φϕ g)155
=
(Φϕ g)255
=
(Φϕ g)355
=
(Φϕ g)455
=
(Φϕ g)517
=
(Φϕ g)537
=
(Φϕ g)555
=
(Φϕ g)567
=
(Φϕ g)578
=
(Φϕ g)657
=
(Φϕ g)757
=
(Φϕ g)855
=
1 a3 , (Φϕ g)157 = (b1 − a1 ), (Φϕ g)177 = b3 , (3.64) 2 1 a4 , (Φϕ g)257 = (b2 − a2 ), (Φϕ g)277 = b4 , 2 1 −a1 , (Φϕ g)357 = (b3 − a3 ), (Φϕ g)377 = −b1 , 2 1 −a2 , (Φϕ g)457 = (b4 − a4 ), (Φϕ g)477 = −b2 , 2 1 1 −(Φϕ g)715 = (b1 + a1 ), (Φϕ g)527 = −(Φϕ g)725 = (b2 + a2 ), 2 2 1 1 −(Φϕ g)735 = (b3 + a3 ), (Φϕ g)547 = −(Φϕ g)745 = (b4 + a4 ), 2 2 1 (b + a)a3 − a7 , (Φϕ g)557 = −b5 , 2 1 1 −(Φϕ g)756 = (b6 + a6 ), (Φϕ g)577 = (b + a)b3 + a7 , 2 2 1 −(Φϕ g)758 = (b8 + a8 ), (Φϕ g)655 = −a8 , 2 1 1 (b6 − a6 ), (Φϕ g)677 = −b8 , (Φϕ g)755 = − (b + a)a1 − b5 , 2 2 1 −a7 , (Φϕ g)777 = − (b + a)b1 + b5 , 2 1 a6 , (Φϕ g)857 = (b8 − a8 ), (Φϕ g)877 = b6 . 2
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From (3.64) we have Theorem 93. The triple (M8 , ϕ, g) is K¨ahler-Norden-Walker if and only if the following PDEs hold: a1 = a2 = a3 = a4 = a6 = a7 = a8 = 0, b1 = b2 = b3 = b4 = b5 = b8 = 0.
(3.65)
Corollary 94. (M8 , ϕ, g) is K¨ahler-Norden-Walker if and only if the matrix B in (3.60) has components a(x5 ) 0 0 0 0 0 0 0 . B= 7 0 0 b(x ) 0 0 0 0 0
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It is well known that, if (M8 , ϕ, g) be an almost Norden manifold, then the Norden metric g is a quasi-K¨ahler-Norden if and only if (Φϕg)(X,Y, Z) + (Φϕg)(Y, Z, X) + (Φϕg)(Z, X,Y ) = 0
(3.66)
for any X,Y, Z ∈ ℑ10 (M2n ) (see Theorem 72). From (3.66) we easily see that a K¨ahler-Norden manifold is a quasi-K¨ahler-Norden. Conversely, quasi-K¨ahlerNorden manifold is a non-K¨ahler-Norden, in general. In (M8 , ϕ, g) particular, let be an almost Norden-Walker 8-manifold. Using (3.64) and (3.66), we have (Φϕ g)155 + (Φϕ g)551 + (Φϕ g)515
=
(Φϕ g)157 + (Φϕ g)571 + (Φϕ g)715
=
(Φϕ g)177 + (Φϕ g)771 + (Φϕ g)717 (Φϕ g)255 + (Φϕ g)552 + (Φϕ g)525
= =
(Φϕ g)257 + (Φϕ g)572 + (Φϕ g)725
=
(Φϕ g)277 + (Φϕ g)772 + (Φϕ g)727 (Φϕ g)355 + (Φϕ g)553 + (Φϕ g)535
= =
(Φϕ g)357 + (Φϕ g)573 + (Φϕ g)735
=
a3 = 0, 1 (b1 − a1 ) = 0, 2 b3 = 0, a4 = 0, 1 (b2 − a2 ) = 0, 2 b4 = 0, −a1 = 0, 1 (b3 − a3 ) = 0, 2
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(3.67)
116
Arif Salimov (Φϕ g)377 + (Φϕ g)773 + (Φϕ g)737 (Φϕ g)455 + (Φϕ g)554 + (Φϕ g)545
= =
(Φϕ g)457 + (Φϕ g)574 + (Φϕ g)745
=
(Φϕ g)477 + (Φϕ g)774 + (Φϕ g)747
=
(Φϕ g)557 + (Φϕ g)575 + (Φϕ g)755
=
(Φϕ g)555
=
(Φϕ g)567 + (Φϕ g)756 + (Φϕ g)675
=
−b1 = 0, −a2 = 0, 1 (b4 − a4 ) = 0, 2 −b2 = 0, 1 − (a + b)a1 − 3b5 = 0, 2 1 (a + b)a3 − a7 = 0, 2 1 (b6 − a6 ) = 0, 2
From (3.64) and (3.67) we see that the triple (M8 , ϕ, g) is quasi-K¨ahlerNorden-Walker if and only if the PDEs in the form (3.65) holds. On the other hand, the equation (3.65) is a K¨ahler condition of almost Norden-Walker manifolds. Thus we have
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Theorem 95. Let (M8 , ϕ, g) be an almost Norden-Walker 8−Manifold. Then there does not exist a (non-K¨ahler) quasi-K¨ahler structure on this manifold.
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Chapter 4
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Applications to the Theory of Lifts In this chapter we concentrate our attention to tensor bundles and we study the various lifts from manifold to its tensor bundle by using tensor operators. In section 4.1 some introductory materials concerning with the tensor bundle of type (p, q) over differentiable manifold Mn are collected. Here some of our notations are fixed too. In section 4.2 is devoted to the some particular types of p vector fields on the tensor bundle Tq (Mn ). Explicit expressions of the vertical lift of tensor fields of type (p, q) and the complete and horizontal lifts of vector fields are presented. In this section we also discuss complete lifts of derivations. These results generalize some of those already obtained for tangent and cotangent bundles. Section 4.3 devoted to the analysis of lifts on the cross-section of the tensor bundle. A tensor field of type (p, q) on the tensor bundle defines a cross-section of the tensor bundle, which is an n−dimensional submanifold p of the tensor bundle Tq (Mn ). Also, the behavior along the cross-section of the complete and horizontal lifts of vector fields was reviewed. In section 4.4 deals p with how affinor fields on Tq (Mn ) can be induced from affinor fields on Mn . We note that the φϕ −operator is an extension of the operator of Lie derivation LX to affinor fields ϕ. Models of the complete lifts of affinor fields along the pure cross-section are found by using of the φϕ −operator, moreover we discuss almost complex property of these lifts. Also we consider almost hyperholomorphic pure submanifolds of tensor bundles. By using of Vishnevskii operator, the problem of the horizontal lifts of affinor fields along the pure cross-section is solved in section 4.4.5. In section 4.4.6, a new φϕ −operator is defined and dis-
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cussed relations between the φϕ −operator and the diagonal lifts of affinor fields along the cross-section of Tqp (Mn ). In section 4.5, the adapted frame which allows the tensor calculus to be done efficiently is inserted in the tensor bundle Tqp (Mn ). The components of Levi-Civita and metric connections of Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done it is shown possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifold.Section 4.6 is devoted to the study of para-Nordenian property of Sasaki metric on cotangent bundles. We also study paraholomorphic property of Cheeger-Gromoll metric on tangent bundles and almost complex structures along the holomorhic cross-section of the cotangent bundle.In the last section 4.7, as applications of Yano-Ako operators we construct model of complete lifts of skew-symmetric tensor field of type (1, 2) on pure cross-sections in the tensor bundle. Through the chapter we always suppose that all functions, vector fields and tensor fields on tensor bundles are of class C∞ .
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4.1. Tensor Bundles Let Mn be a differentiable manifold of class C∞ and finite dimension n. Then p p the set Tq (Mn ) = ∪ Tq (P) is, by definition, the tensor bundle of type (p, q) P∈Mn
p
over Mn , where ∪ denotes the disjoint union of the tensor spaces Tq (Mn ) for all P ∈ Mn . For any point Pe of Tqp (Mn ), the surjective correspondence Pe → P p determines the natural projection π : Tq (Mn ) → Mn . In order to introduce a p manifold structure in Tq (Mn ), we define local charts on it as follows: Let x j be local coordinates in a neighborhood U of P ∈ Mn , then a tensor t at P which i ...i i ...i is an element of Tqp (Mn ) is expressible in the form (x j ,t j11 ... jpq ), where t j11 ... jpq are i ...i
components of t with respect to natural base. We may consider (x j ,t j11... jpq ) = (x j , x j ) = (xJ ), j = 1, ..., n , j = n + 1, ..., n + n p+q , J = 1, ..., n + n p+q as local coordinates in a neighborhood π−1 (U) ⊂ Tqp (Mn ). p It is straightforward to see that Tq (Mn ) becomes an (n + n p+q )−manifold; indeed if x j´ are local coordinates in a neighborhood V of P ∈ Mn , with U ∩V 6= ∅, then the change of coordinates is given by ( x j´ = x j´(x j ), (4.1) i´ ...i´ j ... j i ...i i´ ...i´ (i´) ( j) x j´ = t j´11 ... j´pq = Ai11 ...i pp A j´11 ... j´qq t j11 ... jpq = A(i) A( j´) x j ,
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Applications to the Theory of Lifts where i´ ...i´
(i´) ( j)
j ... j
A(i) A( j´) = Ai11 ...i pp A j´11 ... j´qq , Aii´11 =
119
∂xi´ j1 ∂x j , A j´1 = j´ . ∂xi ∂x
The Jacobian of (4.1) is:
∂xJ´ ∂xJ
=
∂x j´ ∂x j ∂x j´ ∂x j
∂x j´ ∂x j ∂x j´ ∂x j
!
j´
0
Aj
=
(i)
(i´) (k)
(i´) ( j)
t(k)∂ j A(i) A( j´) A(i) A( j´) (i)
!
,
(4.2)
i ...i
where J = ( j, j), J = 1, ..., n + n p+q, t(k) = tk11 ...kpq . We denote by ℑrs (Mn ) the module over F(Mn ) ( F(Mn ) is the ring of C∞ −functions on Mn ) all tensor fields of class C∞ and of type (r, s) on Mn . p If α ∈ ℑq (Mn ) , it is regarded, in a natural way (by contraction), as a function in p Tq (Mn ), which we denote by ια. If α has local expression j ... j
α = αi11...i pq ∂ j1 ⊗ ... ⊗ ∂ jq ⊗ dxi1 ⊗ ... ⊗ dxi p
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in a coordinate neighborhood U(x j ) ⊂ Mn , then ια = α(t) has the local expresj ... j i ...i sion ια = αi11...i pq t j11 ... jpq with respect to the coordinates (x j , x j ) in π−1 (U). For later use, we first shall state following Lemma. e Lemma 96. Let Xe and Ye be vector fields on Tqp (Mn ) such that X(ια) = Ye (ια), p p 1 e = Ye , i.e. Xe ∈ ℑ (Tq (Mn )) is completely deterfor any α ∈ ℑq (Mn ). Then X 0 mined by its action on functions of type ια. e Proof. It is sufficient to prove that if Z(ια) = (Xe − Ye )(ια) = 0, then Ze is zero. J If Ze has components Ze with respect to the coordinates (x j , x j ) in π−1 (U), we have j ... j i ...i j ... j i ...i e Z(ια) = Ze j ∂ j (αi11...i pq t j11 ... jpq ) + Ze j ∂ j (αi11...i pq t j11 ... jpq ) j ... j i ...i i ...i = Ze j t j11 ... jpq ∂ j αi11...i pq + Ze j α j11 ... jpq = 0.
(4.3)
j ... j
If this holds for any α ∈ ℑqp (M), ∂ j αi11...i pq taking any preassigned values at a fixed point, from (4.3) (homogeneous system of linear equations), we have i ...i Ze j t j11 ... jpq = 0, Ze j = 0.
(4.4)
From the first equation of (4.4), it follows that Ze j at all points of Tqp (Mn ) except i ...i possibly those at which all the components t j11 ... jpq are zero: that is, at points of
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the base spaces. However, the components of Ze are continuous and so Ze j is zero at points of the base space. Thus Ze j = 0 at all points of π−1 (U). Therefore, taking account of the second equation of (4.4), we see that Ze is zero in π−1 (U). This completes the proof of Lemma 96.
4.2. Horizontal and Complete Lifts of Vector Fields 4.2.1.
Vertical Lifts of Tensor Fields and γ−operator p
p
Suppose that A ∈ ℑq (Mn ). Then there is a unique vector field V A ∈ ℑ10 (Tq (Mn )) q such that for α ∈ ℑ p (Mn ) [38] V
A(ια) = α(A) ◦ π =V (α(A)),
(4.5)
where V (α(A)) is the vertical lift of the function α(A) ∈ F(Mn ). We note that the vertical lift V f = f ◦ π of the arbitrary function f ∈ F(Mn ) is constant along each fibre π−1 (P). If V A =V Ak ∂k +V Ak ∂k , then we have from (4.5)
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V k ...k
i ...i
k ...k
j ... j
k ...k
h ...h
Ak t j11 ... jpq ∂k αi11...i pq +V Ak αh11 ...hqp = αh11 ...hqp Ak11...kqp . j ... j
But αh11 ...hqp and ∂k αi11...i pq can take any preassigned values at each point. Thus, we have from the equation above V
i ...i
Ak t j11 ... jpq = 0,
V
h ...h
Ak = Ak11...kqp .
p
Hence V Ak = 0 at all points of Tq (Mn ) except possibly those at which all the i ...i components x j = t j11 ... jpq are zero: that is, at points of the base space. Thus we see that the components V Ak are zero a point such that x j 6= 0, that is, on p p p Tq (Mn ) − Mn . However, Tq (Mn ) − Mn is dense in Tq (Mn ) and the components of V A are continuous at every point of Tqp (Mn ). Hence, we have V Ak = 0 at p p all points of Tq (Mn ). Consequently, the vertical lift V A of A to Tq (Mn ) has components V j A 0 V A= V j = (4.6) i ...i A A j11 ... jpq with respect to (x j , x j ) the coordinates in Tqp (Mn ). We complete this section by giving the following definition which is needed later on.
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p
Definition 14. Let ϕ ∈ ℑ11 (Mn ) and ξ ∈ ℑq (Mn ) which locally are represented by ∂ ϕ = ϕij i ⊗ dx j , ∂x ∂ ∂ i ...i ξ = ξ j11 ... jpq i ⊗ ... ⊗ i p ⊗ dx j1 ⊗ ...dx jq . ∂x 1 ∂x p
A vector field γϕ ∈ ℑ10 (Tq (Mn )) is defined by p i1 ...m...i p iλ ∂ γϕ = ( ∑ t j1 ... jq ϕm ) ∂x j , (p ≥ 1, q ≥ 0), λ=1 q
i1 ...i p m ∂ eγϕ = ( ∑ t j1 ...m... jq ϕ jµ ) ∂x j , (p ≥ 0, q ≥ 1) µ=1
p
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with respect to the coordinates (x j , x j ) in Tq (Mn ).
From (4.2) we easily see that the vector fields γϕ and eγϕ determine respecp tively global vector fields on Tq (Mn ). The local expressions of the global vector fields γϕ and eγϕ are as follows: 0 0 γϕ = p i1 ...m...i p iλ and eγϕ = q i1 ...i p . ∑ t j1 ... jq ϕm ∑ t j1 ...m... jq ϕmjµ λ=1
4.2.2.
µ=1
Complete Lifts of Vector Fields
Let LV be the Lie derivation with respect to V ∈ ℑ10 (Mn ). The complete lift p c V = LV of V to Tq (Mn ) is defined by [38] c
V (ια) = ι(LV α)
(4.7)
p
for α ∈ ℑq (M). If cV =c V k ∂k +c V k ∂k , then we have from (4.7) c
i ...i
j ... j
k ...k
V kt j11 ... jpq ∂k αi11...i pq +c V k αh11 ...hqp i ...i
j ... j
q
(4.8) j ...k... jq
= t j11 ... jpq (V k ∂k αi11...i pq − ∑ (∂kV jµ )αi11...i p µ=1
p
j ... j
q + ∑ (∂iλ V k )αi11...k...i ). p
λ=1
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Thus, discussing in the same way as in the case of the vertical lift, from (4.8) we see that, the complete lift cV has components c
Vj V= c j V
c
=
p
Vj i ...m...i p
1 ∑ t j1 ... jq
λ=1
q
i ...i
p m ∂mV iλ − ∑ t j11 ...m... jq ∂ jµ V
µ=1
(4.9)
p
with respect to the coordinates (x j , x j ) in Tq (Mn ) (see [67]).
4.2.3.
