Pseudo-Riemannian Homogeneous Structures [1 ed.] 978-3-030-18151-2

Table of contents :
Introduction......Page 7
Contents......Page 12
1.1.1 Matrix Groups......Page 15
1.1.2 Principal Bundles......Page 16
1.1.3 Connections on Principal Bundles......Page 19
1.1.4 Holonomy......Page 24
1.2.1 Pseudo-Riemannian Connections and G-structures......Page 26
1.2.2 Berger's Theorem......Page 29
1.3.1 Pseudo-Kähler Manifolds......Page 32
1.3.2 Para-Kähler Manifolds......Page 35
1.3.3 Pseudo-quaternion Kähler Manifolds......Page 38
1.3.4 Para-quaternion Kähler Manifolds......Page 42
1.3.5 Sasakian and Cosymplectic Manifolds......Page 45
1.4.1 Definition......Page 48
1.4.2 Invariant Connections......Page 50
2 Ambrose–Singer Connections and Homogeneous Spaces......Page 54
2.1 Symmetric Spaces and Cartan's Theorem......Page 55
2.2 The Ambrose–Singer Theorem......Page 57
2.3 Kiričenko's Theorem......Page 59
2.4 Homogeneous Pseudo-Riemannian Structures......Page 64
3 Locally Homogeneous Pseudo-Riemannian Manifolds......Page 72
3.1.1 Pseudo-Riemannian Manifolds......Page 73
3.1.2 Locally Homogeneous Pseudo-Riemannian Manifolds with Invariant Geometric Structures......Page 83
3.1.3 Local and Global Homogeneity......Page 85
3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds......Page 86
3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces......Page 92
3.4 Examples and the Reductivity Condition......Page 100
4.1 The General Procedure......Page 104
4.2.1 Homogeneous Pseudo-Kähler Structures......Page 109
4.2.2 Homogeneous Para-Kähler Structures......Page 111
4.2.3 Homogeneous Pseudo-Quaternion Kähler Structures......Page 114
4.2.4 Homogeneous Para-Quaternion Kähler Structures......Page 119
4.2.5 Homogeneous Sasakian and Cosymplectic Structures......Page 124
4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds......Page 128
4.3.1 The Riemannian Case......Page 130
4.3.2 The Lorentzian Case......Page 134
4.3.3 Four-Dimensional Homogeneous Pseudo-Riemannian Manifolds......Page 141
5 Homogeneous Structures of Linear Type......Page 145
5.1 Homogeneous Structures in the Class mathcalS1......Page 146
5.2.1 The Non-degenerate Case......Page 148
5.2.2 The Degenerate Case......Page 151
5.2.3 Local Form of the Metrics......Page 158
5.3 Infinitesimal Models, Homogeneous Models and Completeness......Page 166
5.3.1 The Non-degenerate Para-Kähler Case......Page 169
5.3.2 The Non-degenerate Pseudo-Kähler Case......Page 172
5.3.3 The Degenerate Case with λ=-ε2......Page 173
5.3.4 The Degenerate Case with λ=0......Page 176
5.4 Relation with Homogeneous Plane Waves......Page 177
6 Reduction of Homogeneous Structures......Page 182
6.1.1 Reduction of a Homogeneous Structure......Page 183
6.1.2 The Space of Tensors Reducing to a Given Tensor......Page 187
6.2 Reduction in a Principal Bundle......Page 189
6.2.1 Examples......Page 195
6.3 Application to Cosymplectic and Sasakian …......Page 202
7 Where All This Fails: Non-reductive Homogeneous Pseudo-Riemannian Manifolds......Page 207
7.1 Classification and Invariant Metrics in Dimension Four......Page 210
7.2 Geometry of Four-Dimensional Examples......Page 214
7.3 Explicit Invariant Metrics......Page 220
BookmarkTitle:......Page 232
Index......Page 237

Citation preview

Developments in Mathematics

Giovanni Calvaruso Marco Castrillón López

PseudoRiemannian Homogeneous Structures

Developments in Mathematics Volume 59

Series Editors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville, FL, USA Pham Huu Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ, USA Loring W. Tu, Department of Mathematics, Tufts University, Medford, MA, USA

The Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.

More information about this series at http://www.springer.com/series/5834

Giovanni Calvaruso Marco Castrillón López •

Pseudo-Riemannian Homogeneous Structures

123

Giovanni Calvaruso Dipartimento di Matematica e Fisica “Ennio De Giorgi” Università del Salento Lecce, Italy

Marco Castrillón López Facultad de Ciencias Matemáticas Universidad Complutense de Madrid Madrid, Spain

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-030-18151-2 ISBN 978-3-030-18152-9 (eBook) https://doi.org/10.1007/978-3-030-18152-9 Mathematics Subject Classification (2010): 53C30, 53C15, 53C50, 22E60, 53C80 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Maria Grazia, Nicolò, Maria Sofia and the loving memory of my Parents To Tony, Pilar, Patricia, Sergio and Andrés

Introduction

The notion of symmetry, that is, a transformation of a geometric space preserving certain natural properties, plays a major role in Geometry. The attraction that these objects have for mathematicians is not only motivated by the beauty of the constructions themselves, but also because symmetries reveal intrinsic properties of the spaces in which they are defined, which become essential for their deep understanding. In the realm of (pseudo-)Riemannian Geometry, the most natural symmetries are the isometries, that is, diffeomorphisms of the manifold preserving the metric tensor. Most pseudo-Riemannian manifolds do not admit any isometries. For example, the set of positive definite metrics on a compact manifold with no isometries is open and dense for an appropriate topology. This fact shows that the existence of symmetries is reserved for some classes of spaces endowed with special properties. Moreover, one often has to consider manifolds equipped with a richer geometry beyond the one determined by the sole metric tensor. To mention just two main instances, we may think of complex or quaternionic manifolds. Then, isometries are also required to preserve these ingredients and provide more tools for the better understanding of the additional geometric structure. Among the set of manifolds with isometries, homogeneous spaces play a central role. For these spaces, the group of all symmetries may connect any pair of points. In particular, this property implies that the local geometry of the manifold is “everywhere the same”, a rigid behaviour with important consequences that has attracted the attention of many researchers. Homogeneous manifolds are ubiquitous. They can be found in very different situations, such as the set of solutions of differential equations and as plausible models in General Relativity, to mention just two examples. In Physics, the homogeneity condition involves the preservation of the physical laws throughout the space. Regarding this point, the weaker condition of ‘local homogeneity’, where the isometries connecting arbitrary points need not be defined in the whole space but only on neighbourhoods of the points, seems to be enough. Locally homogeneous spaces are also the target of many investigations in Geometry.

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viii

Introduction

A cornerstone for the study of Riemannian homogeneous and locally homogeneous manifolds was laid by W. Ambrose and I. M. Singer in 1958, when they proved the equivalence between homogeneity and the existence of a tensor satisfying certain geometric partial differential equations. In fact, they characterized connected, simply-connected and complete homogeneous Riemannian manifolds 

ðM; gÞ with the existence on M of a linear connection r satisfying 





r g ¼ 0; r R ¼ 0; r S ¼ 0; 

where S ¼ r  r (r being the Levi-Civita connection of g) and R is the curvature of g. These equations are nowadays known as Ambrose-Singer equations. The above result extends É. Cartan’s characterization of locally symmetric spaces and connected, simply-connected and complete symmetric Riemannian manifolds as those Riemannian manifolds whose curvature tensor is invariant under parallel translation. It is interesting to note that the existence of involutive (local) symmetries, which is now the classical definition of (locally) symmetric spaces, was proved by Cartan some years after the true introduction of these spaces as those with parallel curvature. Symmetric spaces are an important subclass of (pseudo-)Riemannian manifolds in general, and of homogeneous spaces in particular. On the one hand they are complicated enough to provide examples for many geometric phenomena, and on the other hand they are sufficiently simple to allow us to work on them explicitly. They constitute a natural generalization of Euclidean spaces, retaining many of their properties while increasing their geometric complexity. The characterization by É. Cartan led to the complete classification of symmetric and locally symmetric Riemannian spaces by É. Cartan himself, through two ingenious group-theoretic formulations of the problem [40, 41]. The first method is based on the holonomy group. More precisely, it exploits the relation between the holonomy algebra and the metric and curvature tensor induced by their parallel translation invariance. The second method is based on the fact that the invariance of the curvature tensor under parallel translation is equivalent to the geodesic symmetry at each point being a local isometry. Passing through a coset representation G=H, the problem is thus reduced to the study of certain involutive automorphisms of semisimple Lie algebras. In the same way, the Ambrose-Singer Theorem is not only a “nice” characterization of homogeneous spaces, but also introduces a new tool for studying the 

geometry of this kind of manifold, namely, the Ambrose-Singer connection r (equivalently, the so-called homogeneous structure (tensor) S). The power of this result became evident afterwards with several publications devoted to the topic and the publication of the book “Homogeneous structures on Riemannian manifolds” by F. Tricerri and L. Vanhecke in 1983 [116]. In this now classical reference, homogeneous structures are classified into eight classes, using only algebraic arguments and a representation theoretical approach. This classification is defined