Horizontal Lifts of Vector Fields
Let ∇ be a symmetric affine connection on Mn . We define the horizontal lift H e V ∈ ℑ1 (Tqp (Mn )) of V ∈ ℑ0 (Mn ) to Tqp (Mn ) by [38] ∇=∇ 0 1 H
V (ια) = ι(∇V α), α ∈ ℑqp (Mn ). p
The horizontal lift H V of V ∈ ℑ01 (Mn ) to Tq (Mn ) has components
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H
V=
Vj q
i ...i
p
i
i ...m...i p
1 p 1 λ V s ( ∑ Γm s jµ t j1 ...m... jq − ∑ Γsm t j1 ... jq
µ=1
λ=1
)
(4.10)
with respect to the coordinates (x j , x j ) in Tqp (Mn ) , where Γkij are local components of ∇ (see [39]).
4.2.4.
Complete Lifts of Derivations ∞
p
We now put F(Mn ) = ∑ ℑq (Mn ), which is the direct sum of all tensor modp,q=0
ules on Mn . A map D : F(Mn ) → F(Mn ) is a derivation on Mn , if a) D is linear with respect to constant coefficients, b) For all p, q, Dℑqp (Mn ) ⊂ ℑqp (Mn ), c) For all tensor fields T1 and T2 on Mn , D(T1 ⊗ T2 ) = (DT1 ) ⊗ T2 + T1 ⊗ (DT2 ), d) D commutes with contraction. For a derivation D on Mn , there exists a vector field P on Mn such that P f = D f , f ∈ F(Mn ).
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If we put D(∂i ) = Qhi ∂h in each coordinate neighborhood U of Mn , then the pair (Ph , Qhi ) is called the components of the derivation D in U [114, p.26]. j ... j Let α be an element of ℑqp (Mn ) with local expression α = αi11...i pq ∂ j1 ⊗ ... ⊗ ∂ jq ⊗ dxi1 ⊗ ... ⊗ dxi p . Then we see that Dα has components of the form Dα = (P
m
j ... j ∂m αi11...i pq
q
+∑
j ...m... j j αi11...i p q Qmµ
p
−
µ=1
j1 ... jq
∑ αi ...m...i 1
p
Qm iλ )
λ=1
in Mn , Ph being the components of P ∈ ℑ10 (Mn ) . Let D be a derivation on Mn . Then there is a unique vector field C D ∈ p p ℑ10 (Tq (Mn )) such that for α ∈ ℑq (Mn ) [8], [38] C
D(ια) = ι(Dα).
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We call C D the complete lift of D to Tqp (Mn ). The C D has components Pj C p i ...m...i D = q i1 ...i p 1 p i ∑ t j1 ...m... jq ϕmjµ − ∑ t j1 ... jq ϕmλ µ=1
(4.11)
λ=1
p
with respect to the coordinates (x j , x j ) in Tq (Mn ) . j ... j Let LV denote the Lie derivation with respect to V . From LV αi11...i pq = j ... j
q
j ...m... jq
V m ∂m αi11...i pq − ∑ (∂mV jλ )αi11...i p λ=1
p
j ... j
q + ∑ (∂iµ V m )αi11...m...i , we see that the Lie p
µ=1
derivation LV is a derivation on Mn having components LV : (V h , −∂iV h ). Using (4.9) and (4.11), we have C (LV ) =C V, where C V is the complete lift of the vector field V to Tqp (Mn ). Let now ∇ be an affine connection on Mn and ∇V denote the covariant derivation with respect to V . By similar devices, we see that the covariant derivation ∇V is a derivation on Mn having components ∇V : (V h ,V s Γhsi ) and C
`
`
(∇V ) =C V − γ(∇V ) + eγ(∇V ) ,
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Arif Salimov `
where ∇ is a new affine connection on Mn defined by `
∇V W = ∇W V + [V,W] , ∀V,W ∈ ℑ10 (M). In particular, if ∇ is a symmetric affine connection, then C
(∇V ) =H V ,
(4.12) p
where H V is the horizontal lift of the vector field V ∈ ℑ10 (Mn ) to Tq (Mn ).
4.2.5.
Derivations DKX Y and Formulas on Lie Derivations
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When a derivation D on Mn satisfies the condition D f = 0 for any f ∈ F(Mn ), D determines an element ϕ ∈ ℑ11 (Mn ) in such a way that DX = ϕX, ∀X ∈ ℑ10 (Mn ) . In such a case, D is denoted by Dϕ and called the derivation determined by ϕ. From Dϕ f = 0 and Dϕ X = ϕX, we easily verify that the Dϕ has local components Dϕ : (0, ϕhi ), Ph = 0, Qhi = ϕhi , (4.13) where ϕhi are local components of ϕ. From (4.11), (4.13) and Definition 4.2.1, we have 0 C q p (Dϕ) = = eγϕ − γϕ i1 ...i p i1 ...m...i p i ∑ t j1 ...m... jq ϕmjµ − ∑ t j1 ... jq ϕmλ λ=1
µ=1
or
C
(Dϕ ) = eγϕ − γϕ.
The Lie derivative LX ∇ of a symmetric affine connection ∇ with respect to X ∈ ℑ10 (Mn ) is, by definition, an element of ℑ12 (Mn ) such that (LX ∇)(Y, Z) = LX (∇Y Z) − ∇Y (LX Z) − ∇ [X,Y ] Z for any Y, Z ∈ ℑ10 (Mn ). We now denote by KX Y the tensor field of type (1, 1), defined by (KX Y )Z = (LX ∇)(Y, Z) = [LX , ∇Y ]Z − ∇[X,Y ] Z, (4.14) where KX Y = [LX , ∇Y ] − ∇[X,Y ] ,
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(4.15)
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125
which is an equation in terms of derivations. If we take the complete lifts of both sides in (4.15), we have eγ(KX Y ) − γ(KX Y ) =C (DKX Y ) =C [LX , ∇Y ] −C (∇[X,Y ] ).
(4.16)
Taking account of [38]
[C D1 ,C D2 ] =C [D1 , D2 ]
for any derivations D1 and D2 , from (4.16), we have eγ(KX Y ) − γ(KX Y ) =
C
(DKX Y )
C
= [ (LX ),C (∇Y )] −C (∇[X,Y ] ) = [C X,H Y ] −H [X,Y ].
Thus, we have Theorem 97. [C X,H Y ] =H [X,Y ] + eγ(KX Y ) − γ(KX Y ) for any X,Y ∈ ℑ10 (Mn ) , where KX Y denotes the tensor field of type (1, 1) defined by (4.14).
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An infinitesimal transformation defined by a vector field X ∈ ℑ10 (Mn ) is said to be an infinitesimal affine transformation with affine connection ∇, if LX ∇ = 0. From (4.14) and Theorem 97 we have Theorem 98. Let X be an infinitesimal affine transformation on Mn . Then [C X,H Y ] =H [X,Y ]. Let ∇ be a Riemannian connection on Mn and ∇X = 0. Then LX g = 0, i.e. X is an infinitesimal isometry or a Killing vector field. We next have LX ∇ = 0 as a consequence of LX g = 0. Since C X =H X (∇X = 0), we have Theorem 99. Let X be a vector field with vanishing Riemannian covariant derivative. Then [H X,H Y ] =H [X,Y ] p
i.e. the operation of taking the horizontal lift H : ℑ10 (Mn ) → ℑ10 (Tq (Mn )) is a homomorphism.
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4.3. Cross-sections in the Tensor Bundle Suppose that there is given a tensor field ξ ∈ ℑqp (Mn ) . Then the correspondence x → ξx , ξx being the value of ξ at x ∈ Mn , determines a mapping σξ : Mn → Tqp (Mn ), such that π ◦ σξ = idMn , and the n dimensional submanifold σξ (Mn ) of p Tq (Mn ) is called the cross-section determined by ξ. If the tensor field ξ has the h ...h local component ξk11...kqp (xk ), the cross-section σξ (Mn ) is locally expressed by (
xk = xk , h ...h xk = ξk11...kqp (xk )
(4.17)
p
with respect to the coordinates (xk , xk ) in Tq (Mn ). Differentiating (4.17) by x j , we see that n tangent vector fields B j to σξ (Mn ) have components (Bkj )
δkj ∂xk =( j)= h ...h ∂x ∂ j ξk11...kqp
(4.18)
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p
with respect to the natural frame {∂k , ∂k } in Tq (Mn ). On the other hand, the fibre is locally expressed by ( xk = const, h1 ...h p h ...h tk1 ...kq = tk11...kqp , h ...h
tk11...kqp being considered as parameters. Thus, on differentiating with respect to i ...i
x j = t j11 ... jpq we see n p+q that tangent vector fields C j to the fibre have components (Ckj )
∂xk 0 =( )= j h δkj11 ...δkqq δhi11 ...δi pp ∂x j
(4.19)
p
with respect to the natural frame {∂k , ∂k } on Tq (Mn ), where δ is the Kronecker symbol. p We consider in π−1 (U) ⊂ Tq (Mn ), n + n p+q local vector fields B j and C j along σξ (Mn ) . They form a local family of frames {B j ,C j } along σξ (Mn ) , which is called the adapted (B,C)−frame of σξ (Mn ) in π−1 (U). From cV =c V h ∂h +c V h ∂h and cV =c V j B j +c V jC j , We easily obtain cV k =c V j Bkj +c V jCkj , c
V k =c V j Bkj +c V jCkj . Now, taking account of (4.9) on the cross-section σξ (Mn ),
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e k = V k , cV e k = −LV ξh1 ...h p . Thus, the and also (4.18) and (4.19), we have cV k1 ...kq complete lift cV has along σξ (Mn ) components of the form c
c
ek V V = c ek V
Vk = h ...h −LV ξk11...kqp
(4.20)
with respect to the adapted (B,C)−frame. From (4.6), (4.18) and (4.19), by using similar way the vertical lift V A also has components V k 0 A V A= V k = (4.21) h ...h Ak11...kqp A with respect to the adapted (B,C)−frame. The lift H V has along σξ (Mn ) components of the form
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H
H
ej V V = Hej V
Vj = i ...i −(∇V ξ) j11 ... jpq
(4.22) i ...i
with respect to the adapted (B,C)−frame , where (∇V ξ) j11 ... jpq are local components of ∇V ξ. Let ϕ ∈ ℑ11 (M) now . We can easily verify that γϕ and eγϕ have along σξ (M) components 0 0 J J p q e e e γϕ = ((γϕ) ) = , γϕ = ((γϕ) ) = j1 ...m... j p i i1 ...i p ∑ ξi1 ...iq ϕmλ ∑ ξ j1 ...m... jq ϕmjµ λ=1
µ=1
(4.23) i ...i with respect to the adapted (B,C)−frame, where ξ j11 ... jpq are local components of ξ. Similarly, if S ∈ ℑ12 (M) , then γS and V (S ◦ ξ) are affinor fields along σξ (M) with components ! 0 0 0 0 , (V (Se oξ)IJ ) = γS = ((eγS)IJ ) = p jλ j1 ...m... j p m j ... j 1 0 ∑ S jm ξi1 ...iq S jjm ξi1 ...i2 q p 0 λ=1
with respect to the adapted (B,C)−frame,
1 where S jjm
(4.24) are local components of S.
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4.4. Lifts of Affinor Fields 4.4.1.
Complete Lifts of Affinor Fields
The complete lift of tensor field of type (1, 1) on Mn is defined as follow. Definition 15. [41], [86] We define a tensor field c ϕ ∈ ℑ11 (Tqp (Mn )) along the ϕ pure cross-section σξ (Mn ) by c c ϕ( V ) =c (ϕ(V )) − γ(LV ϕ) +V ((LV ϕ) ◦ ξ), (i) (4.25) c V ϕ( A) =V (ϕ(A)), (ii) p
p
for all V ∈ ℑ10 (Mn ) and A ∈ ℑq (Mn ), where ϕ(A) ∈ ℑq (Mn ), ((LV ϕ) ◦ ξ)(x1 , ..., xq; α1 , ..., αp ) = ξ(x1 , ..., xq; (LV ϕ)´α1 , ..., αp ) p
and call c ϕ the complete lift of ϕ ∈ ℑ11 (Mn ) to Tq (Mn ), p ≥ 1, q ≥ 0 along ϕ σξ (Mn ). In particular, if we assume that p = 1, q > 0 then we get
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γ(LV ϕ) =V ((LV ϕ) ◦ ξ). Substituting this into (4.25), we find (see [114]) c
ϕ(cV ) =c (ϕ(V )), cϕ(V A) =V (ϕ(A)).
∗ p
p
Let ℑq (Mn ) denotes a module of all the tensor fields ξ ∈ ℑq (Mn ) which are ϕ pure with respect to ϕ. Now, we consider a pure cross-section σξ (Mn ) deter∗ p
mined by ξ ∈ ℑq (Mn ), p ≥ 1, q ≥ 0. We observe that the local vector fields c
h ∂ δj h ∂ c X( j) = ( j ) = (δ j h ) = ∂x ∂x 0 c
and V
X ( j) =
V
(∂ j1 ⊗ ... ⊗ ∂ j p ⊗ dxi1 ⊗ ... ⊗ dxiq ) i
k
(δih11 ...δhqq δkj11 ...δ j pp ∂k1 ⊗ ... ⊗ ∂k p ⊗ dxh1 ⊗ ... ⊗ dxhq ) 0 = i k δih11 ...δhqq δkj11 ...δ j pp =
V
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j = 1, ..., n, j = n + 1, ..., n + n p+q span the module of vector fields in π−1 (U). Hence any tensor field is determined in π−1 (U) by its action of cX( j) and V X ( j) . Remark 25. The equation (4.25) is useful extension of the equation (see [38]) c
L(ια) = ι(LV α), α ∈ ℑqp (Mn ) ϕ
to affinor fields along the pure cross-section σξ (Mn ). Now, let us compute components of the complete lift of tensor fields of type (1, 1) by using Φϕ −operator. ϕ
Theorem 100. Let ϕ ∈ ℑ11 (Mn ) and σξ be a pure cross-section of Tqp (Mn ) with respect to ϕ. Then the complete lift c ϕ ∈ ℑ11 (Tqp (Mn )) of ϕ has along the pure ϕ cross-section σξ (Mn ) components h ...h Cϕ ekl = ϕkl , C ϕ elk = 0, C ϕ ekl = −(Φϕ ξ)lk11 ...kpq , r Cϕ ek = ϕh1 δhs 2 ...δhs p δr1 ...δ q s1
l
2
p
k1
kq
ϕ
with respect to the adapted (B,C)− frame of σξ (Mn ), where Φϕ ξ is the Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
h ...h
s ...s
Tachibana operator and xk = tk11...kqp , xl = tr11...rqp . Proof. Let cϕKL be components of c ϕ with respect to the adapted (B,C)−frame ϕ of the pure cross-section σξ (Mn ). Then from (4.25) we have K K ∼ ∼ c Kc eL c ∼ K eL V = (ϕ(V)) − (γ(LV ϕ)) +V ((LV ϕ) ◦ ξ) , (i) ϕ (4.26) ∼ c eK V eL V ϕL A = (ϕ(A))K , (ii) where
V
∼
K
( (ϕ(A)) ) = V
K
∼
((LV ϕ) ◦ ξ) ∼
γ(LV ϕ)
0 mh ...h p
2 ϕhm1 Ak1 ...k q
!
,
=
0 h ...m...h h (LV ϕmλ )ξk11...kq p
=
0 mh2 ...h p h 1 ((LV ϕ)m )ξk1 ...k q
K
!
,
!
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First, consider the case where K = k. In this case, (i) of (4.26) reduces to C ek C e l ϕl V
∼
k
e l =C (ϕ(V ))K = (ϕ(V))k = ϕklV l . el C V +C ϕ
(4.27)
Since the right-hand side of (4.27) are functions depending only on the base coordinates xi , the left-hand sides of (4.27) are too. Then, since cV l depend on fibre coordinates, from (4.27) we obtain C ek ϕl
= 0.
(4.28)
e l =C ϕ eklCV eklV l = ϕklV l , V i being arbitrary, From (4.27) and (4.28), we have C ϕ which implies C ek ϕl = ϕkl . When K = k, (ii) of (4.26) reduces to C e k V el ϕl A
or
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for all A
∼
el =V (ϕ(A))k ekl V A +C ϕ
mh2 ...h p C e k s1 ...s p ϕl Ar1 ...rq = ϕhm1 Ak1 ...k q p ∈ ℑq (Mn ), which implies
s ...s
c ek ϕl
s s ...s
r
= δrk11 ...δkqq ϕhs11 δhs22 ...δhs pp Ar11 r22 ...rqp r
= δrk11 ...δkqq ϕhs11 δhs22 ...δhs pp ,
h ...h
where xl = tr11...rqp , xk = tk11...kqp .When K = k, (i) of (4.26) reduces to
or
c ek c e l ϕl V
p
∼
k
e l =c (ϕ(V ))k − e l cV +c ϕ
r c el c ek c e l ϕl V + ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq V +
h1 ...l...h p
∑ (LV ϕhl )ξk ...k λ
1
λ=2
p
h1 h2 ...l...h p
∑ (LV ϕhl )ξk ...k λ
1
λ=2
q
q
∼
=c (ϕ(V ))k . (4.29)
ekl . The Now, using the Tachibana operator we will investigate components cϕ ∗ p
Tachibana operator on the pure module ℑq (Mn ) is given by (see Section 1.2) h ...h
(Φϕξ)lk11 ...kpq
∗
h ...h
h ...h
q
h ...h
1 p 1 p 1 p m = ϕm l ∂m ξk1 ...kq − ∂l ξ k1 ...kq + ∑ (∂ka ϕl )ξk1 ...m...kq
a=1
p
h
h ...m...h p
+2 ∑ ∂[l ϕm]λ ξk11...kq
,
λ=1
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where h ...h
h ...h
h ...h
1 p m 1 p m 1 p ϕm k1 ξmk2 ...kq = ϕk2 ξk1 m...kq = ... = ϕkq ξk1 k2 ...m =
mh ...h
h m...h
∗
h
h h ...h
h2 ...m 2 p ϕhm1 ξk1 ...k = ϕhm2 ξk11...kq p = ... = ϕmp ξkh11...k = ξ k11k22...kqp . q q
After some calculations we have h ...h
∗
h ...h
p
h ...h
h ...m...h p
V l (Φϕξ)lk11 ...kpq = (LϕV ξ)k11...kqp − (LV ξ)k11...kqp + ∑ (LV ϕhmλ )ξk11...kq λ=1
or
h ...h
mh ...h
mh ...h
2 p 2 p V l (Φϕ ξ)lk11 ...kpq + ϕhm1 (LV ξ)k1...k + ξk1 ...k (LV ϕ)hm1 q q
p
h ...m...h p
h
− ∑ (LV ϕmλ )ξk11...kq λ=1
h ...h
= (LϕV ξ)k11...kqp
(4.30)
for any V ∈ ℑ10 (Mn ).Using (4.20), from (4.30) we have h ...h
mh ...h
mh ...h
2 p 2 p V l (Φϕ ξ)lk11 ...kpq + ϕhm1 (LV ξ)k1 ...k + ξk1 ...k (LV ϕ)hm1 q q
p
h ...m...h p
− ∑ (LV ϕhmλ )ξk11...kq Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
λ=1
h ...h
s ...s
r
= V (Φϕ ξ)lk11 ...kpq + ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq (LV ξ)r11 ...rqp l
p
h ...m...h p
− ∑ (LV ϕhmλ )ξk11...kq λ=2
=
c
h ...h
r
V l (Φϕ ξ)lk11 ...kpq + ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq cV l p
h ...m...h p
− ∑ (LV ϕhmλ )ξk11...kq λ=2
c
= − (ϕ(V ))k or
h ...h
r
−(Φϕ ξ)lk11 ...kpq cV l + ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq cV l p
h
h ...m...h p
+ ∑ (LV ϕmλ )ξk11...kq
=c (ϕ(V ))k.
λ=2
Comparing (4.29) and (4.31), we get c ek ϕl
h ...h
= −(Φϕ ξ)lk11 ...kpq .
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(4.31)
132
Arif Salimov p
Thus, the complete lift cϕ ∈ ℑ11 (Tq (Mn ) of ϕ has along the pure cross-section ϕ σξ (Mn ) components n h ...h cϕ e kl = 0 , c ϕ ekl = ϕkl , cϕ e kl = −(Φϕξ)lk11 ...kpq , (4.32) r c ek ϕl = ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq ϕ
with respect to the adapted (B,C)−frame of σξ (Mn ), where Φϕξ is the s ...s
h ...h
Tachibana operator and xl = tr11...rqp , xk = tk11...kqp . This completes the proof. Remark 26. c ϕ in the form (4.32) is a unique solution of (4.25). Therefore, if ∗ ∗ p ϕ is element of ℑ11 (Tq (Mn )), such that ϕ(cV ) =c ϕ(cV ) =c (ϕ(V )) − γ(LV ϕ) +V ∗
∗
((LV ϕ) ◦ ξ), ϕ(V A) =c ϕ(V A) =V (ϕ (A)), then ϕ =c ϕ. In particular, if we write p = 1, q = 0, then (4.32) is the formula of the complete lift of affinor fields to tangent bundle along the cross-section σξ (Mn ) (for details, see [114, p. 126]). ϕ Now, on putting B j = C j , we write the adapted (B,C)−frame of σξ (M) as ϕ BJ = {B j , B j }. We define a coframe BeJ of σ (M) by BeI (BJ ) = δIJ . From (4.18),
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ξ
(4.19) and BKJ BeIK = δIJ we see that covector fields BeI have components ( ei = (BeiK ) = (δij , 0) B j ... j k j j Bei = (BeiK ) = (−∂k ξi11...iqp , δki11 ...δiqq δh11 ...δhpp )
(4.33)
with respect to the natural coframe (dxk , dxk ). Taking account of c K ϕL
e JI BJ ⊗ BeI (dxk , ∂L ) ϕ(dxk , ∂L ) =c ϕ c eJ K eI eJI dxK (BH ϕI dx (BJ )BeI (∂L ) =c ϕ J ∂H )BL K eI c e J K eI eJI BH = cϕ J δH BL = ϕ I BJ BL ,
= =
c
and also (4.18), (4.19), (4.32) and (4.33), we see that c ϕ has along the pure ϕ cross-section σξ (Mn ) components of the form c k ϕl = ϕkl , c ϕlk = 0, r c k ϕl = ϕhs11 δhs22 ...δhs pp δrk11 ...δkqq , q h1 ...h p mh2 ...h p c k (4.34) ϕl = (∂l ϕhm1 )ξk1 ...k − (∂kµ ϕm ∑ l )ξk1 ...m...kq q µ=1 p h ...m...h h h − ∑ (∂l ϕmλ − ∂m ϕl λ )ξk11...kq p λ=1
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133
ϕ
with respect to the natural frame {∂h , ∂h } of σξ (Mn ) in π−1 (U) [88].
4.4.2.
p
Almost Complex Structures on Tq (Mn )
Theorem 101. If ϕ is an integrable almost complex structure on Mn , then the ϕ complete lift C ϕ of ϕ to Tqp (Mn ) along the pure cross-section σξ (Mn ) is an alp most complex structure on Tq (Mn ). Proof. Let ϕ ∈ ℑ11 (Mn ) and S ∈ ℑ12 (Mn ) , using (4.20), (4.23), (4.24) and (4.34), we have γ(ϕ ± ψ) = γϕ ± γψ, C
ϕ(γψ) = γ(ϕ ◦ ψ) = γ(ψ ◦ ϕ), (γS)CV = γSV ,
(4.35)
where SV is the tensor field of type (1, 1) on Mn defined by SV (W ) = S(V,W), for any W ∈ ℑ10 (Mn ). If V ∈ ℑ10 (Mn ), then from (4.25) and (4.35), we have
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(C ϕ)2 (CV ) = (C ϕ ◦C ϕ)CV =C ϕ(C ϕ(CV )) =
C
ϕ(C (ϕ(V ))) − γ(LV ϕ) +V ((LV ϕ) ◦ ξ)
=
C
ϕ(C (ϕ(V ))) −C ϕ(γ(LV ϕ)) +C ϕ(V ((LV ϕ) ◦ ξ))
=
C
(ϕ(ϕ(V ))) − γ(Lϕ(V ) ϕ) −C ϕ(γ(LV ϕ))
+C ϕ(V ((LV ϕ) ◦ ξ)) +V ((Lϕ(V ) ϕ) ◦ ξ) =
C
(ϕ(ϕ(V ))) − γ(Lϕ(V ) ϕ) − γ((LV ϕ) ◦ ϕ)
+V (ϕ((LV ϕ) ◦ ξ)) + (Lϕ(V ) ϕ) ◦ ξ) =
C
(4.36)
(ϕ(ϕ(V ))) − γ(Lϕ(V ) ϕ + (LV ϕ) ◦ ϕ)
+V ((Lϕ(V ) ϕ) + (LV ϕ) ◦ ξ) +V ((LV (ϕ ◦ ϕ)) ◦ ξ) =
C
(ϕ ◦ ϕ)(CV ) + γ(LV (ϕ ◦ ϕ)) − γ(Lϕ(V ) ϕ + (LV ϕ) ◦ ϕ)
=
C
(ϕ ◦ ϕ)(CV ) − γ(Lϕ(V ) ϕ − ϕ ◦ (LV ϕ))
+V ((Lϕ(V ) ϕ) − ϕ(LV ϕ) ◦ ξ) =
C
(ϕ ◦ ϕ)(CV ) − γNV +V (NV ◦ ξ)
=
C
(ϕ ◦ ϕ)(CV ) − (γN)(CV ) +V (N ◦ ξ)(CV )
= (C (ϕ)2 − γN +V (N ◦ ξ)(CV ),
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Arif Salimov
where NV = Lϕ(V ) ϕ − ϕ ◦ (LV ϕ) and (Φϕ ϕ)(V,W) = (Lϕ(V ) ϕ − ϕ ◦ (LV ϕ))W = [ϕV, ϕW ]−ϕ[V, ϕW ]−ϕ[ϕV,W]+ϕ2 [V,W] = NV W is nothing but the Tachibana operator or the Nijenhuis-Shirokov tensor N(V,W) ∈ ℑ12 (Mn ) constructed from p ϕ. Similarly, if A ∈ ℑq (Mn ), from (4.25), we have (C ϕ)2 (C A) = (C ϕ ◦C ϕ)C A =C ϕ(C ϕ(C A)) =C ϕ(V ϕ(A)) =V (ϕ(ϕ(A)) =
V
((ϕ ◦ ϕ)(A)) =C (ϕ ◦ ϕ)V A =C (ϕ2 )V A.
(4.37)
If we take the integrability condition of ϕ (Nϕ = 0), then by the Remark 26, (4.36), (4.37) and the linearity of the complete lift, we have (C ϕ)2 =C (ϕ2 ) =C (−I) = −I. Let n−dimensional Mn and m−dimensional Nm be two manifolds with complex structures ϕ and ψ, respectively. A differentiable mapping f : Mn → Nm is called a holomorphic (analytic) mapping, if at each point p ∈ Mn d f p ◦ ϕ p = ψ f (p) ◦ d f p .
(4.38)
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ϕ
As the mapping f : Mn → Nm (m = n + n p+q ) we take a cross-section σξ : p p Mn → Tq (Mn ) determined by the pure tensor field ξ ∈ ℑq (Mn ) with respect ϕ p to ϕ−structure. The pure cross-section σξ : Mn → Tq (Mn ) can be locally expressed by (4.17). In (4.38), if ψ is the almost complex structure C ϕ (see Theorem 101), the condition that the pure cross-section ξ be a holomorphic tensor field is locally given by k C K M ϕm (4.39) l ∂m x = ϕM ∂l x , ϕ
where C ϕKM are components of C ϕ along the pure cross-section σξ (Mn ) with respect to the natural frame {∂k , ∂k }. In the case K = k, by virtue of (4.17) and (4.34) we get the identity ϕkl ≡ ϕkl . When K = k, by virtue of (4.17) and (4.34), (4.39) reduces to h ...h
(Φϕξ)lk11 ...kpq
h ...h
∗
h ...h
q
h ...h
1 p 1 p 1 p m = ϕm l ∂m ξk1 ...kq − ∂l ξ k1 ...kq + ∑ (∂ka ϕl )ξk1 ...m...kq
a=1
p
+2 ∑
λ=1
h h ...m...h ∂[l ϕm]λ ξk11...kq p ,
(4.40)
where Φϕ ξ is the Tachibana operator. Thus, the holomorphic tensor field ξ is given by (4.40) with respect to the natural frame {∂i , ∂i } (see Section 2.5).
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4.4.3.
135
Almost Hyperholomorphic Pure Submanifolds in the Tensor Bundle
By using (4.34), in [65] (see also [69]) proved that if Π = ϕ is almost α
integrable algebraic hypercomplex Π−structure on Mn , then the complete lift p c Π = cϕ of Π to Tq (Mn ) along the pure cross-section σΠ ξ (M) is an algebraic α
p
hypercomplex c Π−structure on Tq (Mn ). For an element X ∈ ℑ10 (Mn ) with local coordinates X k , we denote by BX and CX the vector fields with local components ! Xk BX = = X jB j, h ...h X j ∂ j ξk11...kqp
CX
=
0 s δkr11 ...δkrqq X h1 δsh22 ...δhpp
!
s = δkr11 ...δkrqq X i1 δsh22 ...δhpp
0 jq h1 h j1 δk1 ...δkq δi1 ...δi pp
s
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= δkr11 ...δkrqq X i1 δsh22 ...δhpp C( j) which are tangent to σξ (Mn ) and the fibre, respectively. Then by (4.32), we have ∗
along the pure cross-section σξ (Mn ) determined by ξ ∈ ℑqp (Mn ) that C
ϕ(BX) = B(ϕX) −C((Φϕξ)(X;Y1, ...,Yq, ξ1 , ..., ξp ))
(4.41)
for any X ∈ ℑ10 (Mn ). When C ϕ(BX) is tangent to σξ (Mn ) for any X ∈ ℑ10 (Mn ), c ϕ is said to leave σξ (Mn ) invariant. We have from (4.41) Theorem 102. The complete lift c ϕ of an element ϕ ∈ ℑ11 (Mn ) leaves the pure cross-section σξ (Mn ) invariant if and only if Φϕ ξ = 0. Asubmanifold in an almost algebraic hypercomplex manifold with structure Π = ϕ , is said to be almost hyperholomorphic when ϕ, α = 1, ..., m leaves α
α
the submanifold invariant. Thus, from Theorem 102, we have
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!
136
Arif Salimov
Theorem 103. A necessary and sufficient condition for the pure cross-section σξ (Mn ) in Tqp (Mn ) determined by a pure tensor field ξ ∈ ℑqp (Mn ) in an almost hypercomplex manifold Mn with structure Π = ϕ to be almost hyperholoα
p
morphic submanifoldin the almost algebraic hypercomplex manifold Tq (Mn ) with structure c Π = c ϕ is that the pure tensor field ξ ∈ ℑqp (Mn ) be almost α
hyperholomorphic in Mn .
4.4.4.
Horizontal Lifts of Affinor Fields
Now, let us define the horizontal lifts of tensor fields of type (1, 1) p
Definition 16. [39] Let ϕ ∈ ℑ11 (Mn ). The tensor field H ϕ ∈ ℑ11 (Tq (Mn )) of ϕ ϕ ∈ ℑ11 (Mn ) along the cross-section σξ (Mn ) to tensor bundles of type (p, q) is defined as following: H H ϕ( V ) =H (ϕ(V )), ∀V ∈ ℑ10 (Mn ), (4.42) p H V ϕ( A) =V (ϕ(A)), ∀A ∈ ℑq (Mn ),
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where ϕ(A) = C(ϕ ⊗ A). Using the Vishnevskii operator, we shall compute components of the horp izontal lift H ϕ ∈ ℑ11 (Tq (Mn )) with respect to the adapted (B,C)−frame of ϕ σξ (Mn ). We can state following theorem Theorem 104. The horizontal lift H ϕ of ϕ along the pure cross-section to tensor bundles has the following components l ...l
H ek ϕl
ek = 0, H ϕ ekl = −(Φϕξ)lk1 1 ...kp q , = ϕkl , H ϕ ( ls s s r ϕl11 δl22 ...δl pp δrk11 ...δkqq , p ≥ 1 H ek ϕl = s r δsl11 ...δl pp ϕrk11 δrk22 ...δkqq , q ≥ 1
(4.43)
ϕ
with respect to the adapted (B,C)−frame of σξ (Mn ), where Φϕ ξ is the Vishnevskii operator. eKL be components of H ϕ with respect to (B,C)−frame of the crossProof. Let ϕ ϕ section σξ (M). Then from (4.42) we have ( H eK H e L H e ϕL V = (ϕ(V ))K , (i) (4.44) Hϕ eL =H (ϕ e(A))K , (ii) eKL V A
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where V (ϕ(A)) = H
Applications to the Theory of Lifts 137 ! ! 0 0 ∼ = . Since the horizontal lift l1 ml2 ...l p V k ϕ A (ϕ(A)) m k1 ...kq
V is projectable, and so is H ϕ by virtue (i) of (4.44). H ϕ has components H ek ϕl
H ek ϕl
= ϕkl ,
=0
(4.45)
with respect to (B,C)−frame, which is also projectable. When K = k, (ii) of (4.44) reduces to
p
∼ H e k V el H e k V el V ϕl A + ϕl A = (ϕ(A))k , s ...s ml2 ...l p s r s ...s Hϕ ekl Ar11 ...rqp = ϕlm1 Ak1 ...k = ϕls11 δsl22 ...δl pp δrk11 ...δkqq Ar11 ...rqp q
for all A ∈ ℑq (M). It immediately follows that
s ...s
s
r
H ek ϕl
= ϕls11 δsl22 ...δl pp δrk11 ...δkqq , p ≥ 1,
H ek ϕl
= δsl11 ...δl pp ϕrk11 δrk22 ...δkqq , q ≥ 1.