Introduction

ix

pointwise, by invariance of the orthogonal group, and it extends to the whole manifold by the local homogeneity property. The authors provided a geometric interpretation of some of the classes, as well as some relevant and interesting examples. The book is still a basic reference in the field. This theory was then successfully extended to locally homogeneous Riemannian spaces by several authors, starting from the construction of the canonical Ambrose-Singer connection given by Kowalski and Tricerri in [85]. Since then, several new investigations and contributions have either focused their attention on homogeneous structures or used them as a tool to tackle problems on homogeneous manifolds. In particular, starting from the work of P.M. Gadea and J.A. Oubiña [64], the notion of homogeneous structures has been extended to the case of metrics of arbitrary signature, and the result of Ambrose and Singer has been adapted to pseudo-Riemannian manifolds carrying an additional geometric structure by V.F. Kiričenko [78]. It is important to emphasize the fact that when the metric is not definite, homogeneous structure tensors only describe homogeneous manifolds whose isometry group is reductive, that is, admitting a convenient decomposition of its Lie algebra as a sum of the tangent space of the manifold at a certain point plus the isotropy algebra of the same point. This fact does not yield to any constraint in the Riemannian case, since any homogeneous Riemannian manifold is reductive, but it is indeed an intriguing issue in the pseudo-Riemannian case. The goal of this book is twofold. On the one hand, we intend to describe the state of the art in the study of homogeneous structures from a modern point of view, with special emphasis on holonomy. This collection comprises both classical results and recently published theorems. In our opinion, this will help the reader to understand the power and beauty of homogeneous structures. On the other hand, we shall illustrate a number of applications of homogeneous structures. Sometimes, these applications are motivated by physical problems, though the orientation is mainly geometric. We are particularly interested in manifolds with either complex or quaternionic homogeneous geometries, fields of active current research, together with the interpretation and classification of homogeneous structures. The behaviour of homogeneous structures under reduction is analysed, with unexpected connections with the classification of different geometric structures. The contrast is provided by the last chapter, where we give some insight into non-reductive homogeneous spaces, a field where homogeneous structures are, in principle, useless. We think that this book fills a gap in the literature, between the classical reference [116] mentioned above and the current research activity developed in the last few years. We have tried to make this work as self-contained as possible, and accessible to anyone with a basic knowledge of smooth manifolds, Lie groups and the elementary theory of representations. The potential audience comprises differential geometers, but also any researcher who has to deal with homogeneous spaces, with a special mention to relativists. The outline of the book is the following. In Chap. 1, we first detail some needed information concerning the theory of principal bundles and connections. Moreover,

x

Introduction

we describe the holonomy of a connection, which will play a central role throughout the rest of the book, and present the most relevant results in this context. We then apply this theory to pseudo-Riemannian manifolds and G-structures. After stating Berger’s Theorem, we describe some basic features of the geometric structures we will consider in Chaps. 5 and 6. Chapter 2 is devoted to homogeneous spaces and their Ambrose-Singer connections. We start from Cartan’s characterization of symmetric spaces, which inspired the work of Ambrose and Singer, and then illustrate their characterization of homogeneous spaces in its pseudo-Riemannian version. Finally, we consider Kiričenko’s result, concerning homogeneous spaces carrying some additional structure. Chapter 3 deals with the description of locally homogeneous pseudoRiemannian manifolds in terms of homogeneous structures. In particular, the suitable notion of strongly reductive locally homogeneous pseudo-Riemannian manifolds is described and we show how to reconstruct a strongly reductive locally homogeneous space, possibly equipped with some additional structure P, from the knowledge of its curvature tensor field, the tensor field P, and their covariant derivatives up to some finite order. In Chap. 4, we describe a procedure which leads to the classification of homogeneous structures associated to Ambrose-Singer connections (and Ambrose-Singer-Kiričenko connections whose underlying geometric structure is integrable). We then apply that procedure to the geometric structures that will be treated in subsequent chapters and to the classification of low-dimensional homogeneous pseudo-Riemannian manifolds. Chapter 5 deals with recent results obtained for a special class of homogeneous structures, called of linear type. In particular, we investigate homogeneous "-Kähler and homogeneous "-quaternion Kähler structures of linear type, where pseudo-Kähler and para-Kähler manifolds (respectively, pseudo-quaternion Kähler and para-quaternion Kähler manifolds) are treated in a unified way. In Chap. 6 we study the behaviour of homogeneous structures by reduction under subgroups of the group of isometries. Such a reduction gives rise to new homogeneous structures in the orbit space of the action. Moreover, the reduction process reveals and sheds light on some previously known properties of some homogeneous structures, and this technique allows us to obtain information about homogeneous structures in the unreduced space from homogeneous structures in the orbit space. Applications are described for cosymplectic and Sasakian homogeneous structures of linear type. In the final Chap. 7 we consider a phenomenon peculiar to the proper pseudoRiemannian settings, namely, the existence of homogeneous pseudo-Riemannian manifolds which do not admit any reductive decomposition. Consequently, homogeneous structures cannot be used to investigate them. These spaces appear starting from dimension four. We illustrate the classification, the explicit description of the homogeneous metrics and some aspects of the geometry of the four-dimensional examples.

Introduction

xi

We are indebted to our colleagues and friends, with whom we have enjoyed enlightening conversations about the contents of the book. Our special thanks to Ignacio Luján and Pedro Martínez Gadea for their wise advices and generous help during the preparation of the manuscript. We also wish to thank the anonymous Referees for their valuable suggestions, and the Editors for their careful editing.

Contents

1 G-structures, Holonomy and Homogeneous Spaces . . . . . . . . . . 1.1 Principal Bundles and Connections . . . . . . . . . . . . . . . . . . . 1.1.1 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Connections on Principal Bundles . . . . . . . . . . . . . . . 1.1.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Pseudo-Riemannian Connections, G-structures, and Berger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Pseudo-Riemannian Connections and G-structures . . . 1.2.2 Berger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A Geometric Description of Some G-structures . . . . . . . . . . 1.3.1 Pseudo-Kähler Manifolds . . . . . . . . . . . . . . . . . . . . . 1.3.2 Para-Kähler Manifolds . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Pseudo-quaternion Kähler Manifolds . . . . . . . . . . . . . 1.3.4 Para-quaternion Kähler Manifolds . . . . . . . . . . . . . . . 1.3.5 Sasakian and Cosymplectic Manifolds . . . . . . . . . . . 1.4 Homogeneous Spaces and the Canonical Connection . . . . . . 1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Invariant Connections . . . . . . . . . . . . . . . . . . . . . . .

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2 Ambrose–Singer Connections and Homogeneous Spaces 2.1 Symmetric Spaces and Cartan’s Theorem . . . . . . . . . . 2.2 The Ambrose–Singer Theorem . . . . . . . . . . . . . . . . . 2.3 Kiričenko’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Homogeneous Pseudo-Riemannian Structures . . . . . . .

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3 Locally Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . 3.1 Reductive Locally Homogeneous Manifolds . . . . . . . . . . . . . . . 3.1.1 Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . 3.1.2 Locally Homogeneous Pseudo-Riemannian Manifolds with Invariant Geometric Structures . . . . . . . . . . . . . . . 3.1.3 Local and Global Homogeneity . . . . . . . . . . . . . . . . . . 3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples and the Reductivity Condition . . . . . . . . . . . . . . . . .

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4 Classification of Homogeneous Structures . . . . . . . . . . . . . . . . . 4.1 The General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Homogeneous Pseudo-Kähler Structures . . . . . . . . . . 4.2.2 Homogeneous Para-Kähler Structures . . . . . . . . . . . . 4.2.3 Homogeneous Pseudo-Quaternion Kähler Structures . 4.2.4 Homogeneous Para-Quaternion Kähler Structures . . . 4.2.5 Homogeneous Sasakian and Cosymplectic Structures . 4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Riemannian Case . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Lorentzian Case . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Four-Dimensional Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Homogeneous Structures of Linear Type . . . . . . . . 5.1 Homogeneous Structures in the Class S 1 . . . . . . 5.2 The Pseudo- and Para-Kähler Cases . . . . . . . . . . 5.2.1 The Non-degenerate Case . . . . . . . . . . . . 5.2.2 The Degenerate Case . . . . . . . . . . . . . . . 5.2.3 Local Form of the Metrics . . . . . . . . . . . 5.3 Infinitesimal Models, Homogeneous Models and Completeness . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Non-degenerate Para-Kähler Case . . 5.3.2 The Non-degenerate Pseudo-Kähler Case 5.3.3 The Degenerate Case with ‚ ¼  2 . . . . . 5.3.4 The Degenerate Case with ‚ ¼ 0 . . . . . . 5.4 Relation with Homogeneous Plane Waves . . . . .

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6 Reduction of Homogeneous Structures . . . . . . . . . . . . . . . . . 6.1 Reduction by a Normal Subgroup of Isometries . . . . . . . . 6.1.1 Reduction of a Homogeneous Structure . . . . . . . . 6.1.2 The Space of Tensors Reducing to a Given Tensor 6.2 Reduction in a Principal Bundle . . . . . . . . . . . . . . . . . . . 6.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Application to Cosymplectic and Sasakian Homogeneous Structures of Linear Type . . . . . . . . . . . . . . . . . . . . . . . . 7 Where All This Fails: Non-reductive Homogeneous Pseudo-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . 7.1 Classification and Invariant Metrics in Dimension Four 7.2 Geometry of Four-Dimensional Examples . . . . . . . . . . 7.3 Explicit Invariant Metrics . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Chapter 1

G-structures, Holonomy and Homogeneous Spaces

In the following sections we establish the foundations needed for the subsequent chapters. The reader is assumed to be familiar with the notions of smooth manifolds, tensorial algebra, pseudo-Riemannian geometry, Lie groups and actions of groups. From this starting point, we introduce principal bundles and connections, with special emphasis on the frame bundle and its reductions, that is, the so-called G-structures. The horizontal lift of paths in principal bundles with respect to a connection leads to the notion of holonomy, which is essential throughout this book. From this perspective, both G-structures and holonomy contain the geometry of the manifolds where they are defined. In particular, they will be extensively used and applied to the main characters of this volume: homogeneous spaces. They are defined together with their canonical connections in the final section. We have tried to give a panoramic view of the required results, comprising the classical constructions and new approaches when needed. Regarding the former, the reader can find them in the literature, so we have generally omitted the proofs. They can be found in any classical texts in the field, but in particular we refer the reader to [79], from which we have borrowed the notation. All objects are assumed to be smooth and manifolds are always finite dimensional unless otherwise indicated.

1.1 Principal Bundles and Connections 1.1.1 Matrix Groups We first recall the definitions of the classical groups of matrices. Even though we will follow the standard notation for them, it will be useful to have their precise definitions, for they will be extensively used throughout the book. The general linear

© Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_1

1

2

1 G-structures, Holonomy and Homogeneous Spaces

group G L(n, K) consists of n × n invertible matrices with coefficients in a field K. For our purposes, K = R or C. From this, we have the following subgroups: • The special group S L(n, K) ⊂ G L(n, K) is the subgroup of matrices with determinant 1. • The orthogonal group of signature ( p, q), p + q = n, is defined as O( p, q) = . ., 1, −1, .(q) . ., −1) {A ∈ G L(n, R)/At D A = D}, where the matrix D is diag(1, .( p) t and A denotes the transpose of A. The case of positive bilinear forms, that is q = 0, is simply denoted by O(n). • The special orthogonal group is S O( p, q) = O( p, q) ∩ S L(n, R). In particular, S O(n) = O(n) ∩ S L(n, R). • The unitary group of signature ( p, q), p + q = n, is defined as U ( p, q) = {A ∈ G L(n, C)/ A¯ t D A = D}, where A¯ denotes the complex conjugate of A. For q = 0 the notation is simply U (n). • The special unitary group is SU ( p, q) = U ( p, q) ∩ S L(n, C), SU (n) = U (n) ∩ S L(n, C). • The complex general group G L(n, C) and all its subgroups can be canonically embedded in G L(2n, R) as  G L(n, C)  A →

Re A −Im A Im A Re A

 ∈ G L(2n, R).

• The compact symplectic group of signature ( p, q), p + q = n, can be defined as Sp(q, p) = {A ∈ S O(4 p, 4q)/AI = I A, A J = J A, AK = K A}, where ⎛

0 ⎜Id I =⎜ ⎝0 0

−Id 0 0 0

0 0 0 Id

⎞ 0 0 ⎟ ⎟ −Id⎠ 0 ⎛

0 ⎜0 K =⎜ ⎝0 Id

0 0 Id 0

⎛

0 ⎜0 J =⎜ ⎝Id 0 0 −Id 0 0

0 0 0 −Id

−Id 0 0 0

⎞ 0 Id⎟ ⎟ 0⎠ 0

⎞ −Id 0 ⎟ ⎟. 0 ⎠ 0

The rest of the groups used in this book will be introduced when needed.

1.1.2 Principal Bundles Definition 1.1.1 Let P and M be manifolds, and let G be a Lie group. A principal bundle P(M, G) is a surjective submersion π : P → M such that G acts smoothly, freely and transitively on the right on the fibers of the projection π.

1.1 Principal Bundles and Connections

3

If Ra (u) = a · g, a ∈ G, u ∈ P stands for the action of G on P, we recall that the action is said to be free if given u 1 , u 2 ∈ P, u 1 = u 2 , the equality Ra (u 1 ) = Ra (u 2 ) necessarily implies a = e ∈ G, the neutral element of G. Transitivity means that for u 1 , u 2 ∈ π −1 ( p), p ∈ M, there always exists a ∈ G such that Ra (u 1 ) = u 2 . From these properties, we have that dim P = dim M + dim G. The manifolds P and M are called the total space and the base space respectively, and G is called the structure group. Principal bundles of the structure group G are usually called G-principal bundles. The choice of right actions instead of left action in the definition of principal bundles is consistent with the standard consensus in the literature, and it is motivated by the main Example 1.1.4. In any case, left principal bundles are also perfectly valid objects and useful in other contexts. Principal bundles fall inside the realm of fiber bundles, a notion that encodes a local triviality property of the total space with respect to the projection π and another manifold F, called typical fiber (the reader is referred to the classical work [71]). In fact, local triviality is sometimes included in the definition of principal bundles, however, as the next result shows, the rigid nature of actions and submersions guarantees the existence of local trivializations so that its inclusion is not necessary. Proposition 1.1.2 Given a G-principal bundle π : P → M, around every p0 ∈ M, there is a neighbourhood U which is the domain of a local section s, that is, a smooth map s : U → π −1 (U ) such that π ◦ s = id. Then, φ : U × G → π −1 (U ) ( p, a) → Ra (σ( p)) is a diffeomorphism such that π ◦ φ( p, a) = p and φ( p, b · a) = Ra (φ( p, b)). Proof We consider u 0 ∈ π −1 ( p0 ). As π is a submersion, there are coordinate systems (V ; x 1 , . . . , x n , y 1 , . . . , y m ) and (U : x 1 , . . . , x n ) around u 0 and p0 respectively such that π(x 1 , . . . , x n , y 1 , . . . , y m ) = (x 1 , . . . , x n ). We define the local section as s(x 1 , . . . , x n ) = (x 1 , . . . , x n , y 1 ( p0 ), . . . , y m ( p0 )). The map φ in the statement is obviously smooth and bijective as the action of G is free and transitive. If (ξ, ζ) belongs to the kernel of dφ : T p M × Te G → Ts( p) P at ( p, e), then ξ = dπ(φ(ξ, ζ)) = 0, and ζ = 0 from the freedom of the action. Then dφ is bijective along the image of s, and also in π −1 (U ) by composing with Ra , a ∈ G, so that φ is a diffeomorphism.  Following the construction of the map φ in the previous proposition, we have the following result. Proposition 1.1.3 A principal G-bundle π : P → M admits a global section s : M → P if and only it admits a global trivialization, that is, a diffeomorphism φ : M × G → P such that φ( p, ba) = Ra (φ( p, b)), for p ∈ M, a, b ∈ G.

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1 G-structures, Holonomy and Homogeneous Spaces

The most important example of a principal bundle for our purposes is the so-called bundle of frames. Example 1.1.4 Let M be a manifold of dimension m. We consider the set

 L(M) = u = ( p; u 1 , . . . , u n ) : p ∈ M, (u 1 , . . . , u m ) is a basis of T p M , which is easily seen to have the structure of a differentiable manifold. The natural projection π : L(M) → M defines a principal bundle structure with structure group GL(m, R). The action of a matrix a = (ai j ) ∈ GL

(m, R) on u = ( p; u 1 , . . . , u m ) is u1, . . . , u m ) with u j = i ai j u i . For the sake of simplicity defined by Ra (u) = ( p; we will often omit the point p when describing a frame and we will only write u = (u 1 , . . . , u m ). In addition, it will be very useful to interpret frames as linear isomorphisms u : Rm → Tp M m ηi u i . η → i=1 From this point of view, the right action is simply the composition of linear maps Ra (u) = u ◦ a. Given any vector bundle π : E → M, we generalize the idea to L(E) → M as the bundle of all possible frames of the fibres of π. Hence L(M) = L(T M), the tangent bundle of M. Principal bundles also serve as generators of other fiber bundles. More precisely, given a G-principal bundle π : P → M, and a left action of G on another manifold F, we consider the right action of G on P × F defined as (u, f ) · a = (u · a, a −1 · f ). The quotient E = P ×G F = (P × F)/G together with the projection π E : E → M, π E ([u, v]G ) = π(u) is a fiber bundle called the associated bundle to P(M, G) with fiber F. When F = V is a vector space and G acts linearly on V , the associated bundle E = P ×G V is a vector bundle. In fact, any vector bundle E → M with typical fiber V is the associated bundle of the principal bundle L(E) → M with respect to the natural action of G L(V ) on V . The main construction of associated bundles in our context is the following. Example 1.1.5 Let V = Rm be endowed with the standard left action of GL(m, R). One can easily check that the associated vector bundle E = L(M) ×GL(m,R) Rm is isomorphic to the tangent bundle T M of M. In the same way, the vector bundle Tsr (M) of tensor fields of type (r, s) on M can be modelled as the vector bundle associated to L(M) with fiber V = (⊗s (Rm )∗ ) ⊗ (⊗r Rm ). Given an associated bundle E = P ×G F, there is a one to one correspondence between equivariant maps f : P → V (that is, maps such that f (Ra (u)) = a −1 · f (u)) and sections σ : M → E. We associate to every equivariant map f the section

1.1 Principal Bundles and Connections

5

σ( p) = [u, f (u)]G , where u is any element in π −1 ( p). Conversely, we associate to every section σ the equivariant map f (u) = η, where [u, η]G = σ(π E (u)). A homomorphism between two principal bundles P (M , G ) and P(M, G) is a map  : P → P together with a homomorphism of Lie groups γ : G → G such that  Ra (u ) = Rγ(a ) ( u ). Each homomorphism  of principal bundles induces a map ψ : M → M with π ◦  = ψ ◦ π . Definition 1.1.6 We say that P (M , G ) is a subbundle of P(M, G) if there is a homomorphism i : P (M , G ) → P(M, G) such that i : P → P is an embedding and γ : G → G is a monomorphism. If moreover M = M , and the map induced in the base manifolds is the identity transformation, then P (M , G ) is called a reduction of P(M, G) to the structure group G . For us, the most important examples of reduction are the so-called G-structures, that is, reductions of L(M) to a subgroup G ⊂ GL(m, R). The reason is that under suitable conditions a G-structure will determine a geometric structure on M and vice versa. Example 1.1.7 Let (M, g) be a pseudo-Riemannian manifold with signature (r, s). We consider the set 

O(M) = u ∈ L(M)/ u is an orthonormal basis of (Tπ(u) M, gπ(u) ) . The natural inclusions i : O(M) → L(M) and i : O(r, s) → GL(m, R) determine a reduction of L(M) to the structure group O(r, s). Conversely, every O(r, s)-reduction P of L(M) determines a pseudo-Riemannian metric g on M so that P is the bundle of orthonormal frames of g. Indeed, for every point p ∈ M, g p is the image of the canonical metric on Rm with signature (r, s) under any isomorphism u : Rm → T p M, u ∈ π −1 ( p). Remark 1.1.8 Let P (M, G ) be a reduction of P(M, G). Let V be a vector space on which G acts on the left (hence so does G by restriction). It is easy to see that the associated bundles to P (M, G ) and P(M, G) with fiber V are isomorphic, that is, P ×G V = P ×G V. In particular, this implies that the definition of a section of these associated bundles is equivalent to give either a G-equivariant map P → V , or a G -equivariant map P → V .

1.1.3 Connections on Principal Bundles For every u ∈ P we define the vertical subspace Vu P ⊂ Tu P, u ∈ P, as the tangent space to the fiber π −1 (π(u)) at u.