(4.46)
l ...l
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where x j = tr11...rqp , xk = tk11 ...kpq . By the similar way, we have s
r
When K = k, (i) of (4.44) reduces to H ek H e l ϕl V
k
∼
k H el
el +H ϕ
V =H (ϕ(V))k .
(4.47)
e l , let us introduce Vishnevskii operator defined For obtaining the component H ϕ by (see Section 1.3) ( ml2 ...l p ϕlm1 ∇l ξk1 ...k , p≥1 l1 ...l p l ...l 1 p m q (Φϕξ)lk1 ...kq = ϕl ∇m ξk1 ...kq − (4.48) l ...l 1 p ϕm k1 ∇l ξmk2 ...kq , q ≥ 1 From (4.48), we have l ...l
ll ...l
l ...l
V l (Φϕξ)lk1 1 ...kp q + ϕll1 ∇V ξk12...kqp = ∇ϕ(V ) ξk11 ...kpq , p ≥ 1
(4.49)
Using (4.22), from (4.49) we have l ...l
ll ...l
l ...l
r
s
V l (Φϕ ξ)lk1 1 ...kp q + ϕll1 ∇V ξk12...kqp = V l (Φϕξ)lk1 1 ...kp q + ϕ ls11 δsl22 ...δl pp δrk11 ...δkqq l ...l
s
r
∼
q H el p r1 1 p H el l1 s2 p H k ∇V ξsr11 ...s ...rq = (Φϕξ)lk1 ...kq V − ϕs1 δl2 ...δl p δk1 ...δkq V = − (ϕ(V))
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Arif Salimov
or l ...l
s
r
∼
e l = −H (ϕ(V))k . (Φϕξ)lk1 1 ...kp q H Ve l − ϕls11 δsl22 ...δl pp δrk11 ...δkqq H V
(4.50)
Comparing (4.47) and (4.50), and using (4.46), we get k
l ...l
el + (Φϕ ξ)lk1 1 ...kp q )V l = 0, (H ϕ
for all V ∈ ℑ10 (M), from which
Similarly, we obtain
H ek ϕl
H ek ϕl
l ...l
= −(Φϕ ξ)lk1 1 ...kp q , p ≥ 1. l ...l
= −(Φϕξ)lk1 1 ...kp q , q ≥ 1.
This completes the proof.
Remark 27. Let ϕ be an integrable ϕ−structure in Mn and ∇ϕ = 0. If ξ ∈ p ℑq (Mn ) is a pure tensor field with respect to ϕ−structure, then the equation l ...l (Φϕξ)lk1 1 ...kp q = 0 is the condition for ξ to be holomorphic.
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Remark 28. The formula (4.43) is valid if and only if Φϕ ξ is the Vishnevskii ∗
operator, i.e. in the form (4.43) is a unique solution of (4.42). Therefore, if ϕ is ∗ ∗ p an element of ℑ11 (Tq (Mn )), such that ϕ(H V ) =H ϕ(H V ) =H (ϕ(V)), ϕ(V A) =H ∗
ϕ(V A) =V (ϕ(A)), then ϕ =H ϕ.
4.4.5.
Diagonal Lifts of Affinor Fields
In this section we defined a new operator for obtaining components of the diagonal lift D ϕ along the cross-section σξ (Mn ) to tensor bundles with respect to the adapted (B,C)−frame Firstly, let us define the diagonal lift of tensor field of type (1, 1). Definition 17. Let ϕ ∈ ℑ11 (Mn ). We define a tensor field D ϕ of type (1, 1) on Tqp (Mn ) along the cross-section σξ (Mn ) by D H ϕ( V ) =H (ϕ(V )), ∀V ∈ ℑ10 (Mn ), (4.51) D ϕ(V A) = −V (ϕ(A)), ∀A ∈ ℑ p (M ), q n p
where ϕ(A) = C(ϕ ⊗ A) ∈ ℑq (Mn ) and call D ϕ the diagonal lift of ϕ ∈ ℑ11 (Mn ) p to Tq (Mn ) along σξ (Mn ).
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139 p
Theorem 105. Let ϕ ∈ ℑ11 (Mn ) and σξ be cross-section of Tq (Mn ). Then, the diagonal lift D ϕ of ϕ has along the cross-section σξ (Mn ) components in the form l ...l
ek = 0, D ϕ ekl = −(Φϕ ξ)lk1 1 ...kp q , = ϕkl , D ϕ ( ls s s r −ϕl11 δl22 ...δl pp δrk11 ...δkqq , p ≥ 1 D ek ϕl = s r −δsl11 ...δl pp ϕrk11 δrk22 ...δkqq , q ≥ 1
D ek ϕl
(4.52)
with respect to the adapted (B,C)−frame of σξ (Mn ). Proof. From (4.51) we have D eK H e L H ∼ K ϕL V = (ϕ(V )) , ∼ Dϕ eL = −V (ϕ(A))K , eK V A
(i)
0
∼
where V (ϕ(A)) =
∼
V (ϕ(A))k
!
0
=
(4.53)
(ii) !
L
. Substituting K = k in (i)
ml2 ...l p ϕhm1 Ak1 ...k q
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of (4.53) and K = k, (ii) of (4.53), calculating in the same way as in Section 4.4.4, we obtain
s ...s
Dϕ ekl Dϕ e kl
Dϕ e kl = ϕkl , D ϕ e lk = 0, s r = −ϕsl11 δsl22 ...δl pp δrk11 ...δkqq , s r = −δsl11 ...δl pp ϕrk11 δrk22 ...δkqq ,
p≥1
(4.54)
q≥1
l ...l
where xr = tr11...rqp , xk = tk11 ...kpq . k
el of the diagonal lift D ϕ. In the case Now, we will study the component D ϕ when K = k, (i) of (4.53) reduces to D ek H e l ϕl V
p
k H el
el +D ϕ
∼
V =H (ϕ(V ))k .
Let ξ ∈ ℑq (Mn ). We consider a new Φϕ −operator [21] ( ml2 ...l p , p ≥ 1, ϕlm1 ∇l ξk1 ...k l1 ...l p l1 ...l p m q (Φϕξ)lk1...kq = ϕl ∇m ξk1 ...kq + l ...l 1 p ϕm k1 ∇l ξmk2 ...kq , q ≥ 1.
(4.55)
(4.56)
From (4.56), we have ml ...l
l ...l
l ...l
2 p 1 p l1 l V l (Φϕξ)lk1 1 ...kp q = (V l ϕm l )∇m ξk1 ...kq + ϕmV (∇l ξk1 ...kq ), p ≥ 1,
l ...l
ml ...l
l ...l
2 p ) = ∇ϕ(V )ξk11 ...kpq . V l (Φϕξ)lk1 1...kp q − ϕlm1 (∇V ξk1 ...k q
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(4.57)
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Arif Salimov
Using (4.22), from (4.57) we have l ...l
r
(Φϕ ξ)lk1 1 ...kp q H V l + ϕls11 δls22 ...δlspp δrk11 ...δkqq H V l = −H (ϕ(V ))k, l ...l
(4.58)
r
−(Φϕ ξ)lk1 1 ...kp q H V l − ϕls11 δls22 ...δlspp δrk11 ...δkqq H V l =H (ϕ(V ))k. Comparing (4.55) and (4.58) and making use of (4.54), we get
Similarly, we obtain
l ...l
D ek ϕl
= −(Φϕξ)lk1 1 ...kp q , p ≥ 1.
D ek ϕl
= −(Φϕ ξ)lk1 1...kp q , q ≥ 1.
l ...l
This completes the proof.
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h ...h
Remark 29. In the case of ∂m ξk11...kqp , (B,C)−frame is considered as a natural frame {∂h , ∂h } of σξ (Mn ). Then, from (4.56) we obtain components of D ϕ along the cross-section ( r −ϕls11 δls22 ...δlspp δrk11 ...δkqq , p ≥ 1, D k k D k D k ϕl = ϕl , ϕl = 0, ϕl = r −δls11 ...δlspp ϕrk11 δrk22 ...δkqq , q ≥ 1, p q l ...l p lλ l1 ...s...l p ϕm ξ (− Γ + Γsmkµ ξk11 ...s...k )+ ∑ ∑ ms l k1 ...kq q µ=1 λ=1 q p ml2 ...l p s...l p l ml2 ...s...l p l1 s +ϕ ( Γ ξ − Γlsλ ξk1 ...k − Γm ∑ ∑ m lk k ...s...k ls ξk1 ...kq ), p ≥ 1, µ 1 q q µ=1 λ=2 D k ϕl = p q l1 ...s...l p l ...l p l m λ ϕ (− Γ ξ + Γsmkµ ξk11 ...s...k )+ ∑ ∑ ms l k1 ...kq q µ=1 λ=1 q p l1 ...l p l l1 ...s...l p s ξl1 ...l p ( Γ − Γlsλ ξmk + Γslm ξs...k ), q ≥ 1 ∑ ∑ +ϕm k1 lkµ mk2 ...s...kq 2 ...kq q µ=2
λ=1
p
with respect to the natural frame {∂h , ∂h } of σξ (M) in π−1 (U) ⊂ Tq (Mn ) [21].
4.5. Lifts of Metrics In this section, we first introduce adapted frame to overcome the difficulty, which also allows the tensor calculation to do efficiently.
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Applications to the Theory of Lifts
4.5.1.
141
Adapted Frames
In each local chart U(xh ) of Mn , we put X( j) = ∂x∂ j = δhj ∂x∂h ∈ ℑ10 (Mn ), A( j) = ∂i1 ⊗ ... ⊗ ∂i p ⊗ dx j1 ⊗ ... ⊗ dx jq k j j p = δki11 ...δi pp δh11 ...δhqq ∂k1 ⊗ ... ⊗ ∂k p ⊗ dxh1 ⊗ ... ⊗ dxhq ∈ ℑq (Mn ), j = n + 1, ..., n + n p+q. Then from (4.6) and (4.10), we see that these vector fields have, respectively, local expressions p H
k
k ...s...k p
X( j) = δhj ∂h + (− ∑ Γ jsλ th11...hq
q
k ...k
p + ∑ Γsjhµ th11...s...h )∂h , q
(4.59)
µ=1
λ=1
k
j
j
A( j) = δki11 ...δi pp δh11 ...δhqq ∂h (4.60) n o n o p with respect to the natural frame ∂x∂H = ∂x∂h , ∂h on Tq (Mn ), where xh = V
∂x
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k ...k th11...hqp ,
j δi −Kronecker
n + n p+q
delta. These vector fields are linear independent and generate, respectively, the horizontal distribution of ∇ and the vertical o n p H V ( j) the frame adapted to the distribution of Tq (Mn ). We call the set X( j), A
affine connection ∇ in π−1 (U) ⊂ Tqp (Mn ). On putting e( j) =H X( j) , e( j) =V A( j)
o n we write the adapted frame as eβ = e( j) , e( j) . The indices α, β, γ, ... run over the range {1, ..., n, n + 1, ...,n + n p+q} and indicate the indices with respect to the adapted frame. Using (4.59) and (4.60), we have H
X =
= X
X j δhj p
k
k ...s...k p
−X( ∑ Γ jsλ th11...hq λ=1
j
q
k ...k
p ) − ∑ Γsjhµ th11...s...h q
µ=1
δhj p
−( ∑ λ=1
k k ...s...k Γ jsλ th11...hq p
q
− ∑ µ=1
k ...k p ) Γsjhµ th11...s...h q
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= X j e( j) ,
142
Arif Salimov 0 0 V A = = k1 k p j1 k ...k j k ...k Ah11 ...hpq δi1 ...δi p δh1 ...δhqq Ah11 ...hpq 0 k1 ...k p k ...k = Ah1 ...hq k1 k p j1 = Ah11 ...hpq e( j) , j δi1 ...δi p δh1 ...δhqq
i.e. the lifts H X and V A have respectively components H X j X j H X = HXβ = H j = , (4.61) X 0 V A j 0 Ve V β A= A = V j = (4.62) i ...i A j11 ... jpq A i ...i with respect to the adapted frame eβ , X j and A j11 ... jpq being local components of X and A in Mn , respectively.
4.5.2.
Sasakian Metrics on the Tensor Bundles
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For each p ∈ Mn the extension of scalar product g (denoted also by g) is defined p on the tensor space π−1 (p) = Tq (P) i ...i
t ...t
g(A, B) = gi1t1 ...gi pt p g j1 l1 ...g jqlq A j11 ... jpq Bl11 ...lpq for all A, B ∈ ℑqp (P). Definition 18. A Sasakian metric s g (or a diagonal lift of g ) is defined on p Tq (Mn ) by the three equations s
g(V A,V B) =V (g(A, B)), A, B ∈ ℑqp (Mn ), s
s
(4.63)
g(V A,H Y ) = 0,
(4.64)
g(H X,H Y ) =V (g(X,Y)), X,Y ∈ ℑ10 (Mn ).
(4.65)
These equations are easily seen to determine s g on Tqp (Mn ) with respect to which the horizontal and vertical distributions are complementary and orthogonal. From (4.63)-(4.65) we see that the Sasakian metric s g has components s g jl s g jl g jl 0 t ...t s geβγ = s , xl = tl11...lqp , (4.66) = g jq lq g jl s g jl 0 gi1 t1 ...gi pt p gej1 l1 ...e
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s βγ
g =
s g jl s jl
g
Applications to the Theory of Lifts 143 ! s g jl g jl 0 i ...i = , x j = t j11 ... jpq (4.67) i1t1 i pt p s jl 0 g ...g g e ...e g j l j l qq g 11
with respect to the adapted frame, gi j and gi j being local covariant and contravariant components g of on Mn . Sasakian metrics on tangent bundle were introduced in 1958 by the Japanese geometer Sasaki [87]. Sasakian metrics on tangent and cotangent bundles were also studied in [28], [73], [74], [114]. In a more general case of tensor bundles of type (1, 1), (1, q) and (0, q), Sasakian lifting of metrics are considered in [78], [79]. The Sasakian metrics on frame bundle was first considered by Mok [54] (for details, see [10]). We now consider local 1−forms ωα defined by e αdxB ωα = A B
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in π−1 (U), where eα = A B
eh A eh A j j eh A eh A j j
!
=
δhj
p
k
λ=1
AβA =
=
q
(k)
− ∑ Γslhµ th1 ...s...hq µ=1
k
j
j
δki11 ...δi pp δh11 ...δhqq
(4.68)
is the inverse matrix of the matrix !
A hj A hj A hj A hj
k ...s...k p
1 ∑ Γlsλ t(h)
0
p
δhj k
k ...s...k
0 q
k ....k
p − ∑ Γlsλ th11....hq p + ∑ Γslhµ th11...s...h q
λ=1
µ=1
k
j
j
δki11 ...δi pp δh11 ...δhqq
(4.69) of frames changes eβ = AβA ∂A . These n + n p+q 1−forms ωα are linearly indep pendent on Tq (Mn ). We call the set {ωα } the dual adapted coframe. For various types of indices, we have ! p q k ...s...k (k) k 1 p e = A A∂ = ∂ + − + ∑ Γsjhµ th1 ...s...hq ∂h , ∑ Γjsλ t(h) j j j A (4.70) µ=1 λ=1 e j = A Aj ∂A = ∂ j and
(
e j dxB = dx j , ωj = A B e j dxB = δt i1 ...i p , ωj = A B j1 ... jq
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(4.71)
144
Arif Salimov i ...i
i ...i
p
i
q
i ...m...i p
i ....i
1 p k − ∑ Γm kjµ t j1 ...m... jq )dx .