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1 G-structures, Holonomy and Homogeneous Spaces

Definition 1.1.9 Let g be the Lie algebra of G and let A ∈ g. The fundamental vector field A∗ associated to A is defined by A∗u

 d  = Rexp(t A) (u), dt t=0

u ∈ P.

As the action of G is free and transitive, we have that A → A∗u is an isomorphism between g and Vu P, for any u ∈ P. On the other hand, it is easy to see that [A∗ , B ∗ ] = [A, B]∗ , so we have an injective Lie algebra homomorphism between g and X(P). Furthermore, a direct calculation shows that (Rb )∗ (A∗ ) = (Ad(b−1 )A)∗ . Remark 1.1.10 In the literature, the name “fundamental vector field” always refers to the infinitesimal generators with respect to an action on a manifold. Such an action can be either “right”(as for principal bundles) or “left” (as the usual actions on manifolds by isometries). The latter leads us to define the fundamental vector field X ∗ on G/H associated to X ∈ g as X ∗p =

 d  L exp(t X ) ( p), dt t=0

p ∈ G/H,

(1.1.1)

from which one gets [X ∗ , Y ∗ ] = −[X, Y ]∗ ,

 ∗ Adg (X ) L g ( p) = (L g )∗ p (X ∗p ).

To avoid misunderstandings and at the same time to keep the notations largely used in literature, we shall use the same notation (super star) in both cases, but we shall refer to the one introduced in the above Definition 1.1.9 as the “fundamental vector field”, whereas we shall call the infinitesimal generator of the action (or vector field defined by the subgroups of isometries exp(t A)) the vector field defined by (1.1.1). Definition 1.1.11 A connection  on a principal bundle P(M, G) is a G-equivariant distribution H P complementary to the vertical distribution VP, that is, for every u ∈ P we can write Tu P = Hu P ⊕ Vu P smoothly with respect to u and such that (Ra )∗ (Hu P) = H Ra (u) P,

u ∈ P, a ∈ G.

H P is called the horizontal distribution. Given X u ∈ Tu P, we can write X u = X uh + X uv , where X uh and X uv denote respectively the horizontal and vertical part of X u with respect to . We define a 1-form ω on P with values in g by ωu (X u ) = A, where A is the unique element of g whose fundamental vector field satisfies A∗u = X uv . The form ω is called the connection form of . As a simple inspection shows, there is a one to one correspondence between connections  on P(M, G) and 1-forms ω on P with values in g, satisfying 1. ω(A∗ ) = A for every A ∈ g.   2. ω is G-equivariant, i.e., (Ra )∗ ω = Ad a −1 ω for every a ∈ G.

1.1 Principal Bundles and Connections

7

Definition 1.1.12 Let X p ∈ T p M and let u ∈ π −1 ( p). We define the horizontal lift of X p to u as the unique vector X uH ∈ Hu P such that π∗ (X uH ) = X p . We thus have that for every vector field X ∈ X(M) there is a unique horizontal vector field X H such that it is G-equivariant and π∗ (X H ) = X . In addition one has [X H , Y H ]h = [X, Y ] H . A C 1 curve on P is called horizontal if its tangent vectors are horizontal at every point. In this way, for every C 1 curve τt on M and every u 0 ∈ P, there is a unique horizontal curve τ¯t on P such that τ¯0 = u 0 and π(τ¯t ) = τt . The curve τ¯t is called the horizontal lift of τt to u 0 with respect to the connection . The endpoint u 1 = τ¯1 will be a point in the fiber π −1 (τ1 ). This defines a map (which we will also denote by τ ) τ : π −1 (τ0 ) → π −1 (τ1 ) u 0 → u 1 called the parallel transport along the curve τ with respect to the connection . It is immediate that the parallel transport commutes with the action of G, that is, Ra ◦ τ = τ ◦ Ra , and that it is independent of the parametrization of τ . In addition, the parallel transport along the inverse curve of τt is the map τ −1 (in particular, τ is an isomorphism) and the parallel transport along the composition of two curves is the composition of the corresponding maps. Definition 1.1.13 Let  be a connection on P(M, G) and ω its connection form. The 2-form  with values in g defined by (X, Y ) = dω(X h , Y h ) is called the curvature form of .  is horizontal and satisfies Ra∗  = Ad(a −1 ). Theorem 1.1.14 (Structure equation) Let  be the curvature of a connection ω. Then (X, Y ) = dω(X, Y ) + [ω(X ), ω(Y )], where the bracket is the Lie algebra bracket of g. Remark 1.1.15 Note that if X, Y are horizontal vectors then (X, Y ) = −ω([X, Y ]), so that the curvature form gives the vertical part of the bracket of two horizontal vector fields. In particular, the curvature of a connection can be regarded as the obstruction to the local integrability, in the sense of Fröbenius, of its horizontal distribution. Let  : P (M , G ) → P(M, G) be a homomorphism of principal bundles with homomorphism of Lie groups γ : G → G, and with ψ : M → M a diffeomorphism. Let  be a connection on P (M , G ) with connection form and curvature form ω and  respectively. Then there is a unique connection  on P(M, G) such that  takes the horizontal subspaces of P with respect to  to the horizontal subspaces of P with respect to . Moreover, let ω and  be the connection form and curvature form of . Then,  ∗ ω = γ ◦ ω and  ∗  = γ ◦  . Under these conditions we say that  takes  to . In the particular case when P (M , G ) is a

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1 G-structures, Holonomy and Homogeneous Spaces

reduction of P(M, G), we say that  is reducible to P (M , G ). On the other hand, if an automorphism  of a principal bundle P(M, G) maps a connection  to itself, we say that  is invariant under . We now relate the notion of a connection on a principal bundle to the well-known notion of covariant derivative on a vector bundle. Let P(M, G) be a principal bundle and  a connection on P(M, G). Let E be an associated vector bundle with fiber a vector space V . We can endow E with a notion of parallel transport inherited from  in the following way. For w ∈ E we define the vertical subspace Vw E ⊂ Tw E as the tangent space to the fiber π −1 E (π E (w)) at w. In order to define a horizontal subspace we consider the natural projection P × V → E = P ×G V (u, η) → (u, η)G , and we take a point (v, ξ) such that w = (v, ξ)G . Fixing ξ, we consider the map P→E u → (u, ξ)G . Then, Hw E is defined as the image of Hv P under the differential of this map (which is independent of the choice of (v, ξ)). It is easy to see that Tw E = Vw E ⊕ Hw E. In this way, a curve in E is said to be horizontal if its tangent vectors are horizontal at every point. As expected, given a curve γt in M and a point w0 in the fiber of γ0 , there is a unique horizontal lift γ¯ t in E starting at w0 . Therefore, the parallel transport along a curve γt , 0 ≤ t ≤ 1, is defined analogously to the case of principal −1 bundles, resulting in this case in a linear isomorphism γ : π −1 E (γ0 ) → π E (γ1 ). A section φ : M → E will be called parallel whenever φ∗ (T p M) ⊂ Hφ( p) E for every p ∈ M, or equivalently, if the parallel transport along any curve γt maps φ(γ0 ) to φ(γ1 ). We shall denote by γtt12 the parallel transport along γ between γt1 and γt2 . Definition 1.1.16 Let φ be a section of E and γt , − ≤ t ≤ , a curve in M. The covariant derivative of φ along γ at γ0 is given by 1 ∇γ˙ 0 φ = lim [γt0 (φ(γt )) − φ(γ0 )] ∈ π −1 E (γ0 ). t→0 t The covariant derivative of φ at p ∈ M in the direction of a tangent vector X p ∈ T p M is just defined as the covariant derivative of φ along a curve γt at γ0 , where γ0 = p and γ˙ 0 = X p . In addition, the covariant derivative of φ in the direction of a vector field X is the section ∇X φ : M → E p → ∇ X p φ. On the other hand, recall that sections of E can be interpreted as G-equivariant maps φ : P → V . Then, the G-equivariant map corresponding to the section ∇ X φ is

1.1 Principal Bundles and Connections

9

given by X H φ : P → V , that is, the horizontal lift of X differentiating the function φ : P → V. The covariant derivative satisfies the following properties: 1. 2. 3. 4.

∇ X +Y φ = ∇ X φ + ∇Y φ. ∇ X (φ + ψ) = ∇ X φ + ∇ X ψ. ∇ f X φ = f ∇ X φ, for all f ∈ C ∞ (M). ∇ X ( f φ) = f (x)∇ X φ + X ( f )φ(x), for all function f ∈ C ∞ (M).

We now focus on the so-called linear connections. They are connections defined on the principal bundle L(M). Recall that the bundle Tsr (M) of tensors of type (r, s) can be seen as an associated bundle to L(M). In this way one can recover the usual covariant derivative of a tensor field. Hereafter we will interchangeably interpret a reference u ∈ L(M) as a basis of Tπ(u) M or as a linear isomorphism u : Rm → Tπ(u) M, and we will not distinguish between the covariant derivative ∇ and the linear connection . Definition 1.1.17 Let ∇ be a linear connection on M. We define the curvature tensor (field) of ∇ as the (1, 3) tensor field R(X, Y )Z = ∇[X,Y ] Z − ∇ X (∇Y Z ) + ∇Y (∇ X Z ), and the torsion (field) of ∇ as the (1, 2) tensor field T (X, Y ) = ∇ X Y − ∇Y X − [X, Y ]. As tensor fields, R and T are associated to GL(m, R)-equivariant functions from L(M) to the corresponding space of tensors. We now see which functions these are. We define the contact form θ of L(M) as the Rm -valued 1-form given by θ(X u ) = −1 u (π∗ (X u )), for X u ∈ Tu L(M). One can check that θ satisfies Ra∗ θ = a −1 · θ for a ∈ GL(m, R). To every η ∈ Rm we can associate in a unique way a horizontal vector B(η)u ∈ Tu L(M) such that π∗ (B(η)) = u(η). The vector field B(η) is called the standard vector field associated to η. It is obvious that standard vector fields depend on the chosen connection, they are nowhere vanishing for η = 0 and satisfy θ(B(η)) = η, and (Ra )∗ (B(η)) = B(a −1 η) for a ∈ GL(m, R). In addition, for A ∈ gl(m, R) and η ∈ Rm one has [A∗ , B(η)] = B(Aη). The torsion form  of a linear connection  is defined as (X, Y ) = dθ(X h , Y h ). In particular, Ra∗  = a −1 ·  for a ∈ GL(m, R), and it satisfies the structure equation  = dθ + ω ∧ θ. The proof of the following proposition follows immediately. Proposition 1.1.18 Let ∇ be a linear connection on M. 1. The equivariant function associated with the torsion tensor field T of ∇ is

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t : L(M) → (Rm )∗ ⊗ (Rm )∗ ⊗ Rm u → t (u)(η, ξ) = u (B(η)u , B(ξ)u ). 2. The equivariant function associated with the curvature tensor field R of ∇ is r : L(M) → (⊗3 (Rm )∗ ) ⊗ Rm u → r (u)(η, ξ, ζ) = u (B(η)u , B(ξ)u )ζ. Definition 1.1.19 Let M and M be manifolds with linear connections  and  . We say that f : M → M is an affine map if f ∗ : T M → T M takes horizontal curves with respect to  to horizontal curves with respect to  . If f : M → M is moreover a diffeomorphism, then it is called an affine transformation. An affine map satisfies in particular f ∗ (∇ X Y ) = ∇ f∗ X f ∗ Y , f ∗ (R(X, Y )Z ) = R ( f ∗ X, f ∗ Y ) f ∗ Z , and f ∗ (T (X, Y )) = T ( f ∗ X, f ∗ Y ). Any transformation f : M → M induces a transformation of principal bundles f : L(M) → L(M) given by f (u) = ( f ∗ (u 1 ), . . . , f ∗ (u m )). In particular, f preserves fundamental vector fields and the contact form θ. If f is moreover an affine transformation, then f ∗ ω = ω.

1.1.4 Holonomy For every p ∈ M we denote by C( p) the space of C ∞ loops based at p. The subset of contractible loops based at p will be denoted by C 0 ( p). Given τ ∈ C( p) (or τ ∈ C 0 ( p)), we consider the parallel transport along τ with respect to , which is a G-equivariant diffeomorphism τ : π −1 ( p) → π −1 ( p). The set of all parallel transports along loops based at p forms a group with the composition of maps as the group operation. Definition 1.1.20 The group

Hol( p) = {τ : π −1 ( p) → π −1 ( p)/ τ ∈ C( p)} is called the holonomy group of  at p. The subgroup

Hol 0 ( p) = {τ : π −1 ( p) → π −1 ( p)/ τ ∈ C 0 ( p)} is called the restricted holonomy group of  at p. Remark 1.1.21 For simplicity, the loops in C( p) and C 0 ( p) are assumed to be C ∞ , although all the holonomy constructions can be carried out within the class C k ,

1.1 Principal Bundles and Connections

11

0 ≤ k ≤ ∞. However, (see [79, Chap. II, Sect. 7 ]), this extension does not entail any relevant geometrical consequence. It will be very convenient to regard these groups as subgroups of the structure group G in the following way. Let τ ∈ C( p) and u 0 ∈ π −1 ( p) be a fixed point. Hence, there is an element a ∈ G such that τ (u 0 ) = Ra (u 0 ). We can thus identify the automorphism τ with the element a ∈ G, so that Hol( p) is seen as a subgroup Hol(u 0 ) of G, called the holonomy group of  with base point u 0 . Indeed, if μ ∈ C( p) is any other loop with μ(u 0 ) = Rb (u 0 ), since the parallel transport commutes with the action of G, we have μ ◦ τ (u 0 ) = μ(Ra (u 0 )) = Ra (μ(u 0 )) = Ra (Rb (u 0 )) = Rba (u 0 ). In addition, one can see that the inverse automorphism τ −1 determines the inverse element a −1 ∈ G. Considering contractible loops one defines the restricted holonomy subgroup Hol 0 (u 0 ) ⊂ Hol(u 0 ) ⊂ G. A third way to define the holonomy group of  is to consider an equivalence relation: u ∼ v if and only if u and v can be joined by a horizontal curve. Then it is immediate that Hol(u 0 ) = {a ∈ G/ u 0 ∼ Ra (u 0 )}. For u, v ∈ P, if π(u) and π(v) can be joined by a curve, then there is an element a ∈ G with u ∼ Ra (v), so that Hol(u) and Hol(v) are conjugate. The same holds for the restricted groups. The following theorem is one of the most important results in Holonomy Theory. For the proof we refer once again to [79]. Theorem 1.1.22 Let P(M, G) be a principal bundle endowed with a connection , where M is connected and paracompact. Let Hol(u) and Hol 0 (u) be the holonomy group and the restricted holonomy group of  based at u ∈ P. Then, 1. Hol 0 (u) is a connected Lie subgroup of G. 2. Hol 0 (u) is a normal subgroup of Hol(u) and Hol(u)/Hol 0 (u) is countable. These imply that Hol(u) is a Lie subgroup of G, whose connected component containing the identity is Hol 0 (u). Concerning the behavior of the holonomy groups under homomorphisms of principal bundles, the following result holds. Proposition 1.1.23 Let  : P (M , G ) → P(M, G) be a homomorphism of principal bundles. Let γ : G → G and ψ : M → M be the corresponding maps. 1. If ψ is a diffeomorphism and (u ) = u, then γ takes Hol(u ) to Hol(u) and Hol 0 (u ) to Hol 0 (u).

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2. If γ is an isomorphism and (u ) = u, then γ takes Hol(u ) to Hol(u) and Hol 0 (u ) to Hol 0 (u). For u ∈ P fixed, we consider the set P(u) = {v ∈ P/ v ∼ u}. It turns out that P(u) is a principal bundle called the holonomy bundle of  based at u. It is obvious that P(u) = P(v) if and only if u ∼ v, and if u  v then P(u) ∩ P(v) = ∅. As the action of G takes horizontal curves to horizontal curves, we have that Ra : P(u) → P(Ra (u)) is an isomorphism of principal bundles with the corresponding isomorphism of Lie groups Ad(a −1 ) : Hol(u) → Hol(Ra (u)), for any a ∈ G. Since for every u, v ∈ P there exists an element a ∈ G such that u ∼ Ra (v), the holonomy bundles P(u) and P(v) are isomorphic for every u, v ∈ P. Theorem 1.1.24 (Reduction Theorem) Let P(M, G) be a principal bundle with a connection , and let u 0 ∈ P. Then P(u 0 ) is a reduction of P(M, G) to the group Hol(u 0 ). Moreover, the connection  is reducible to P(u 0 ). Theorem 1.1.25 (Holonomy Theorem) Let P(M, G) be a principal bundle with a connection . Let  be the curvature form of  and P(u) its holonomy bundle with base point u ∈ P. Then the Lie algebra of Hol(u) is the subalgebra hol(u) ⊂ g spanned by all the elements of the form v (X, Y ), where v ∈ P(u) and X, Y ∈ Hv P.

1.2 Pseudo-Riemannian Connections, G-structures, and Berger’s Theorem 1.2.1 Pseudo-Riemannian Connections and G-structures Let (M, g) be a pseudo-Riemannian manifold with signature (r, s), and O(M) be the corresponding bundle of orthonormal frames. Definition 1.2.1 A linear connection is called metric if it is reducible to O(M). Proposition 1.2.2 A linear connection ∇ is metric if and only if ∇g = 0. This proposition is a special case of a more general result given at the end of the section. The following theorem is a well-known result. Theorem 1.2.3 There is a unique metric and torsionless linear connection on (M, g), called the Levi-Civita connection of g. It is uniquely determined by 2g(∇ X Y, Z ) = X (g(Y, Z )) + Y (g(X, Z )) − Z (g(X, Y )) + g([X, Y ], Z ) + g([Z , X ], Y ) − g([Y, Z ], X ).

1.2 Pseudo-Riemannian Connections, G-structures, and Berger’s Theorem

13

Unless otherwise specified, hereafter ∇ will denote the Levi-Civita connection of (M, g). Let R be the curvature tensor field of ∇. Since ∇ is uniquely determined by g, we will refer to R as the curvature tensor field of g. We will also interpret R both as a (1, 3)-tensor field and as a (0, 4)-tensor field by means of the formula R X Y Z W = g(R X Y Z , W ). It satisfies the following symmetries: R X Y Z W = −RY X Z W , RXY Z W = RZ W XY ,

SR

XY ZW

= 0 (first Bianchi identity),

XYZ

S(∇

X R)Y Z W V

(second Bianchi identity).

XYZ

The Ricci tensor field  and the scalar curvature τ of g are defined by contraction as m m   X Y = εi Rei X ei Y , ø= ε j e j e j , (1.2.2) i=1

j=1

where {ei } is any orthonormal reference and εi = g(ei , ei ) = ±1. Let p ∈ M, and let π be a 2-plane of T p M. We say that π is non-degenerate if g p restricted to π is non-degenerate. In that case we define K x (π) = R X Y X Y , where {X, Y } is any orthonormal basis of π. The function K p is called the sectional curvature at p. If K p is constant for every non-degenerate 2-plane of T p M and for every p ∈ M, we say that (M, g) is a space of constant sectional curvature. It is a well-known result that in that case the curvature tensor field takes the form R X Y Z W = k (g(X, W )g(Y, Z ) − g(X, Z )g(Y, W )) , where k ∈ R is the value of the sectional curvature. We will see later that the spaces of constant curvature are of great importance in our present work. When (M, g) is in addition connected, simply-connected and complete, it is called a space form. Definition 1.2.4 Let (M, g) and (M , g ) be pseudo-Riemannian manifolds. A map f : M → M is called an isometry if it is a diffeomorphism and the differential f ∗ p : (T p M, g p ) → (T f ( p) M , g f ( p) ) is a linear isometry at every point p ∈ M. We will say that (M, g) and (M , g ) are isometric if there is an isometry between them. We will say that (M, g) and (M , g ) are locally isometric if for every pair of points p ∈ M and q ∈ M there are neighbourhoods U and V of p and q respectively, and an isometry f : U → V with f ( p) = q. Proposition 1.2.5 A diffeomorphism f : M → M is an isometry if and only if the induced map f : L(M) → L(M) restricts to a map f : O(M) → O(M). Proposition 1.2.6 Let f : (M, g) → (M , g ) be a diffeomorphism.