λ where δt j11 ... jpq = dt j11 ... jpq + ( ∑ Γkm t j11 .... jq
λ=1
µ=1
Since the adapted frame field {eα } is non-holonomic, we put γ
[eα, eβ ] = Ωαβ eγ , from which we have e α. Ωαγβ = (eγAβA − eβ AγA )A A
Thus, according to (4.68), (4.69) and (4.70), the components of non-holonomic object Ωαγβ are given by p q vp v vλ j1 jq v1 s r = −Ωr = Ω Γ δ ...δ ...δ δ ...δ − Γslrµ δrj11 ...δsjµ ...δrjqq δvi11 ...δi pp , ∑ ∑ lj rq i1 iλ ip ls r1 jl λ=1
µ=1
q
Ωrl j = ∑
µ=1
(v) Rsl jrµ tr1 ...s...rq
p
v
v ...s...v p
− ∑ Rljsλ t(r)1 λ=1
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(4.72) all the others being zero, with respect to the adapted frame, Rhijk being local components of the curvature tensor R of Mn with metric g. If s ∇ denote the Levi-Civita connection of s g, from T (X,Y ) =s ∇X Y −s ∇Y X − [X,Y ] = 0, ∀X,Y ∈ ℑ10 (Tqp (Mn )) we have s α Γγβ −s Γαβγ
= Ωαγβ
(4.73)
with respect to the adapted frame, where s Γαγβ are components of the Levi-Civita connection s ∇. The equation (s∇X s g)(Y, Z) = 0, ∀X,Y, Z ∈ ℑ10 (Tqp (Mn )) has form eδ s gγβ −s Γεδγ s gεβ −s Γεδβ s gγε = 0
(4.74)
with respect to the adapted frame. Thus, we have from (4.73) and (4.74) s α Γγβ
=
1 1 s αε g (eα s gεβ + eβ s gγε − eε s gγβ ) + (Ωγβ α + Ωα γβ + Ωα βγ), (4.75) 2 2
where Ωαγβ =s gαε s gδβ Ωδεγ . Taking account of (4.69), (4.72) and (4.75), for various types of indices, we find
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Applications to the Theory of Lifts
s r Γl j
=
145
p 1 [(− ∑ gt1 i1 ...∇X gtλ iλ ...gi pt p )e g j1 l1 ...e g jqlq 2 λ=1 q
+( ∑ gej1 l1 ...∇X gejµ lµ ...e g jqlq )gi1t1 ...gi pt p ] µ=1
= 0,
s r Γl j p s r Γl j
=
q
v
v
λ=1
s r Γl j s r Γl j
v
∑ Γlsλ δrj11 ...δrjqq δvi11 ...δsiλ ...δipp − ∑ Γslrµ δrj11 ...δsjµ ...δrjqq δvi11 ...δipp ,
(4.76)
µ=1
s r Γl j
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= 0, s Γrl j = 0, s Γrl j = Γrl j ,
p 1 q (v) v v ...s...v p = ( ∑ Rsl jrµ tr1...s...rq − ∑ Rljsλ t(r)1 ), 2 µ=1 λ=1 q
(k)
p
k
k ...s...k p
1 λ = 12 gxr gi1 t1 ...gi pt p gej1 h1 ...e g jq hq ( ∑ Rsxlhµ th1 ...s...hq − ∑ Rxls t(h)
µ=1 q
(k)
λ=1 p
k
k ...s...k p
1 = 12 gxr gt1 k1 ...gt pk p gel1 h1 ...e glqhq ( ∑ Rsx jhµ th1 ...s...hq − ∑ Rxλjst(h)
µ=1
), )
λ=1
with respect to the adapted frame, where Γhji denote of the Levi-Civita connection components constructed with g on M with respect to the natural frame {∂i }.
4.5.3.
Geodesics on Tensor Bundles
In this section, we shall characterize the geodesics on the tensor bundles with respect to the Levi-Civita connections of Sasaki metric and the horizontal lift H ∇ of a affine connection ∇. Definition 19. Let eγ = eγ(t) be a curve in Tqp (Mn ) and suppose that eγ is locally v1...v p r R R r r expressed n by o x n= x (t),oi.e. x = x (t), x = tr1...rq (t) with respect to the natural ∂ ∂xl
∂ , ∂ ∂xi ∂xi
, t being a parameter an arc length of eγ. Then the curve γ = π ◦ eγ on M is called the projection of the curve eγ and denoted by πeγ which is expressed locally by xr = xr (t). frame
=
Let ∇ be a Riemannian connection on Mn . Then a curve eγ is, by definition, p a geodesic in Tq (Mn ) with respect to s ∇ if and only if it satisfies the differential equations
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146
Arif Salimov
δ2 xR d 2 xR s R dxC dxB = + ΓCB = 0. (4.77) dt 2 dt 2 dt dt We find it more convenient to refer equations (4.77) to the adapted frame. Using (4.71), we now put v1 ...v p ωr dxr ωr δtr1 ...rq = , = , (4.78) dt dt dt dt along a curve eγ. Using (4.78) equations (4.77) can be transformed into d ωε s α ωγ ωβ ( ) + Γγβ =0 dt dt dt dt
(4.79)
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with respect to the adapted frame. By means of (4.76), (4.79) reduces to d ωr dxl dx j ( ) + Γrl j dt dt dt dt p 1 k1 ...k p + gexr gt1 k1 ...gt p k p gel1 h1 ...e glq hq (− ∑ Rx js kλ t(h) 2 λ=1 q
+∑
µ=1
(4.80)
t ...t δl11 ...lqp s (k) Rx jhµ th1...s...hq )
dx j dt dt
p 1 k1 ...k p + gexr gi1 k1 ...gi pk p gej1 h1 ...e g jqhq (− ∑ Rxls kλ t(h) 2 λ=1 q
+∑
µ=1
dxl (k) Rxlhµ sth1 ...s...hq ) dt
l ...l
δt j11 ... jpq dt
= 0,
v1 ...v p
p l j 1 q d δtr1 ...rq k1 ...k p dx dx (k) ) ( ) + ( ∑ Rl jhµ sth1 ...s...hq − ∑ Rl js kλ t(h) dt dt 2 µ=1 dt dt λ=1 p
+( ∑
λ=1
l v Γlsλ δrj11 ...δrjqq δli11 ...δsiλ ...δi pp
q
−∑
(4.81)
l l dx Γslrµ δrj11 ...δsjµ ...δrjqq δli11 ...δi pp )
µ=1
= 0.
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dt
i ...i
δt j11 ... jpq dt
Applications to the Theory of Lifts
147
We now transform (4.80) as follows: q δ2 xr (k) l1 h1 lqhq + g ...g g e ...e g ( t1 k1 tpk p ∑ Rrjhµ sth1...s...hq dt 2 µ=1 p
−∑
λ=1
t ...t δ1 p r kλ k1 ...s...k p l1 ...lq R js t(h) ) dt
(4.82)
dx j dt
= 0. p
Using the identity
− ∑ λ=1
k k1 ...s...k p Rl λjst(h)
q
+ ∑ µ=1
(k) Rsl jhµ th1 ...s...hq
!
dxl dx j dt dt
= 0,
transform of (4.81) as follow: v ...v
δ2 tr11...rqp = 0. dt 2
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Thus we have p Theorem 106. Let eγ be a geodesic on Tq (Mn ) of s ∇. Then the tensor field v1 ...v p tr1 ...rq (t) defined along γ satisfies the differential equations (4.82) and has vanishing second covariant derivative. p
Let now eγ be a geodesic of s ∇. If eγ lies on a fibre π−1 (P) = Tq (P), P = P(xh )
given by xh = ch = const, then
v ...v p δ2tr11...rq dt 2
= 0 reduces to
v ...v
d 2tr11...rqp = 0, (dxh = 0) dt 2
from which we have xr = ar t + br , r = n + 1, ..., n + n p+q , ar and br being constant. Hence we have p
Theorem 107. If a geodesic e γ lies in a fibre of Tq (Mn ) with respect to the metric s g, the geodesic e γ is expressed by linear equations xh = ch , h x = ah t + bh , where ch , ah and bh are constant.
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148
Arif Salimov Since the parameter t in (4.77) is the arc length of eγ, we have gβγ(
ωβ ωγ )( ) = 1. dt dt
Using (4.66) and (4.71) we find i ...i
t ...t
1 p δt 1 p δt dx j dxl j1 l1 jqlq j1 ... jq l1 ...lq g jl + gi1 t1 ...gi pt p ge ...e g = 1. dt dt dt dt On the other hand, (4.83) implies
i ...i
(4.83)
t ...t
δt 1 p δt 1 p δ j1 l1 jqlq j1 ... jq l1 ...lq (gi t ...gi pt p ge ...e g ) = 0, dt 1 1 dt dt t ...t
i ...i
i.e. gi1t1 ...gi pt p gej1 l1 ...e g jq lq
p δt j1 ... jpq δtl 1 ...lq 1 1 dt dt
ds dt
2
= const. Then we find that
= g jl
dx j dxl = const. dt dt
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and hence: Theorem 108. If t and s are the arc lengths of the geodesic eγ of s ∇ and the projection curve γ = πeγ, respectively, then s and t are related linearly: s = αt + β, where α and β constants. i ...i
Next, let γ be a curve on M expressed locally by xh = xh (t) and S j11 ... jpq (t) be p a tensor field of type (p, q) along γ. Then, on the tensor bundle Tq (Mn ) over the H Riemannian manifold Mn , we define a curve γ by ( xh = xh (t), k ...k xh = Sh11...hpq (t).
If the curve H γ satisfies at all the points the relation k ...k
δSh11 ...hpq dt
= 0,
i ...i
(4.84)
i.e. S j11 ... jpq (t) is a parallel tensor field along γ, then the curve H γ is said to be a horizontal lift of γ. From (4.82) and (4.84), we obtain δ2 xh = 0. dt 2 Thus we have
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149
Theorem 109. The horizontal lift of a geodesic on M is always geodesic on Tqp (M) with respect to the connection s ∇. Definition 20. We define the horizontal lift H ∇ of Levi-Civita connection on M to Tqp (M) by the conditions H ∇V V B = 0, H ∇V H Y = 0, A A (4.85) H ∇H X V B =V (∇X B) , H ∇H X H Y =H (∇X B) p
for any X,Y ∈ ℑ10 (M) and A, B ∈ ℑq (M). Denoting by T and Te, respectively, the torsion tensors of ∇ and H ∇, we have from (4.85) Te V A,V B =H ∇V A V B −H ∇V B V A − [V A,V B] = 0,
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Te
Te
V
A,H Y
H
X,H Y
= −Te H Y,V A =H ∇V A H Y −H ∇H Y V A − [V A,H Y ] = −V (∇Y A) +V (∇Y A) = 0, X − [H X,H Y ] = H (∇X Y ) −H (∇Y X) −H [X,Y ] + (eγ − γ) R(X,Y ) = H (T (X,Y )) + (eγ − γ) R(X,Y ). =
H
∇H X H Y −H ∇H Y
H
p
Since the vector fields H X and V A for any X ∈ ℑ10 (Mn ) and A ∈ ℑq (Mn ) span the module of vector fields on Tqp (M), then the tensor field Te is determined by its action of H X and V A. Therefore, we have Theorem 110. The connection H ∇ has nontrivial torsion even for the LeviCivita connection ∇ determined by g, unless g is locally flat. We have, from (4.63)-(4.65) and (4.85) (H ∇V C s g)(V A,V B) =V CV (g(A, B)) = 0, (H ∇V C s g)(V A,V B) = =
H V
Z (g(A, B)) −s g(V (∇Z A),V B) −s g(V A,V (∇Z B))
V
(Zg(A, B)) − g(∇Z A, B) − g(A, ∇Z B) i ...i
t ...t
g jq lq ) = 0, = A j11 ... jpq Bl11 ...lpq (∇Z (gi1t1 ...gi pt p gej1 l1 ...e Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
150
Arif Salimov (H ∇V C s g)(V A,H Y ) = 0, (H ∇H Z s g)(V A,H Y ) = 0, (H ∇V C s g)(H X,V B) = 0, (H ∇H Z s g)(H X,V B) = 0, (H ∇V C s g)(H X,H Y ) =V CV (g(X,Y)) = 0, (H ∇H Z s g)(H X,H Y ) =V ((∇Z g)(X,Y)) = 0
for any A, B,C ∈ ℑqp (Mn ) and X,Y, Z ∈ ℑ10 (Mn ). Thus, from last equations we have Theorem 111. The horizontal lift H ∇ of the Levi-Civita connection ∇ to p Tq (Mn ) is a metric connection with respect to the Sasakian metric, i.e. H ∇ s g. We now put H
∇α =H ∇eα ,
where {eα } = {ei , ei }−adapted frame. Then
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H
γ
∇α eβ =H Γαβ eγ .
(4.86)
Thus, taking account of (4.85) and writing expression (4.86) for the different indices, we find H r Γl j =H Γrl j =H Γrl j =H Γrl j =H Γrl j =H Γrl j = 0, H r Γl j = Γrl j , (4.87) p q lp lp vλ j1 jq l1 s j1 jµ jq l1 H r s Γ = ∑ Γ δ ...δ δ ...δ ...δ − ∑ Γ δ ...δ ...δ δ ...δ lj
λ=1
ls r1
rq i1
iλ
ip
µ=1
lrµ r1
s
rq i1
ip
with respect to the adapted frame. A curve eγ in Tqp (Mn ) is a geodesic with respect to the horizontal connection H ∇ when it satisfies d ωε H ε ωγ ωβ ( ) + Γγβ =0 (4.88) dt dt dt dt with respect to the adapted frame. As a consequence of (4.87), the equation (4.88) is equivalent to l dx j d 2 xr + Γrl j dx dt dt = 0, dt 2 v ...v p lp d δtr11...rqp vλ j1 jq l1 s dt ( dt ) + ( ∑ Γls δr1 ...δrq δi1 ...δiλ ...δi p − (4.89) λ=1 i1 ...i p q δt l l p dx j1 ... jq j1 jµ jq l1 s − ∑ Γlrµ δr1 ...δs ...δrq δi1 ...δi p ) dt dt = 0, µ=1
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151
where second equation of (4.89) reduces to v ...v
δ2 tr11...rqp = 0. dt 2 Thus we have p Theorem 112. Let eγ be a geodesic on Tq (Mn ) with respect to the metric conH s nection ∇ of Sasakian metric g. Then the projection γ of eγ is a geodesic with v ...v respect to the Levi-Civita connection ∇ on Mn and the tensor field tr11...rqp (t) defined along γ has vanishing second covariant derivative.
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4.5.4.
Jacobi Tensor Fields
We consider other possible connections on Tqp (Mn ). The complete lift c ∇ of p Levi-Civita connection ∇ to Tq (Mn ), defined by c ∇V AV B = 0, c ∇V A H Y = 0, c ∇H V B =V (∇ B), (4.90) X c XH ∇H X Y =H (∇X Y ) + (γ − eγ)(R( , X)Y ) p
for any A, B ∈ ℑq (Mn ) and X,Y ∈ ℑ10 (Mn ), where R( , X)Y denotes a tensor field ϕ of type (1, 1) on M, such that ϕ(Z) = R(Z, X)Y for any Z ∈ ℑ10 (Mn ), R being the curvature tensor of ∇. If we put e Ye ) =H ∇ eYe −C ∇ eYe S(X, X X p
e Ye ∈ ℑ1 (Tq (Mn )), then the tensor field S of type (1, 2) satisfies the for any X, 0 conditions V V S( A, B) = 0, S(V A,H Y ) = 0, (4.91) S(H X,V B) = 0, S(H X,H Y ) = (γ − eγ)(R( , X)Y ) for any A, B ∈ ℑqp (Mn ) and X,Y ∈ ℑ10 (Mn ). Therefore S has components Sεγβ such that p
v ...m...v p vλ Rml j +
Shlj = − ∑ tr11...rq λ=1
q
v1 ...v p
∑ tr ...m...r Rmr l j , 1
q
µ
µ=1
all the others being zero, with respect to the adapted frame.
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(4.92)
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Since the components of H ∇ are given by (4.87), it follows from (4.91) and (4.92) that complete lift C ∇ has components c r c r c r c r c r c r Γ = Γl j = Γl j = Γl j = Γl j = Γl j = 0 , lj c Γr = Γr , lj lj p q v ...v p v1 ...m...v p vλ r c Γ = ∑ tr ...r Rm R − ∑ tr 1...m...r lj
λ=1
1
ml j
q
µ=1
1
(4.93)
rµ l j
q
with respect to the adapted frame. p A curve eγ is, by definition, a geodesic in Tq (Mn ) with respect to C ∇ if and only if it satisfies the differential equation d ωα c α ωγ ωβ ( ) + Γγβ = 0. dt dt dt dt
By means of (4.93), (4.94) reduces to δ dxi dt ( dt ) = 0 , Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
i ...i p
δ2 t j1 ... jq 1
dt 2
p
i ...m...i p
+ ( ∑ t j11 ... jq λ=1
i
q
i ...i
(4.94)
l
1 p dx λ m Rml j − ∑ t j1 ...m... jq R jµ l j ) dt
µ=1
dx j dt
= 0.
(4.95)
i ...i
The second equation of (4.95) shows that the tensor field t j11 ... jpq (t) on Mn defined along γ = πeγ is a Jacobi tensor field ( a Jacobi vector field for p = 1, q = 0) along γ, where γ is a geodesic on Mn . Thus we have
Theorem 113. Let eγ be a geodesic on Tqp (Mn ) with respect to the C ∇ =H ∇ + S, where H ∇ is a metric connection of Sasakian metric s g, S is a tensor field of type (1, 2) defined by conditions (4.91). Then the projection γ = πeγ is a geodesic on i ...i M with respect to ∇ and the tensor field t j11 ... jpq (t) defined along γ is a Jacobi tensor field along the geodesic γ.
4.6. Some Special Cases In this section, we present some results concerning the study of φϕ −operators in the geometry of tangent and cotangent bundles.