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1. If f is an isometry, then it is an affine map with respect to the Levi-Civita connections of g and g . and ∇ be metric connections on (M, g) and (M , g ). If M is connected 2. Let ∇ and ∇ and f ∗ p is a linear isometry for and f is an affine map with respect to ∇ some point p ∈ M, then f is an isometry. The following result exhibits the rigidity of isometries. Proposition 1.2.7 Let f, h : (M, g) → (M , g ) be two isometries between connected manifolds. If there is a point p ∈ M such that f ( p) = h( p) and f ∗ p = h ∗ p , then f = h. A vector field X ∈ X(M) is called an infinitesimal isometry or a Killing vector field if its one-parameter group of local transformations consists of local isometries. Analogously, X is called an infinitesimal affine transformation if its one-parameter group consists of local affine maps. Proposition 1.2.8 Let X ∈ X(M). The following are equivalent: 1. X is a Killing vector field. 2. L X g = 0. 3. The horizontal lift X H of X with respect to the Levi-Civita connection is tangent to O(M). The set of all isometries f : (M, g) → (M, g) has a group structure with the usual composition of maps. This group is called the isometry group of (M, g), and will be denoted by Isom(M, g) or simply Isom(M). One of the main results concerning the isometry group is the following. Theorem 1.2.9 The isometry group of a pseudo-Riemannian manifold (M, g) with a finite number of connected components is a Lie group with the compact-open topology. We have seen the relation between pseudo-Riemannian metrics and reductions of L(M) to the structure group O(r, s), and how a special connection can be defined in that reduction. This idea can be generalized to other geometries related to some G-structures. We will see under which conditions a connection on L(M) can be reduced to a G-structure.   Let K 0 ∈ (⊗r (Rm )∗ ) ⊗ ⊗l Rm be a tensor of type (r, l). Let H ⊂ GL(m, R) be the stabilizer of K 0 under the action of GL(m, R), that is, H = {a ∈ GL(m, R)/ a · K 0 = K 0 }. Suppose that there is tensor field K of type (r, l) on M, such that the associated equivariant map     k : L(M) → ⊗r (Rm )∗ ⊗ ⊗l Rm

1.2 Pseudo-Riemannian Connections, G-structures, and Berger’s Theorem

15

takes values in the GL(m, R)-orbit of K 0 . Then the set Q = k −1 (K 0 ) defines a reduction of L(M) to the structure group H , that is, an H -structure (see [87, 107]). It is worth noting that Q is the set of references with respect to which K is expressed as K 0 . In addition, if K 0 and K 0 are in the same GL(m, R)-orbit, then their stabilizers H and H are conjugate, and the H -structure defined by K 0 and the H -structure such defined by K 0 are isomorphic. Conversely, let Q be an H -structure  that H is  the stabilizer inside GL(m, R) of a tensor K 0 ∈ (⊗r (Rm )∗ ) ⊗ ⊗l Rm . We define the following H -equivariant map   k : Q → (⊗r (Rm )∗ ) ⊗ ⊗l Rm u → K 0 . This map can be extended to L(M) by GL(m, R)-equivariance, defining in this way a tensor field on M. We have thus proved the following Proposition 1.2.10 Let H be the stabilizer inside GL(m, R) of a tensor K 0 . There is a one to one correspondence between H -structures and tensor fields K on M such that the corresponding equivariant map takes values in the GL(m, R)-orbit of K 0 . Moreover, the following result holds (see for instance [107, Lemma 1.3]). Proposition 1.2.11 Let Q be an H -structure with H the stabilizer inside GL(m, R) of a tensor K 0 . Let K be the associated tensor field on M. A linear connection ∇ K = 0. reduces to Q if and only if ∇ We will say that a G-structure P(M, G) is integrable if there is a linear connection with vanishing torsion which reduces to P(M, G).

1.2.2 Berger’s Theorem We begin this section by showing the relation between Proposition 1.2.11 and the holonomy of a pseudo-Riemannian manifold. We refer the reader to [18, p. 282] for a proof of the following result. Proposition 1.2.12 (Equivalence Principle) Let (M, g) be a pseudo-Riemannian manifold. Let H be the stabilizer inside O(r, s) of a tensor K 0 . The following statements are equivalent: 1. There is a tensor field K on M whose equivariant map k takes values in the O(r, s)-orbit of K 0 and such that ∇ K = 0. 2. There is a reduction Q(M, H ) of O(M) which is integrable. 3. H ol(u 0 ) ⊂ H for u 0 ∈ O(M). Corollary 1.2.13 The holonomy group of a pseudo-Riemannian manifold is contained in O(r, s).

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1 G-structures, Holonomy and Homogeneous Spaces

The celebrated Berger’s Theorem [16, 17] provides a list (which was refined later by several authors) of the possible groups appearing as the holonomy group of an irreducible non-locally symmetric pseudo-Riemannian manifold. This result, in conjunction with the decomposition theorem of de Rham and Wu (see below), gives a classification of the possible geometric structures admitted by a pseudo-Riemannian manifold. Before stating Berger’s Theorem we need some definitions and facts. Definition 1.2.14 Let G be a group and V a vector space. 1. A representation (ρ, V ) of G on V is said to be irreducible if there is no proper invariant subspace of V . 2. Let V be endowed with an inner product ·, ·. A representation (ρ, V ) is said to be orthogonal with respect to ·, · if every automorphism ρ(a) : V → V , a ∈ G, is an isometry with respect to ·, ·. In that case, (ρ, V ) is said to be indecomposable if ·, · is degenerate on every proper invariant subspace of V . In the literature indecomposable representations are also said to be weakly irreducible. It is evident that an orthogonal irreducible representation is always indecomposable. The converse holds only for definite inner products. Let (M, g) be a pseudo-Riemannian manifold of signature (r, s), and let p ∈ M. The parallel transport along a loop based at p with respect to the Levi-Civita connection gives a transformation of O(T p M), which can be identified with O(r, s) by fixing an orthonormal basis. The holonomy group Hol( p) of the Levi-Civita connection is thus seen as a subgroup of O(r, s) which acts orthogonally on (T p M, g p ). We refer to this representation as the holonomy representation. When the holonomy representation is irreducible we say that (M, g) is irreducible, and when the holonomy representation is indecomposable we say that (M, g) is indecomposable. Recall that if (M1 , g1 ) and (M2 , g2 ) are pseudo-Riemannian manifolds, the product M1 × M2 with the metric g = g1 + g2 is a pseudo-Riemannian manifold, whose holonomy group is Holg ( p1 , p2 ) = Holg1 ( p1 ) × Holg2 ( p2 ) acting on T p1 M1 ⊕ T p2 M2 as the product representation. The converse result is stated in the following theorem. Theorem 1.2.15 (de Rham, Wu) Let (M, g) be a pseudo-Riemannian manifold and p ∈ M. Then there exists an orthogonal decomposition of T p M into invariant subspaces T p M = E 0 ⊕ E 1 ⊕ . . . ⊕ El , such that Hol( p) acts trivially on E 0 and indecomposably on E 1 , . . . , El , and

Hol( p) = {id} × Hol( p)| E1 × . . . × Hol( p)| El . Furthermore, if (M, g) is simply-connected and complete, then it is isometric to the product (N0 , g0 ) × (N1 , g1 ) × . . . × (Nl , gl ),

1.2 Pseudo-Riemannian Connections, G-structures, and Berger’s Theorem

17

where (N0 , g0 ) is flat, T p Ni = E i , gi = g| Ei , and Holgi ( p) = Hol( p)| Ei for i = 1, . . . , l. If (M, g) is not simply-connected or complete the previous decomposition holds locally. The previous result was proved in [20, 54] for the Riemannian case, and then extended for metrics with signature in [122]. Definition 1.2.16 A pseudo-Riemannian manifold (M, g) is called locally symmetric if ∇ R = 0, where R is the curvature tensor field of g. Although this is not the original definition of locally symmetric spaces, but rather the characterization achieved by É. Cartan (which will be analyzed in the next chapter), for the sake of simplicity it will be enough for the moment. We are now in position to enounce Berger’s Theorem, the proof of which can be found with geometric arguments in [111]. Theorem 1.2.17 (Berger’s Theorem) Let (M, g) be a pseudo-Riemannian manifold of signature (r, s). If (M, g) is irreducible and non-locally symmetric, then the restricted holonomy group is one of the following: • • • • • • • • • • • • • •

SO(r, s), U( p, q), r = 2 p, s = 2q, SU( p, q), r = 2 p, s = 2q, Sp( p, q), r = 4 p, s = 4q, Sp( p, q)Sp(1), r = 4 p, s = 4q, SO(r, C), r = s, Sp( p)SL(2, R), r = s = 2 p, Sp( p, C)SL(2, C), r = s = 4 p, G2 , r = 0, s = 7, G∗2(2) , r = 4, s = 3, GC2 , r = s = 7, Spin(7), r = 0, s = 7, Spin(4, 3), r = s = 4, Spin(7)C , r = s = 8.

Remark 1.2.18 The initial list of Berger [16, 17] was refined and completed by Bryant, Chi, Merkulov and Schwachhöfer (see [23, 49, 92]). In particular, the initial list of Berger [16] included the group Spin(9) for m = 16, but manifolds with that holonomy are locally symmetric, as claimed by Alekseevsky in [2] and proved by Brown and Gray in [21]. Examples of Riemannian manifolds with holonomy group U( m2 ), SU( m2 ), Sp( m4 ), and Sp( m4 )Sp(1) were constructed by Calabi, Yau and Alekseevsky, and years later Bryant constructed the first examples of Riemannian manifolds with exceptional holonomy G2 and Spin(7) in [22]. Many of these constructions can be found in the book [76]. Note that in the Riemannian setting, the notions of irreducible manifold and indecomposable manifold coincide. Therefore, Theorems 1.2.15 and 1.2.17 provide a

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1 G-structures, Holonomy and Homogeneous Spaces

complete classification of non-locally symmetric Riemannian manifolds and their possible geometric structures. However, this is not the case when metrics with signature are considered. There is a gap between Theorem 1.2.17 (which deals with irreducible manifolds) and Theorem 1.2.15 (which refers to indecomposable manifolds). This problem can be solved by obtaining a classification of indecomposable representations of Lie algebras g ⊂ so(r, s), but the difficulty of this problem is considerably higher than the case of irreducible representations. So far the solution is only known for Lorentzian manifolds and manifolds of index 2 (see [67]), making the Holonomy Theory of pseudo-Riemannian manifolds still an active field of intense research.

1.3 A Geometric Description of Some G-structures Theorem 1.2.17 suggests the study of G-structures with G a group appearing in its list. For practical reasons (to be applied in the following chapters), here we focus our attention to the groups U( p, q), Sp( p, q) and Sp( p, q)Sp(1) by reviewing some of their main geometric features, with special emphasis on the integrability of the G-structures. We want to note that there are other G-structures the structure groups of which do not appear in Theorem 1.2.17. This is because they define geometries that are either locally decomposable or symmetric. However, these possibilities for G still include several geometric structures of great interest. For example, this is the case for para-Kähler and para-quaternion Kähler structures or Sasakian and cosymplectic structures, which will be described below since they will appear in the following chapters.

1.3.1 Pseudo-Kähler Manifolds For a detailed introduction to complex manifolds, we refer to [79, Chap. IX]. An almost complex structure on a manifold M of dimension m = 2n is a reduction of L(M) to the structure group GL(n, C) ⊂ GL(2n, R). This subgroup is the stabilizer of the (1, 1)-tensor on R2n  0 −Idn . Idn 0

 J0 =

Equivalently, an almost complex structure is a (1, 1)-tensor field J on M satisfying J 2 = −Id. This tensor field, seen as a section of End(T M), provides a splitting of the complexified tangent bundle and the complexified cotangent bundle

1.3 A Geometric Description of Some G-structures

T c M = T 1,0 ⊕ T 0,1 ,

19

T ∗c M = T ∗1,0 ⊕ T ∗0,1 ,

corresponding to the eigenspaces of J with eigenvalues ±i respectively. The second splitting defines a bigraduation r (M, C) =



 p,q (M, C)

p+q=r

in the space of complex r -forms. A complex r -form belonging to  p,q (M, C) is said to be of type ( p, q). In the same way, a section of T 1,0 (respectively, T 0,1 ) is called a complex vector field of type (1, 0) (respectively, (0, 1)). The celebrated Newlander–Nirenberg Theorem [96] asserts that the following statements are equivalent: 1. M is a complex manifold, that is, M admits an atlas {Uα } of complex valued coordinates ϕα : Uα → Cn with holomorphic transition functions. 2. M admits an integrable GL(n, C)-structure. 3. M admits a complex structure, that is, an almost complex structure J with vanishing Nijenhuis tensor field N (X, Y ) = J [J X, Y ] + J [X, J Y ] + [X, Y ] − [J X, J Y ]. For a complex manifold (M, J ), d( p,q (M, C)) ⊂  p+1,q (M, C) ⊕  p,q+1 (M, C), so that we have differential operators ∂ :  p,q (M, C) →  p+1,q (M, C),

∂¯ :  p,q (M, C) →  p,q+1 (M, C),

defined by the corresponding projections. These satisfy ∂ 2 = 0, ∂¯ 2 = 0, ∂ ◦ ∂¯ + ∂¯ ◦ ∂ = 0. A function f : M → C is said to be holomorphic if ∂¯ f = 0. In the same way, ¯ = 0. A holomorphic vector field a complex ( p, 0)-form ω is holomorphic if ∂ω is a complex vector field Z of type (1, 0) such that Z ( f ) is holomorphic whenever f : M → C is holomorphic. An analogous definition can be made for antiholomorphic functions, (0, q)-forms, and vector fields of type (0, 1). There is a Lie algebra isomorphism between the set of infinitesimal automorphisms of J (i.e. L X J = 0) and the set of holomorphic vector fields given by X → 21 (X − i J X ). Finally, a mapping f : (M, J ) → (M , J ) between complex manifolds is called holomorphic if J ◦ f ∗ = f ∗ ◦ J . A pseudo-Hermitian metric on (M, J ) is a pseudo-Riemannian metric g such that g(J X, J Y ) = g(X, Y ),

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1 G-structures, Holonomy and Homogeneous Spaces

or equivalently, a reduction to the group structure U( p, q), where the signature of g is (2 p, 2q), p + q = n. In that case, (M, g, J ) is called an almost pseudo-Hermitian manifold, and if J is complex then it is called a pseudo-Hermitian manifold. Definition 1.3.1 An almost pseudo-Hermitian manifold (M, g, J ) is called pseudoKähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) The U( p, q)-structure is integrable. (b) ∇ J = 0, where ∇ is the Levi-Civita connection of g. (c) The holonomy group of g is contained in U( p, q). A pseudo-Kähler manifold (M, g, J ) is in particular a complex manifold with complex structure J . Moreover, it is a symplectic manifold with symplectic form ω(X, Y ) = g(X, J Y ). This relation between symplectic, complex, and pseudoKähler manifolds can be read from U( p, q) = O(2 p, 2q) ∩ GL(n, C) = O(2 p, 2q) ∩ Sp(n, R) = Sp(n, R) ∩ GL(n, C).

Concerning the curvature of a pseudo-Kähler manifold, since the holonomy algebra of g is contained in u( p, q), the curvature tensor field R and the Ricci tensor field  have the following additional symmetries: R J X Y + R X J Y = 0, RXY J Z = J RXY Z , J X J Y = X Y . A real form β is said to be of type (1,1) if β(J X, J Y ) = β(X, Y ). It is easy to see that in that case the complexification of β is a complex form of type (1, 1). The map b → β = b(·, J ·) defines a linear isomorphism between the space of symmetric J invariant bilinear forms (i.e., b(J X, J Y ) = b(X, Y )) and the space of real 2-forms of type (1, 1), and the image of the Ricci tensor field under this isomorphism is called the Ricci form of (M, g, J ). It is a well-known result that the Ricci form of a pseudo-Kähler manifold is closed. Let x ∈ M, and let π be a non-degenerate 2-plane of Tx M. We say that π is complex if it is invariant under J . In that case we define K x (π) = R X J X X J X , where X is a unitary vector of π. The function K x is called the holomorphic sectional curvature at x. If K x is constant for every non-degenerate complex 2-plane of Tx M and for every x ∈ M, we say that (M, g, J ) is of constant holomorphic sectional curvature. One proves that in this case the curvature tensor field takes the form R(X, Y, Z , W ) =

k g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) 4 − g(X, J Z )g(Y, J W ) + g(X, J W )g(Y, J Z )  − 2g(X, J Y )g(Z , J W ) ,

1.3 A Geometric Description of Some G-structures

21

where k ∈ R is the value of the holomorphic sectional curvature. It is straightforward to adapt the arguments from the well-known case of definite metrics to prove that two spaces of constant and equal holomorphic sectional curvature are locally holomorphically isometric (see for instance [10]). When (M, g, J ) has constant holomorphic sectional curvature and is in addition connected, simply-connected and complete, then it is called a complex space form. In that case (M, g, J ) is flat or holomorphically isometric to the symmetric spaces CPnp (k) if k > 0 or CHnp (k) if k < 0, where

SU(n + 1 − p, p) SU(n − p, p + 1) , CHnp (k) = , S(U(n − p, p) × U(1)) S(U(n − p, p) × U(1)) (1.3.3) are endowed with a suitable metric such that its holomorphic sectional curvature is constant and equal to k. For CPpn (k) we recall that S(U (n − p, p) × U (1)) denotes (U (n − p, p) × U (1)) ∩ SU (n + 1 − p, p), where U (n − p, p) × U (1) sits in U (n + 1 − p, p) in a natural way, and the factor U (1) is considered to have positive signature. For CH pn (k) the factor U (1) has negative signature. CPnp (k) =

n Remark 1.3.2 There is a diffeomorphism between CHnp (k) and CPn− p (−k) (for k < 0), which is an isometry up to a change of sign.

1.3.2 Para-Kähler Manifolds For a complete introduction to para-complex geometry we refer for instance to the surveys [3, 53]. Let C = R + eR be the set of para-complex numbers, where e denotes the paracomplex imaginary unit, i.e., e2 = 1. An almost para-complex structure on a 2nC), dimensional manifold M is a reduction of L(M) to the structure group GL(n, seen as the stabilizer of the (1, 1)-tensor on R2n  J0 =

0 Idn Idn 0



inside GL(2n, R). Equivalently, an almost para-complex structure is a (1, 1)-tensor field J on M satisfying J 2 = Id, J = Id, and such that the eigenspaces of Jx , seen as an endomorphism of Tx M, corresponding to the eigenvalues ±1, have the same dimension for every x ∈ M. J provides a splitting of the para-complexified tangent bundle and the para-complexified cotangent bundle T c M = T 1,0 ⊕ T 0,1 ,

T ∗c M = T ∗1,0 ⊕ T ∗0,1 ,

corresponding to the eigenspaces of J with eigenvalues ±e respectively. The second splitting defines a bigraduation