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153
Para-Nordenian Structures in Cotangent Bundles
Let Mn be an n−dimensional differentiable Riemannian manifold of class C∞ and with metric g, C T (Mn ) its cotangent bundle, and π the natural projection C T (Mn ) → Mn . A system of local coordinates (U, xi ), i = 1, ..., n in Mn induces on C T (Mn ) a system of local coordinates (π−1 (U), xi , xi = pi ), i = n + i = n + 1, ..., 2n, where xi = pi is the components of covectors p in each cotangent space C Tx(Mn ), x ∈ U with respect to thenatural coframe {dxi }. We denote by ℑrs (Mn ) ℑrs (C T (Mn )) the modul over F(Mn )(F(C T (Mn ))) of C∞ tensor fields of type (r, s), where F(Mn )(F(C T (Mn ))) is the ring of realvalued C∞ functions on Mn (C T (Mn )) . Let X = X i ∂x∂ i and ω = ωi dxi be the local expressions in U ⊂ Mn of a vector and covector (1−form) fields X ∈ ℑ10 (Mn ) and ω ∈ ℑ01 (Mn ), respectively. Then the complete and horizontal lifts C X,H X ∈ ℑ10 (C T (Mn )) of X ∈ ℑ10 (Mn ) and the vertical lift V ω ∈ ℑ10 (C T (Mn )) of ω ∈ ℑ01 (Mn ) are given, respectively, by X = Xi
∂ ∂ − ph ∂i X h , ∂xi ∑ ∂xi i
(4.96)
X = Xi
∂ ∂ − ph Γhij X j , ∂xi ∑ ∂xi i
(4.97)
C
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H
V
ω = ∑ ωi i
∂ ∂xi
(4.98)
with respect to the natural frame { ∂x∂ i , ∂x∂ i }, where Γhij are components of the Levi-Civita connection ∇g on Mn (see [114] for more details). For each x ∈ Mn the scalar product g−1 = (gi j ) is defined on the cotangent space π−1 (x) =C T (Mn ) by g−1 (ω, θ) = gi j ωi θ j for all ω, θ ∈ ℑ01 (Mn ). A Sasakian metric s g is defined on C T (Mn ) by the three equations S
g(V ω,V θ) =V (g−1 (ω, θ)) = g−1 (ω, θ) ◦ π , S S
(4.99)
g(V ω,H Y ) = 0,
(4.100)
g(H X,H Y ) =V (g(X,Y )) = g(X,Y) ◦ π
(4.101)
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for any X,Y ∈ ℑ10 (Mn ) and ω, θ ∈ ℑ01 (Mn ). Since any tensor field of type (0, 2) on C T (Mn ) is completely determined by its action on vector fields of type H X and V ω (see [114, p.280]), it follows that S g is completely determined by equations (4.99), (4.100) and (4.101). We now see, from (4.96) and (4.97), that the complete lift C X of X ∈ ℑ10 (Mn ) is expressed by C X =H Y −V (p(∇X)), (4.102) where p(∇X) = pi (∇h X i )dxh . Using (4.99), (4.100), (4.101) and (4.102), we have S
g(C X,C Y ) =V (g(X,Y )) +V (g−1 (p(∇X), p(∇Y ))),
(4.103)
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where g−1 (p(∇X), p(∇Y )) = gi j (pl ∇i X i )(pk ∇ jY k ). Since the tensor field S g ∈ ℑ02 (C T (Mn )) is completely determined also by its action on vector fields of type C X and C Y (see [114, p.237]), we have an alternative characterization of S g : Sasakian metric S g on C T (Mn ) is completely determined by the condition (4.103). In U ⊂ Mn , we put X(i) =
∂ , θ(i) = dxi , i = 1, ..., n. ∂xi
Then from (4.97) and (4.98) we see that H X(i) and V θ(i) have respectively local expressions of the form ee(i) =H X(i) =
∂ ∂ , + ∑ pa Γahi i ∂x ∂xh h
(4.104)
∂ . ∂xi
(4.105)
ee(i) =V θ(i) =
o n o n We call the set ee(α) = e e(i) , ee(i) = H X(i) ,V θ(i) the frame adapted to the Levi-Civita connection ∇g . The indices α, β, ... = 1, ..., 2n indicate the indices with respect to the adapted frame. We now, from equations (4.97), (4.98), (4.104) and (4.105) see that H X and V ω have respectively components i X H i H H α X = X ee(i) , X =( X )= , (4.106) 0
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Applications to the Theory of Lifts 155 0 V V ω = ∑ ωi ee(i), ω = (V ωα ) = (4.107) ωi i with respect to the adapted frame ee(α) , where X i and ωi being local components of X ∈ ℑ10 (Mn ) and ω ∈ ℑ01 (Mn ), respectively. Let S ∇ be the Levi - Civita connection determined by the Sasakian metric S g. The components of S ∇ are given by (see Section 4.5)
S h Γ ji = Γhji , S Γhji =S Γhji =S Γhji = 0, S h Γ ji = 12 pm Rh. j.im , S Γhji = 12 pm Rh. i.im , S Γh = 1 p R m , S Γh = −Γi ji jh 2 m jih ji
(4.108)
with respect to the adapted frame ee(α) , where Rl jı α being local components of the curvature tensor R of ∇g . e Ye ∈ ℑ1 (C T (Mn )) and Xe = Xeα eeα , Ye = Ye β eeβ . The covariant Let now X, 0 S derivative ∇Ye Xe along Ye has components
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S
e α = Ye γ eeγ X e α +S Γα XeβYe γ ∇Ye X γβ
(4.109)
with respect to the adapted frame ee(α) . Using (4.106), (4.107), (4.108) and (4.109), we have
Theorem 114. Let Mn be a Riemannian manifold with metric g and S ∇ be the Levi – Civita connection of the cotangent bundle C T (Mn ) equipped with the Sasakian metric S g. Then S ∇ satisfies i) S ∇V ω V θ = 0, ii) S ∇V ω H Y =
1H e )), (P(g−1 ◦ R( ,Y )ω 2
iii) S ∇H X V θ =V (∇X θ) +
1H (P(g−1 ◦ R( , X)e θ)), 2
iv) S ∇H X H Y =V (∇X Y ) +
1V (PR(X,Y)) 2
e = g−1 ◦ ω ∈ for all X,Y ∈ ℑ10 (Mn ) and ω, θ ∈ ℑ01 (Mn ), where ω 1 1 −1 2 e ∈ ℑ1 (Mn ), g ◦ R( , X)ω e ∈ ℑ0 (Mn ), ℑ0 (Mn ), R( , X)ω
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Let (C T (Mn ),S g) be a cotangent bundle with the Sasakian metric S g. We define a tensor field F of type (1, 1) on C T (Mn ) by H F X =V Xe , (4.110) e F V ω =H ω
e = g−1 ◦ ω ∈ for any X ∈ ℑ10 (Mn ) and ω ∈ ℑ01 (Mn ), where Xe = g ◦ X ∈ ℑ01 (Mn ), ω 1 ℑ0 (Mn ). Then we obtain F 2 = I. In fact, we have by virtue of (4.110) e =H F 2 (H X) = F(F H X) = F(V X) e ) =V F 2 (V ω) = F(F V ω) = F(H ω
e e =H X, X e e =V ω ω
for any X ∈ ℑ10 (Mn ) and ω ∈ ℑ01 (Mn ), which implies F 2 = I.
Theorem 115. The triple (C T (Mn ),S g, F) is an almost para-Nordenian manifold.
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Proof. We put e Ye) =S g(F X, e Ye ) −S g(X, e FYe ) A(X,
e Ye ∈ ℑ1 (C T (Mn )). From (4.99), (4.100), (4.101) and (4.110), we have for any X, 0 A(H X,H Y ) =
g(F H X,H Y ) −S g(H X, F H Y ) e H Y ) −S g(H X,V Ye ) = 0, = S g(V X,
A(H X,V ω) =
S
S
g(F H X,V ω) −S g(H X, F V ω) e V ω) −S g(H X,H ω e) = S g(V X,
= g−1 (g ◦ X, ω) − g(X, g−1 ◦ ω) = 0, A(V ω,H Y ) = −A(H Y,V ω) = 0, A(V ω,V θ) =
S
g(FV ω,V θ) −S g(V ω, FV θ) e ,V θ) −S g(V ω,H e = g−1 (H ω θ) = 0,
i.e. S g is pure with respect to F. Thus Theorem 115 is proved.
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We now consider the covariant derivative of F. Taking account i)-iv) of Theorem 114 and (4.110), we obtain S ∇H X F H Y = S ∇H X F H Y − F S ∇H X H Y =S ∇H X V Ye − F S ∇H X H Y 1H V 1 −1 H V p g ◦ R ( , X)Y − F (∇X Y ) + (pR(X,Y )) = ∇X Ye + 2 2 1H pg−1 ◦ (R ( , X)Y − R(X,Y )) , = 2 S
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S
∇H X F
S
∇V ωF
V
H
Y
θ =
∇V ω F
∇V ω F H Y − F S ∇V ω H Y 1 e = S ∇V ωV Ye − F H p g−1 ◦ R ( ,Y ) ω 2 1V e ), = − (pR( ,Y )ω 2 S
=
∇H X F V θ − F S ∇H X V θ 1 H −1 S He V e p g ◦ R ( , X) θ = ∇H X θ − F (∇X θ) + 2 1V = H ∇X e θ + pR X, e θ −H g−1 ◦ (∇X θ) 2 V 1 pg ◦ g−1 ◦ R( , X)e θ − 2 1V = pR(X, e θ) − pR( , X)e θ , 2 V
S
θ
∇V ω F V θ − F S ∇V ωV θ 1 H −1 e . θ= = S ∇V ω H e p g ◦ R ,e θ ω 2
=
S
Using Theorem 44, from last equations we have
Theorem 116. The cotangent bundle of a Riemannian manifold is paraKahlerian (paraholomorphic Nordenian) with respect to the metric S g and almost paracomplex structure F defined by (4.102) if and only if the Riemannian manifold is flat.
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Arif Salimov A vector field Xe ∈ ℑ10 C T (Mn ) with respect to which the almost paraNordenian structure F has a vanishing Lie derivative LXe F = 0 is said to be almost paraholomorphic. It is well known that [114, p. 277] C H [ X, Y ] =H [X,Y ] +V (p (LX ∇)Y ) , (4.111) [C X,V ω] =V (LX ω),
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where (LX ∇)Y = ∇Y ∇X + R(X,Y ) and (LX ∇) (Y, Z) = LX (∇Y X) − ∇Y (LX Z) − ∇[X,Y ] Z . A vector field X ∈ ℑ10 (Mn ) is called a Killing vector field (or an infinitesimal isometry) if LX g = 0, and X is called an infinitesimal affine transformation if LX ∇g = 0. A Killing vector field is necessarily an infinitesimal affine transformation, i.e. we have LX ∇g = 0 as a consequence of LX g = 0. We now consider the Lie derivative of F with respect to the complete lift C X. Taking account of (4.110) and (4.111), we obtain
(LC X F)V θ = LC X F V θ − F LC X V θ = LC X H e θ − F V (LX θ)
(4.112)
= LC X H e θ −H g−1 ◦ (LX θ) = V [X, e θ] +V p (LX ∇) e θ −H g−1 ◦ (LX θ) = H L g−1 ◦ θ − g−1 ◦ (LX θ) +V p (LX ∇) e θ ,
(LC X F)H Y
= LC X F H Y − F LC X H Y = LC X V Ye − F H [X,Y ] +V (p (LX ∇)Y ) .
(4.113)
Let now be a Killing vector field (LX g = 0). Then by virtue of LX ∇ = 0 from (4.112) and (4.113) we have LC X F = 0, i.e. C X is a paraholomorphic with respect to F. If we assume that LC X F = 0 and calculate the equation (4.113) at i x , 0 , pi = 0 we get LX (g ◦Y ) = g ◦ LX Y from, which follows that LX g = 0. We hence have
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Theorem 117. An infinitesimal transformation X of Riemannian manifold (Mn , g) is a Killing vector field if and only if its complete lift C X to cotangent bundle C T (Mn ) is an almost paraholomorphic vector field with respect to the S almost para-Nordenian structure F, g .
Let now R ∇ ∈ ℑ02 C T (Mn ) be a Riemannian extension of the connection ∇g defined by [2], [114, p. 268] R ∇ C X,C Y = −p (∇X Y + ∇Y X) , X,Y ∈ ℑ10 (Mn ). The metric R ∇ has components
j
R
∇=
−2pa Γaji δi 0 δij
!
(4.114)
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with respect to the natural frame {∂i , ∂i }. From (4.97), (4.98) and (4.118) we easily see that R ∇ H X,H Y = 0, R ∇ V ω,V θ = 0, R ∇ H X,V θ =V (θ (X)) , (4.115)
i.e. the metric R ∇ is completely determined also by conditions (4.115). Using (4.101), (4.110) and (4.115), we have R e H Y =V X(Y e ) ∇ ◦ F H X,H Y = R ∇ F H X,H Y =R ∇ V X, = V (g(X,Y )) =S g H X,H Y , R e V θ =S g H X,V θ = 0, ∇ ◦ F H X,V θ =R ∇ F H X,V θ =R ∇ V X, R e ,H Y =S g V ω,H Y = 0, ∇ ◦ F V ω,H Y =R ∇ F V ω,H Y =R ∇ H ω R
∇◦F
V
ω,V θ =
e ,V θ =V (θ (ω e )) ∇ F V ω,V θ =R ∇ H ω = V g−1 (ω, θ) =S g V ω,V θ , R
i.e. R ∇ ◦ F =S g. Thus we have
Theorem 118. The almost para-Nordenian structure F determined by the con−1 S dition (4.110) has an expression in the form F = R ∇ ◦ g.
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4.6.2.
Arif Salimov
Paraholomorphic Cheeger-Gromoll Metric in the Tangent Bundle
In [9], Cheeger and Gromoll study complete manifolds of nonnegative curvature and suggest a construction of Riemannian metrics useful in that context. Inspired by a paper of Cheeger and Gromoll, in [55] Musso and Tricerri defined a new Riemannian metric CG g on tangent bundle of a Riemannian manifold which they called the Cheeger-Gromoll metric. The Levi-Civita connection of CG g and its Riemannian curvature tensor are calculated by Sekizawa in [91] (for more details see [27], [28]). The main purpose of this section is to investigate a paraholomorphic Cheeger-Gromoll metric with respect to the natural paracompex structure on the tangent bundle. Let Mn be a Riemannian manifold with metric g, and let T (Mn ) be a tangent bundle of Mn , and π the projection π : T (Mn ) → Mn . Let the manifold Mn be a covered by a system of coordinate neighbourhoods (U, xi ), where (xi ), i = 1, ..., n is a local coordinate system defined in the neighbourhood U. Let (yi ) be the cartesian coordinates in each tangent spaces Tp (Mn ) at p ∈ Mn with respect to the natural base { ∂x∂ i }, P being an arbitrary point in U whose coordinates are xi . Then we can introduce local coordinates (xi , yi ) in the open set π−1 (U) of T (Mn ). We call them coordinates induced in π−1 (U) from (U, xi ). The projection π is represented by (xi , yi ) → (xi ). We use the notations xl = (xi , xi ) and xi = yi . The indices I, J, ... run from 1 to 2n, the indices i, j, ... from 1 to n and the indices i, j, ... from n + 1 to 2n. Let X ∈ ℑ10 (Mn ), which locally are represented by X = X i ∂i
(∂i =
∂ ). ∂xi
Then the vertical and horizontal lifts V X and H X of X are given by V
X = X i ∂i (∂i =
∂ ) ∂xi
and H
X = X i ∂i − Γijk x j X k ∂i ,
where Γijk are the coefficients of the Levi-Civita connection on Mn . Suppose that we are given in U ⊂ Mn a vector field X ∈ ℑ10 (Mn ) and a covector field gX = (gi j X i dx j ). Then we define a function γgX in π−1 (U) ⊂ T (Mn ) by
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γgX = x j gi j X i with respect to the induced coordinates (xi , xi ). The function γgX defined in each π−1 (U) determine global function on T (Mn ), which is denoted also by γgX . Now, let r be the norm a vector y = (yi ) = (xi ), i.e. r2 = gi j xi x j . The Cheeger-Gromoll metric CG g on the tangent bundle T (Mn ) is given by (i) CG g(H X,H Y ) =V (g(X,Y)), (ii) CG g(H X,V Y ) = 0 (ii) CG g(V X,V Y ) =
1 V [ (g(X,Y )) + (γgX )(γgY )] 1 + r2
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for all vector field X,Y ∈ ℑ10 (Mn ) , where V (g(X,Y)) = (g(X,Y)) ◦ π. The diagonal lift D ϕ on T (Mn ) is defined by D H ϕ X =H (ϕX), D ϕV X = −V (ϕX) for X ∈ ℑ10 (Mn ) any and ϕ ∈ ℑ11 (Mn ). The diagonal lift D I of the identity tensor field I ∈ ℑ11 (Mn ) has the components ! δij 0 D I= j −2yt Γti −δij with respect to the induced coordinates and satisfies (DI)2 = IT (Mn ). Thus D I is an almost paracomplex structure determining the horizontal distribution and the distribution consisting of the tangent planes to fibres. We put e Ye ) =CG g(DI Xe , Ye) −CG g(X, e D IYe ). S(X,
e Ye ) = 0 for all vector fields Xe and Ye which are of the form V X,V Y If S(X, or H X,H Y, then S = 0. By virtue of D IV X = −V X, D I H X =H X and (i)-(iii) we have S(V X,V Y ) =CG g(−V X,V Y ) −CG g(V X, −V Y ) = 0, S(V X,H Y ) =CG g(−V X,H Y ) −CG g(V X,H Y ) = 0, S(H X,V Y ) =CG g(H X,V Y ) −CG g(H X, −V Y ) = 0, S(H X,H Y ) =CG g(H X,H Y ) −CG g(H X,H Y ) = 0, i.e. CG g is pure metric with respect to D I. We hence have:
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Theorem 119. (T (Mn ),D I,CG g) is an almost paracomplex Riemannian manifold. Let CG ∇ be the Levi-Civita connection of CG g. Using properties of V X,H X and γR(X,Y ) = ys Rkijs X iY j ∂k (see [28] and [114]), i.e. ∂x
V
X V (g(Y, Z)) = 0, [V X,V Y ] = 0,
[V X,H Y ] =V [X,Y ] −V (∇X Y ) = −V (∇Y X), H CG ∇H
X
VY
=
X V (g(Y, Z)) =V (Xg(Y, Z)),
H 1 (R(xi ,Y )X 2(1+r 2)
+V (∇X Y ) (R-curvature tensor field of)
[H X,H Y ] =H [X,Y ] − γR(X,Y ), D
IγR(X,Y) = −γR(X,Y),
CG
g(γR(X,Y ),H Z) = 0.
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Using Tachibana operator, we have i)(ΦDI CG g)(V X,V Y,V Z) = (DIV X)(CGg(V Y,V Z)) −V X(CGg(D IV Y,V Z)) +CG g((LV Y D I)V X,V Z) +CG g(V Y, (LV Z D I)V X) = −V X(CG g(V Y,V Z)) +V X(CG g(V Y,V Z)) +CG g(LV Y (D IV X) −D ILV Y V X,V Z) +CG g(V Y, LV Z (DIV X) −D ILV Z V X) =
CG
g([V X,V Y ] −D I[V Y,V X],V Z)
+CG g(V Y, [V X,V Z] −D I[V Z,V X]) =
CG
g(0,V Z) +CG g(V Y, 0) = 0.
ii)(ΦDI CG g)(V X,V Y,H Z) = (DIV X)(CGg(V Y,H Z)) −V X(CG g(DIV Y,H Z)) +CG g((LV Y D I)V X,H Z) +CG g(V Y, (LH Z D I)V X)
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Applications to the Theory of Lifts = −V X.0 +V X.0 +CG g(LV Y (D IV X) −D ILV Y V X,H Z) +CG g(V Y, LH Z (D IV X) −D ILH Z V X) =
CG
g(−[V Y,V X]D I.0,H Z) +CG g(V Y, −[H Z,V X] −D I[H Z,V X])
=
CG
g(0,H Z)
+CG g(V Y,V [X, Z] −V (∇X Z) +D I(V [X, Z] −V (∇X Z)) =
CG
g(V Y, 0) = 0.
iii)(ΦDICG g)(V X,H Y,V Z) = (DIV X)(CGg(H Y,V Z)) −V X(CG g(DI H Y,V Z)) +CG g((LHY D I)V X,V Z) +CG g(HY, (LV Z D I)V X) = −V X.0 +V X.0 +CG g(LHY (D IV X) −D ILH Y V X,V Z) +CG g(H Y, LV Z (D IV X) −D ILV Z V X)
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=
CG
g(−[H Y,V X] −D I[H Y,V X],V Z)
+CG g(H Y, −[V Z,V X] −D I[V Z,V X]) = CG g(V [X,Y ] −V (∇X Y ) +D I(V [X,Y ] −V (∇X Y )),V Z) +CG g(H Y, 0) =
CG
g(V [X,Y ] −V (∇X Y ) −V [X,Y ] +V (∇X Y )),V Z)
=
CG
g(0,V Z) = 0.
iv)(ΦDI CG g)(H X,V Y,V Z) = (D I H X)(CGg(V Y,V Z)) −H X(CG g(DIV Y,V Z)) +CG g((LV Y D I)H X,V Z) +CG g(V Y, (LV Z D I)H X) = H X CG g(V Y,V Z) +H X CG g V Y,V Z +CG g LV Y D I H X −D I LV Y H X ,V Z +CG g V Y, LV Z D I H X −D ILV Z H X = 2H X CG g V Y,V Z +CG g LV Y H X −D I V [Y, X] −V (∇Y X) ,V Z +CG g V Y, LV Z H X −D I V [Z, X] −V (∇Z X)
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Arif Salimov = 2H X CG g
= =
=
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=
Y,V Z
[Y, X] −V (∇Y X) +V [Y, X] −V (∇Y X),V Z +CG g V Y,V [Z, X] −V (∇Z X) +V [Z, X] −V (∇Z X) 2H X CG g V Y,V Z + 2CG g V [Y, X] −V (∇Y X) ,V Z +CG g V Y,V [Z, X] −V (∇Z X) 2H X CG g V Y,V Z +2 CG g −V (∇Y X) ,V Z +CG g V Y, −V (∇Z X) 2H X CG g V Y,V Z H 1 CG i CG V V +2( g (R(x ,Y )X) − ∇H X Y, Z 2(1 + r2 ) H 1 CG V i CG V + g Y, (R(x , Z)X) − ∇H X Z 2(1 + r2 ) 2 H X CG g V Y,V Z −CG g CG ∇H X V Y,V Z −CG g V Y,CG ∇H X V Z 2 CG ∇H X CG g V Y,V Z = 0.
+CG g =
V
V
v)(ΦDI CG g)(H X,H Y,H Z) = (DI H X)(CG g(HY,H Z)) −H X(CG g(D I H Y,H Z)) +CG g((LHY D I)H X,H Z) +CG g(H Y, (LH Z D I)H X) X V g(Y, Z) +H X V (g (Y, Z)) H +CG g LH Y D I X −D I LH Y H X ,H Z +CG g H Y, LH Z D I H X −D ILH Z H X = CG g [H Y, H X] −D I[H Y,H X],H Z +CG g H Y, [H Z,H X] −D I[H Z, H X] =
H
[Z, X] − γR(Y, X) ,H Z +CG g H Y,H [Z, X] − γR(Z, X) −D I H [Z, X] − γR(Z, X) = CG g H [Y, X] − γR(Y, X) −H [Y, X] +D IγR(Y, X),H Z +CG g H Y,H [Z, X] − γR(Z, X) −H [Z, X] +D IγR(Z, X) = −2 CG g γR(Y, X),H Z −CG g H Y, γR(Z, X) = 0.
=
CG
g
H
[Y, X] − γR(Y, X) −D I
H
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Applications to the Theory of Lifts vi)(ΦDI CG g)(V X,H Y,H Z) = (DIV X)(CGg(H Y,H Z)) −V X(CG g(D I H Y,H Z)) +CG g((LHY D I)V X,H Z) +CG g(H Y, (LH Z D I)V X) = −2V X CG g H Y,H Z +CG g LH Y D IV X −D I[H Y,V X],H Z +CG g H Y, LH Z D IV X −D I[H Z,V X] = −2V X CG g H Y,H Z +CG g −[H Y,V X] −D I V [Y, X] −V (∇Y X) ,H Z +CG g H Y, −[H Z,V X] −D I V [Z, X] −V (∇Z X) = −2V X CG g H Y,H Z +CG g −[H Y,V X] +V [Y, X] −V (∇Y X) ,H Z +CG g H Y, −[H Z,V X] +V [Z, X] −V (∇Z X)
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= −2V X V (g(Y, Z))
+CG g −V [Y, X] +V (∇Y X) +V [Y, X] −V (∇Y X),H Z +CG g H Y, −V [Z, X] +V (∇Z X) +V [Z, X] −V (∇Z X) = −2V X V (g(Y, Z)) +CG g 0,H Z +CG g H Y, 0 = 0. vii)(ΦDI CG g)(H X,H Y,V Z)
= (DI H X)(CG g(H Y,V Z)) −H X(CG g(DI H Y,V Z)) +CG g((LHY D I)H X,V Z) +CG g(H Y, (LV Z D I)H X) = H X CG g(H Y,V Z) −H X CG g(HY,V Z) +CG g((LHY H X −D I[H Y,H X]),V Z) +CG g(H Y, (LV Z H X −D I[V Z,H X]) =
CG
g([HY,H X] −D I
H
[Y, X] − γR(Y, X ),V Z)
+CG g(H Y, ([V Z,H X] +D IV (∇Y Z)) =
CG
g(H [Y, X] − γR(Y, X) −H [Y, X] − γR(Y, X),V Z) +CG g(H Y, −2V (∇X Z))
= −2CG g(γR(Y, X),V Z). viii) (ΦDI CG g)(H X,V Y,H Z) = −2CG g V Y, γR(Z, X) is analogoue to vii). Therefore, by virtue of Theorem 44 we have
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Arif Salimov
Theorem 120. [75] The almost paracomplex Riemannian manifold (T (Mn ), D I,CG g) is paraholomorphic if and only if M is flat. n
4.6.3.
On Almost Complex Structures in Tangent Bundles
Let ϕ ∈ ℑ11 (Mn ) and ω ∈ ℑ01 (Mn ). It is well known that for an almost holomorphic 1−form ω on a manifold with an almost complex structure ϕ, we have the following equation (see Section 1.2.3 and Capter 2): ω ◦ Nϕ = 0.
(4.116)
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Let ϕ ∈ ℑ11 (Mn ). Then, the complete lift C ϕ of ϕ along the cross-section ω to T ∗ (Mn ) has components of the form ϕhi 0 C ϕ= (∂i ϕai − ∂h ϕai ) − ϕti ∂t ωh + ϕth ∂i ωt ϕhi with respect to the adapted (B,C)−frame [114, p. 308]. We consider that the local vector fields i ∂ X h ∂ C C C X(i) = = δi h = i 0 ∂x ∂x and V
X
(i)
V
i
V
= (dx ) =
(δih dxh ) =
0 δih
i = 1, ..., n; i = n + 1, ..., 2n span the module of vector fields in π−1 (U). Hence, any tensor fields is determined in π−1 (U) by their actions on C X and V θ for any X ∈ ℑ10 (Mn ) and θ ∈ ℑ01 (Mn ). The complete lift C ϕ has the properties C C ϕ( X) =C (ϕ(X)) + γ(LX ϕ), (4.117) C V ϕ( θ) =V (ϕ(θ)), which characterize C ϕ, where ϕ(θ) ∈ ℑ01 (Mn ). Theorem 121. Let Mn be a manifold with an almost complex structure ϕ. Then the complete lift C ϕ ∈ ℑ11 (T ∗ (Mn )) of ϕ, when restricted to the cross-section determined by an almost analytic 1−form on Mn , is an almost complex structure.
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Proof. Let ϕ, ψ ∈ ℑ11 (Mn ) and N ∈ ℑ12 (Mn ). Using (4.20)-(4.24) and (4.117), we have γ(ϕ ± ψ) = γ(ϕ) ± γ(ψ), C
(4.118)
ϕ(γψ) = γ(ψ ◦ ϕ),
(γN)(C X) = γNX , where NX is the tensor field of type (1, 1) on Mn defined by NX (Y ) = N(X,Y ) for any Y ∈ ℑ10 (Mn ).If X ∈ ℑ10 (Mn ), then from (4.117) and (4.118), we have (C ϕ)2 (C X) = (C ϕ ◦C ϕ)(C X) =C ϕ(C ϕ(C X)) =
C
ϕ( ϕ(X)) + γ(LX ϕ)
=
C
ϕ(C (ϕ(X))) +C ϕ(γ(LX ϕ))
= =
C
(ϕ(ϕ(X))) + γ(LϕX ϕ) + γ((LX ϕ) ◦ ϕ) (ϕ ◦ ϕ(X)) + γ(LϕX ϕ + (LX ϕ) ◦ ϕ)
C
(4.119)
C
= (C ϕ)2 (C X) − γ(LX (ϕ ◦ ϕ)) + γ(LϕX ϕ + (LX ϕ) ◦ ϕ) = (C ϕ)2 (C X) + γ(LϕX ϕ − ϕ(LX ϕ))
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= (C ϕ)2 (C X) + γ(Nϕ,X ) = (C ϕ)2 (C X) + (γNϕ )(C X), where Nϕ,X
= (LϕX ϕ − ϕ(LX ϕ))(Y) = [ϕX, ϕY ] − ϕ[X, ϕY ] − ϕ[ϕX,Y ] + ϕ2 [X,Y ] = Nϕ(X,Y )
is nothing but the Nijenhuis tensor constructed by and has local coordinates of the form 0 0 γNϕ = (ω ◦ Nϕ )i j 0 (see (4.24)).Similarly, if θ ∈ ℑ01 (Mn ), then by (4.117), we have (C ϕ)2 (V θ) = (C ϕ ◦C ϕ)(V θ) = =
C
=
V
(ϕ(ϕ(θ)))
=
V
((ϕ ◦ ϕ)(θ))
=
C
(ϕ2 )(V θ).
C
C
V
ϕ( ϕ( θ)) ϕ(V (ϕ(θ))
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By virtue of (4.116), we can easily say that γNϕ = 0. From (4.119), (4.120) and linearity of the complete lift, we have (C ϕ)2 =C (ϕ2 ) =C (IMn ) = −IT ∗ (Mn ) . This completes the proof.
4.7. Complete Lift of a Skew-Symmetric Tensor Field Suppose now that S ∈ ℑ12 (Mn ) is a skew-symmetric tensor field of type (1, 2) with local components Skij , that is, S(V,W) = −S(W,V), ∀V,W ∈ ℑ10 (Mn ). A tensor field ξ ∈ ℑqp (Mn ) is called pure with respect to S ∈ ℑ12 (Mn ), if (
m...h
h
h ...h
h ...h
m...h
h
h ...h
h ...h
...m 1 p 1 p m Shml1 2 ξk1 ...kpq = ... = Smlp 2 ξkh11...k = Sm k1 l2 ξm...kq = ... = Skq l2 ξk1 ...m , q ...m 1 p 1 p m Shl11m ξk1 ...kpq = ... = Sl1pm ξkh11...k = Sm l1 k1 ξm...kq = ... = Sl1 kq ξk1 ...m . q
∗ p
p
Let ℑq (Mn ) denote a module of all the tensor fields ξ ∈ ℑq (Mn ) which are pure with respect to S. We consider a pure cross-section σSξ (Mn ) determined by Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.
∗
ξ ∈ ℑqp (Mn ). We define a tensor field c S ∈ ℑ12 (Tqp (Mn )) along the pure crosssection σSξ (Mn ) by c c c c (S (V ,V )) + γ (L S) S( V , V ) = − γ (L S) − γ S 1 2 1 2 V V [V ,V ] 2 1 V V 1 2 1 2 −V ((LV2 S)V1 ◦ ξ) +V ((LV1 S)V2 ◦ ξ) +V ( S[V1 ,V2 ] ◦ ξ), c S(V A,c V ) =V (S (A)), 2 V2 c c S( V1 ,V B) =V (SV1 (B)), c S(V A,V B) = 0, (4.121) p 1 (B), ((L for any V1 ,V2 ∈ ℑ0 (Mn ), A, B ∈ ℑq (Mn ), where SV2 (A), SV1 V1 S)V2 ◦ p c ξ), ((LV2 S)V1 ◦ ξ), (S[V1 ,V2 ] ◦ ξ) ∈ ℑq (Mn ) and call S the complete lift of p S ∈ ℑ12 (Mn ) to Tq (Mn ), p ≥ 1, q ≥ 0 along σSξ (Mn ). Let c SKL1 L2 be components of c S with respect to the adapted (B,C)−frame of the pure cross-section σSξ (Mn ). Then, from (4.21), (4.23) and (4.121) we have
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∼ ∼ ∼ ∼ ∼ cS e K cV L1 cV L2 =c (S (V1 ,V2 ))K + γ((LV S)V )K − γ((LV S)V )K 2 1 1 2 L1 L2 1 2 ∼ ∼ ∼ K V K V K −γ(S[V1 ,V2 ] ) − ((LV2 S)V1 ◦ ξ) + ((LV1 S)V2 ◦ ξ) ∼ +V ( S[V1 ,V2 ] ◦ ξ), (i) ∼ ∼ cS e K V AL1 cV L2 =V (SV (A))K , (ii) 2 L L 2 1 2 ∼ ∼ cS e K cV L1 V BL2 =V (SV (B))K , (iii) L L 1 1 1 2 cS e K V AL1 V BL2 = 0, (iv) L1 L2 (4.122) where ∼ ∼ 0 0 V V (SV1 (B)) = h1 j mh2 ...h p , (SV2 (A)) = h1 j mh2 ...h p , S jmV1 ξk1 ...kq Sm jV2 ξk1 ...kq ∼ 0 V ((LV1 S)V2 ◦ ξ) = h1 h2 ...l...h p j , h ξk1 ...kq V2 (LV1 Slmλ ) ∼ 0 V ((LV2 S)V1 ◦ ξ) = h1 h2 ...l...h p j , h ξk1 ...kq V1 (LV2 Slmλ ) ∼ 0 V ( S[V1 ,V2 ] ◦ ξ) = h1 h2 ...l...h p hλ . ξk1 ...kq Slm[V1 ,V2]m
Substituting K = k in of (4.122) and calculating in the same way as in Section 4.4 we obtain cek (4.123) S l l =c Se kl l =c Se kl l = 0, c Se kl1 l2 = Skl1 l2 1 2
1 2
1 2
When K = k, of (4.122) reduces to ∼ ∼ c e k c l1 c l2 S l1 l2 V 1 V 2
∼
+c Se kl l cV 1 2
∼ l 1 c l2 1 V 2
∼
+c Se kl l cV 1 2
∼ l1 c l 2 1 V 2
∑
∑
∑
=
(4.124)
lh ...h lh ...h lh ...h −ξk12...kq p V1m (LV2 S)hlm1 + ξk12...kq p V2m (LV1 S)hlm1 + ξk12...kq p Shlm1 [V1 ,V2 ]m p p h h2 ...l...h p m h h2 ...l...h p m h hλ V2 (LV1 Slmλ ) ) − ξk11...k S V (L + ξk11...k V 1 2 lm q q λ=2 λ=2 p h h2 ...l...h p hλ Slm [V1,V2 ]m − ξk11...k q λ=2 ∼ c (S (V1 ,V2))k .
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Now, we will study components c Se kl1 l2 ,c Se kl l ,c Se kl l and cSe kl l of the complete 1 2 1 2 1 2 lift c S. When K = k, (ii) and (iv) of (4.122) can be rewritten by virtue of (4.21) and (4.123) as 0 = 0. For a case where K = k, of (4.122) we have c Se kl l = 0. 1 2
When K = k, from of (4.122) we write
c e k V l1 c e l2 S l1 l2 A V2
+ c Se kl l
1 2
V
=
∼
V
+ c Se kl l
e l 2 + c Se k Al1 cV 2 l l
c e k V l1 c e l2 S l l A V2 1 2
e l2 Al 1 cV 2
1 2
(SV2 (A))k
or s ...s cek S l l Ar11 ...rqp V2l2 1 2
V
1 2
V
e l2 Al 1 cV 2
∼
=V (SV2 (A))k ,
mh ...h
s ...s
r
2 p = Shm1jV2j Ak1 ...k = δrk11 ...δkqq Shs11l2 δhs22 ...δhs pp Ar11 ...rqp V2l2 q
which implies
s ...s
r
cek Sll 1 2
= δrk11 ...δkqq Shs11l2 δhs22 ...δhs pp ,
cek Sll 1 2
= δrk11 ...δkqq Shl11s1 δhs22 ...δhs pp ,
(4.125)
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where xl 1 = tr11...rqp . We also have by of (4.122)
s ...s
where xl 2 = tr11...rqp .
r
(4.126)
Thus, by virtue of (4.125) and (4.126), (4.124) reduces to c e k c e l1 c e l2 S l1 l2 V V2
r e l2 e l 1 cV + δrk11 ...δkqq Shs11l2 δhs22 ...δhs pp cV 2
(4.127)
r e l1 cV e l 2 − ξ lh2 ...h p V1m (LV2 S)h1 +δrk11 ...δkqq Shl11s1 δhs22 ...δhs pp cV 2 k1 ...kq lm lh ...h
lh ...h
+ξk12...kq p V2m (LV1 S)hlm1 + ξk12...kq p Shlm1 [V1 ,V2]m p
h h ...l...h p
2 + ∑ ξk11...k q
λ=2 p
h
p
h h ...l...h p
2 V1m (LV2 Slmλ ) − ∑ ξk11...k q
λ=2
h h ...l...h p hλ Slm [V1,V2 ]m
2 − ∑ ξk11...k q
λ=2
=
c
h
V2m (LV1 Slmλ )
∼
(S (V1 ,V2))k .
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Now, using the Yano-Ako operator we will investigate components c Se kl1 l2 . The ∗
Yano-Ako operator on the pure module ℑqp (Mn ) is given by (see Section 1.6) h ...h
p (ΨS ξ)l11l2 k1 ...k q
h ...h
h ...h
h ...h
1 p 1 p 1 p m m = Sm l1 l2 ∂m ξk1 ...kq − ∂l1 (Sk1 l2 ξmk2 ...kq ) − ∂l2 (Sl1 k1 ξmk2 ...kq )
q
q
h ...h
h ...m...h p
hb hb 1 p 1 + ∑ (∂ka Sm l1 l2 )ξk1 ...m...kq + ∑ (∂l1 Sml2 − ∂m Sl1 l2 )ξk1 ...kq a=1 q
b=1
h ...m...h p
+ ∑ (∂l2 Slh1cm − ∂m Shl1cl2 )ξk11...kq
.
c=1
After some calculations we have h ...h
mh ...h p
2 V1l1 V2l2 (ΨS ξ)l11l2 k1 p...kq +V1l1 Shl11m (LV2 ξ)k1 ...k q
mh ...h
(4.128)
mh ...h
2 p 2 p l1 +V2l2 Shml1 2 (LV1 ξ)k1 ...k + ξk1 ...k V1 (LV2 S)hl11m q q
mh ...h
mh ...h
2 p l2 2 p h1 −ξk1 ...k V2 (LV1 S)hl21m − ξk1 ...k Sl2 m [V1,V2 ]l2 q q
p
h h ...m...h p
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2 − ∑ ξk11...k q
b=1 p
p
h h ...m...h p
2 V1l1 (LV2 Shl1bm ) + ∑ ξk11...k q
b=1
V1l2 (LV1 Shl2bm )
h ...m...h p hb Sl2 m [V1 ,V2]l2
+ ∑ ξk11...kq b=1
h ...h
= (LS(V1 ,V2 ) ξ)k11...kqp . for any V1 ,V2 ∈ ℑ10 (Mn ). Using (4.20), (4.126) and (4.127), (4.128) reduces to r h ...h e l2 e l 1 cV V1l1 V2l2 (ΨS ξ)l11l2 k1 p...kq − δrk11 ...δkqq Shs11l2 δhs22 ...δhs pp cV 2
(4.129)
r e l1 cV e l 2 + ξmh2 ...h p V l1 (LV2 S)h1 −δrk11 ...δkqq Shl11s1 δhs22 ...δhs pp cV 2 1 l1 m k1 ...kq mh ...h
mh ...h
2 p h1 2 p l2 Sl2 m [V1,V2 ]l2 V2 (LV1 S)hl21m − ξk1 ...k −ξk1 ...k q q
p
h h ...m...h p
2 − ∑ ξk11...k q
b=2 p
p
h h ...m...h p
2 V1l1 (LV2 Shl1bm ) + ∑ ξk11...k q
b=2
V2l2 (LV1 Shl2bm )
h ...m...h p hb Sl2 m [V1 ,V2]l2
+ ∑ ξk11...kq b=2
∼
= −c (S (V1 ,V2))k .
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Comparing (4.127) and (4.129) we have cek S l1 l2
h ...h
= − (ΨS ξ)l11l2 k1 p...kq .
Thus, the complete lift c S ∈ ℑ12 (Mn ) (S(V,W) = −S(W,V)) has along the pure cross-section σSξ (Mn ) components [40], [86] cS e k =c S k , c Se k = − (ΨS ξ)h1 ...h p , l1 l2 l1 l2 l1 l2 l1 l2 k1 ...kq c Se k =c Sek =c Se k =c Se k = 0, l 1 l2 l1 l 2 l1l2 l1 l 2 rq h 1 h 2 r1 hp cek = δ S ...δ S δ ...δ sp , k1 kq s1 l2 s2 l 1 l2 r r q h h cS e k = δ 1 ...δ S 1 δs 2 ...δhs p k1 kq l1 s1 2 p l l
(4.130)
1 2
h ...h
p with respect to the adapted (B,C)− frame of σSξ (Mn ), where (ΨS ξ)l11l2 k1 ...k is q the Yano-Ako operator.
Remark 30. c S in the form (4.130) is a unique solution of (4.121). Therefore,
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∗
if S is element of ℑ12 (Tqp (Mn )), such that ∗ S(cV1 ,c V2 ) =c (S(V1,V2 )) + γ((LV2 S)V1 ) − γ((L V1 S)V2 ) − γ(S[V1 ,V1 2 ] ) V V V − ((L S) ◦ ξ) + ((L S) ◦ ξ) + ( S ◦ ξ), ∀V1 ,V2 ∈ ℑ0 (Mn ), V V [V ,V ] 2 1 V1 V2 1 2 ∗ p S(V A,c V2 ) =V (SV2 (A)), ∀A ∈ ℑq (Mn ), ∗ S(cV1 ,V B) =V (SV1 (B)), ∀B ∈ ℑqp (Mn ), ∗ S(V A,V B) = 0, ∗
then S =c S.
Remark 31. The equation (4.121) is useful extension of the equation c L(ια) = p ι(LV α), α ∈ ℑq (Mn ) (see Section 4.2) to tensor fields of type (1,2) along the pure cross-section σSξ (Mn ). Remark 32. Let T and Nϕ be a torsion tensor field of connection ∇ and the Nijenhuis tensor field of ϕ ∈ ℑ11 (Mn ), respectively. From (4.130) we easily see that the complete lifts c T and c Nϕ to Tqp (Mn ) along σξ (Mn )are zero if T and Nϕ are zero in the base manifold Mn .
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h ...h
Remark 33. In the case of ∂m ξk11...kqp = 0, (B,C)−frame is considered as a natural frame {∂h , ∂eh } of σSξ (Mn ). Then, from (4.130) we obtain components of c S along the pure-cross section c k Sl1 l2 = Skl1 l2 , Se lk l =c Se kl l =c Se kl l = 0, 1 2 1 2 1 2 cS e k = δr1 ...δrq Sh1 δh2 ...δh p ,c Se k = δr1 ...δrq Sh1 δh2 ...δh p , sp sp k1 k1 kq s1 l2 s2 kq l1 s1 s2 l1 l 2 l 1 l2 h ...h
h ...h
q
h ...h
1 p 1 p 1 p m m c k Sl1 l2 = ∂l1 (Sm k1 l2 )ξmk2 ...kq + ∂l2 (Sl1 k1 )ξmk2 ...kq − ∑ (∂ka Sl1 l2 )ξk1 ...m...kq − a=1 q q h1 ...m...h p h1 ...m...h p hb hb hc h − (∂ S − ∂ S )ξ (∂ S − ∑ l1 ml2 ∑ l2 l1 m − ∂m Sl1cl2 )ξk1 ...kq . m l1 l2 k1 ...kq
b=1
c=1
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with respect to natural frame {∂h , ∂eh } of σSξ (Mn ) in π−1 (U) [68].
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[66] A.A. Salimov, Almost -holomorphic tensors and their properties, Russian Acad. Sci. Dokl. Math. 45 (1992), 602-605. [67] A.A. Salimov, A new method in the theory of liftings of tensor fields in a tensor bundle, translation in Russian Math. (Iz. VUZ) 38 (1994), 67-73. [68] A.A. Salimov, Generalized Yano-Ako operator and the complete lift of tensor fields, Tensor (N.S.) 55 (1994), 142-146. [69] A.A. Salimov, Lifts of poly-affinor structures on pure sections of a tensor bundle, Russian Math. (Iz. VUZ) 40 (1996), 52-59. [70] A.A. Salimov, Nonexistence of Para-Kahler-Norden warped metrics, Int. J. Geom. Methods Mod. Phys. 6 (2009), 1097-1102. [71] A.A. Salimov, On operators associated with tensor fields, J. Geom. 99 (2010), 107-145. [72] A.A. Salimov, A note on the Goldberg conjecture of Walker manifolds, Int. J. Geom. Methods Mod. Phys. 8 (2011), 925-928. [73] A.A. Salimov and F. Agca, On para-Nordenian structures, Ann. Polon. Math. 99 (2010), 193-200.
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[80] A.A. Salimov and M. Iscan, Some properties of Norden-Walker metrics, Kodai Math. J. 33 (2010), 283-293. [81] A.A. Salimov and M. Iscan, On Norden-Walker 4-manifolds, Note Mat. 30 (2010), 111-128. [82] A.A. Salimov and M. Iscan, On the geometry of B-manifolds, Conference on Differential Geometry dedicated to 80-th anniversary of Prof. V.V.Vishnevskii, Kazan. Gos. Univ. Ucen. Zap. 151 (2009), 231-239. [83] A.A. Salimov, M. Iscan and K. Akbulut, Some remarks concerning hyperholomorphic B-manifolds, Chin. Ann. Math. Ser. B 29 (2008), 631340. [84] A.A. Salimov, M. Iscan and K. Akbulut, Notes on para-Norden-Walker 4-manifolds, Int. J. Geom. Methods Mod. Phys. 7 (2010), 1331-1347. [85] A.A. Salimov, M. Iscan and F. Etayo, Paraholomorphic B-manifold and its properties, Topology Appl. 154 (2007), 925-933. [86] A.A. Salimov and A. Magden, Complete lifts of tensor fields on a pure cross-section in the tensor bundle, Note Mat. 18 (1998), 27-37.
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Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
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Index B-metrics, 70 B-tensor, 70 C-holomorphic, 51 '-connections, 18 '-structure, 46 4 ' -operator, 22 ' ;' -operator, 29 S -operator, 15 ' -operator, 32 S -operator, -structure, 46 -operator, 120 Adjoint operator, 2 Adapted frames, 46, 128 Adapted (B,C)-frame, 126 Admissible, 48 A…nor …elds, 2 Algebra of pracomplex numbers, 46 Algebraic -structures on manifolds, 46 Almost complex structure, 51 Associative algebras, 40 Almost product structure, 79 Almost Norden-Walker metrics 93 Anti-Hermitian, 76 Associated 1-form, 9 Cauchy-Riemann conditions, 44 Cheeger-Gromoll metric, 160 Closed 1-form, 9 Complete lift, 128 Complex analitic manifold, 52 Commutative Algebras, 42 Conjugation, 42 Cotangent bundle, 110 Cross-sections, 126 Decomposable, 79 Derivation, 122 Diagonal lift, 138 Dual algebra, 45 Dual-Kähler-Norden,
86
Exact 1-form, 9 Exterior di¤erentiation, Frobenius algebra,
29
42
Generalized Yano-Ako operator, Geodesic, 145 Goldberg conjecture, 102
36
Harmonic, 95 Hessian, 89 Hessian metric, 89 Hybrid tensor, 70 Hyperbolic, 108 Hypercomplex algebra, 40 Hypercomplex connection, 59 Hypercomlex function, 43 Hypercomplex structure, 47 Hypercomplex tensor, 57 Hyperholomorphic submanifold, 136 Hyper-Kähler-Norden manifolds, 73 Holomorphic manifold, 51 Holomorphic mapping, 134 Holomorphic Norden metric, 71 Holomorphic Norden manifold, 71 Holomorphic Norden-Walker metrics, 96 Holomorphic tensor …eld, 57 Horizontal lift, 122 In…nitesimal automorphism, 58 In…nitesimal isometry, 125 Interior product, 4 Integrable -structure, 46 Integrable regular - structure, 50 Invariant, 135 Invariant submanifold, 22 Isotropic Kähler, 99 Jacobi operator, 3 Jacobi tensor …elds, 152 Jordan tensor, 110
Tensor Operators and their Applications, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,
186
Index
Kähler-Norden manifolds, 76 Kähler-Norden-Walker, 96 Killing vector …eld, 125 Kruchkovich tensor, 55 Locally ‡at, 67 Locally product, Module,
80
50
Nijenhuis tensor, 6 Nijenhuis- Shirokov tensor, 6 Non-holonomic object, 144 Nonregular, 53 Norden metric, 70 Norden manifold, 70 Norden-Hessian, 89 Norden-Walker metric, 93 Norden-Walker structure, 94 Norden-Walker 8-manifolds, 113
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Opposite,
105
Para-Cauchy-Riemann conditions, 46 Para-Kähler-Norden, 80 Para-Norden-Walker manifold, 108 Paraholomorphic Cheeger-Gromoll metric, 160 Paraholomorphic, 81 Proper almost complex structure, 94 Pseudo-Riemannian metric, 3 Pure connections, 19 Pure curvature tensors, 64 Pure product, 4 Pure tensor …elds of mixed kind, 21 Pure tensor …elds, 2
Pure torsion, Quasi-Kähler,
61 101
Regular -structure, 47 Regular representation, 41 Ricci operator, 3 Riemann extension, 110 Rigid '-structure, 46 Sasakian Metric, 142 Sche¤ers conditions, 43 Self-adjoint operators, 3 Slebodzinski tensor, 8 Submanifold, 21 Symplectic manifold, 82 Synectic function, 46 Tachibana operators, 4 Tangent bundle, 52 Tensor bundle, 118 Torsion tensors, 61 Totally geodesic, 109 Twin-Norden metric, 76 Vertical lift, 120 Vishnevskii operator, 15 Vranceanu space, 67 Walker metric, 93 Warped product, 83 Warping function, 83 Warped metric, 83 Yano-Ako Operators,
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29