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1 G-structures, Holonomy and Homogeneous Spaces

r (M, C) =



 p,q (M, C)

p+q=r

in the space of para-complex r -forms. A para-complex r -form belonging to  p,q (M, C) is said to be of type ( p, q). In the same way, a section of T 1,0 (resp. T 0,1 ) is called a para-complex vector field of type (1, 0) (resp. (0, 1)). An analogous result to the Newlander–Nirenberg Theorem asserts that the following statements are equivalent: 1. (M, J ) is para-complex, that is, there is an atlas {Uα } of para-complex valued Cn with para-holomorphic transition functions. coordinates ϕα : Uα → C)-structure is integrable. 2. The GL(n, 3. The almost para-complex structure J is para-complex, i.e. it satisfies N = 0, where N (X, Y ) = J [J X, Y ] + J [X, J Y ] − [X, Y ] − [J X, J Y ]. It is worth recalling that one of the differences between complex and para-complex manifolds is that para-complex coordinates may not be real analytic. For a para-complex manifold (M, J ), C)) ⊂  p+1,q (M, C) ⊕  p,q+1 (M, C), d( p,q (M, so that there are differential operators C) →  p+1,q (M, C), ∂ :  p,q (M,

∂¯ :  p,q (M, C) →  p,q+1 (M, C),

defined by the corresponding projections, satisfying ∂ 2 = 0, ∂¯ 2 = 0, ∂ ◦ ∂¯ + ∂¯ ◦ ∂ = 0. A function f : M → C is said to be para-holomorphic if ∂¯ f = 0. In the same ¯ = 0. A para-holomorphic way, a para-complex ( p, 0)-form ω is holomorphic if ∂ω vector field is a para-complex vector field Z of type (1, 0) such that Z ( f ) is paraholomorphic whenever f : M → C is para-holomorphic. An analogous definition can be made for anti-para-holomorphic functions, vector fields of type (0, 1), and (0, q)-forms. There is a Lie algebra isomorphism between the set of infinitesimal automorphisms of J and the set of para-holomorphic vector fields given by X → 21 (X + e J X ). Finally, a map f between two para-complex manifolds (M, J ) and (M , J ) is called para-holomorphic if f ∗ ◦ J = J ◦ f ∗ . A para-Hermitian metric on (M, J ) is a pseudo-Riemannian metric g such that g(J X, J Y ) = −g(X, Y ), or equivalently, a reduction to the structure group

GL(n, R) = O(n, n) ∩ GL(n, C) =



  B 0 /B ∈ Gl(n, R) . 0 (B −1 )T

1.3 A Geometric Description of Some G-structures

23

In that case the signature of g is (n, n), and (M, g, J ) is called an almost paraHermitian manifold. If J is para-complex then (M, g, J ) is called a para-Hermitian manifold. Definition 1.3.3 An almost para-Hermitian manifold (M, g, J ) is called paraKähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) The GL(n, R)-structure is integrable. (b) ∇ J = 0, where ∇ is the Levi-Civita connection of g. (c) The holonomy group of g is contained in GL(n, R). A para-Kähler manifold (M, g, J ) is in particular a para-complex manifold with para-complex structure J . Moreover, it is a symplectic manifold with symplectic form ω(X, Y ) = g(X, J Y ). This relation between symplectic, para-complex, and para-Kähler manifolds can be read from

GL(n, R) = O(n, n) ∩ GL(n, C) = O(n, n) ∩ Sp(n, R) = Sp(n, R) ∩ GL(n, C). Concerning the curvature of a para-Kähler manifold, since the holonomy algebra of g is contained in gl(n, R), the curvature tensor field R and the Ricci tensor field  have the following symmetries: R J X Y + R X J Y = 0, RXY J Z = J RXY Z ,  J X J Y = − X Y . Real forms of type (1, 1) are defined analogously to the complex case, and in the same way, the Ricci form (X, Y ) → (X, J Y ) is a form of type (1, 1), which is closed if the manifold is para-Kähler. Let x ∈ M, and let π be a non-degenerate 2-plane of Tx M. We say that π is paracomplex if it is invariant under J . In that case we define K x (π) = R X J X X J X , where X is a unitary vector of π. The function K x is called the para-holomorphic sectional curvature at x. If K x is constant for every non-degenerate para-complex 2-plane of Tx M and for every x ∈ M, we say that (M, g) is of constant para-holomorphic sectional curvature. It is a well-known result that in this case the curvature tensor field takes the form R(X, Y, Z , W ) =

k g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) 4 + g(X, J Z )g(Y, J W ) − g(X, J W )g(Y, J Z )  + 2g(X, J Y )g(Z , J W ) ,

where k ∈ R is the value of the para-holomorphic sectional curvature. It is straightforward to adapt the arguments from the well-known case of definite metrics to prove that two spaces of constant and equal para-holomorphic sectional curvature

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1 G-structures, Holonomy and Homogeneous Spaces

are locally para-holomorphically isometric [62]. When (M, g, J ) is in addition connected, simply-connected and complete, then it is called a para-complex space form. In that case (M, g, J ) is flat or para-holomorphically isometric to the symmetric space SL(n + 1, R) , (1.3.4) CPn (k) = S(GL(n, R) × GL(1, R)) endowed with a suitable metric such that its para-holomorphic sectional curvature is constant and equal to k.

1.3.3 Pseudo-quaternion Kähler Manifolds An almost quaternionic structure on a manifold M of dimension m = 4n is a reduction of L(M) to the structure group

GL(n, H)Sp(1) = (GL(n, H) × Sp(1))/{±Id} ⊂ SO(4n), where H denotes the set of quaternions. The group GL(n, H) × Sp(1) acts on R4n as (B, q)v = Bv q, ¯

B ∈ GL(n, H), q ∈ Sp(1),

where v is regarded as a vector in Hn and q¯ stands for the conjugate of the unitary quaternion q. This action is not faithful since (−B, −q) and (B, q) induce the same orthogonal transformation, a fact that explains the quotient in the definition of GL(n, H)Sp(1). This subgroup of SO(4n) is the stabilizer of the three-dimensional subspace of End(R4n ) generated by ⎛

0 ⎜Id I0 = ⎜ ⎝0 0

−Id 0 0 0

0 0 0 Id

⎞ 0 0 ⎟ ⎟, −Id⎠ 0 ⎛

0 ⎜0 K0 = ⎜ ⎝0 Id

0 0 Id 0

⎛

0 ⎜0 J0 = ⎜ ⎝Id 0 0 −Id 0 0

0 0 0 −Id

−Id 0 0 0

⎞ 0 Id⎟ ⎟, 0⎠ 0

⎞ −Id 0 ⎟ ⎟. 0 ⎠ 0

Note that {I0 , J0 , K 0 } generates an algebra isomorphic to the one of the imaginary quaternions. For this reason an almost quaternionic structure on M is equivalent to the existence of a 3-rank subbundle Q ⊂ End(M), such that there is a local basis {J1 , J2 , J3 } satisfying J12 = J22 = J32 = −Id,

J1 J2 = J3 .

1.3 A Geometric Description of Some G-structures

25

Two local bases {J1 , J2 , J3 } and {J1 , J2 , J3 } are related by Ja = 3b=1 Cab Jb for certain matrix (Cab ) ∈ SO(3). An almost pseudo-quaternion Hermitian structure on M is an almost quaternionic structure Q and a pseudo-Riemannian metric g of signature (4 p, 4q) such that g(Ja X, Y ) + g(X, Ja Y ) = 0,

a = 1, 2, 3,

that is, Q is a subbundle of so(M). This is equivalent to a reduction to the smaller structure group Sp( p, q)Sp(1) ⊂ GL(n, H)Sp(1). Setting ωa = g(·, Ja ·),

a = 1, 2, 3,

it is easily seen that the 4-form  = ω1 ∧ ω 1 + ω 2 ∧ ω 2 + ω 3 ∧ ω 3 is globally defined. This way, the group Sp( p, q)Sp(1) can be seen as the stabilizer inside SO(4 p, 4q) of a 4-form on R4n constructed from {I0 , J0 , K 0 } and the standard metric of signature (4 p, 4q), analogously to how  is constructed from J1 , J2 , J3 and g. Definition 1.3.4 An almost pseudo-quaternion Hermitian manifold (M, g, Q) is called pseudo-quaternion Kähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) (b) (c) (d)

The Sp( p, q)Sp(1)-structure is integrable. ∇ = 0, where ∇ is the Levi-Civita connection of g. The holonomy group of g is contained in Sp( p, q)Sp(1). For every local basis {J1 , J2 , J3 } of Q ∇ Ja =

3 

cab Jb ,

a = 1, 2, 3,

b=1

with (cab ) a matrix of 1-forms in so(3). The fact that (d) is equivalent to (a), (b) and (c) is actually a result stated by Ishihara [73, 74]. However, since the authors have not found a complete proof of this result, we exhibit a simple proof here. Proof of Ishihara’s Theorem Suppose that (M, g, Q) has holonomy contained in Sp( p, q)Sp(1). Let P be the holonomy bundle, which is a reduction of O(M) to the structure group Sp( p, q)Sp(1). The group Sp(1) can be seen as the group S 3 ⊂ H of quaternions of norm 1 (that is, q q¯ = 1), so that it acts on the imaginary quaternions Im(H) as L q : Im(H) → Im(H) z → qzq.

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1 G-structures, Holonomy and Homogeneous Spaces

Under the identification Im(H) = R3 , L q is an element of SO(3) determining a twofold covering Sp(1) → SO(3). We consider the associated bundle Q = P ×Sp( p,q)Sp(1) R3 , where Sp( p, q)Sp(1) acts on R3 through the action of Sp(1) described above. This vector bundle is seen as a 3-rank subbundle of End(M) by P × Sp(n)Sp(1) R3 → P × Sp(n)Sp(1) End(R4n ) [v, ξ] → (v, ξ1 I0 + ξ2 J0 + ξ3 K 0 ) Sp(n)Sp(1) . Let {e1 , e2 , e3 } be the canonical basis of R3 . It is easy to see that a local section σ of P determines a local basis J1 = (σ, e1 ) Sp(n)Sp(1) , J2 = (σ, e2 ) Sp(n)Sp(1) , J3 = (σ, e3 ) Sp(n)Sp(1) of Q which satisfies J12 = J22 = J32 = −Id,

J1 J2 = J3 ,

that is, Q is an almost quaternionic structure. Let τ be a curve in M and S0 = (u 0 , ξ) Sp(n)Sp(1) ∈ Q. The horizontal lift of τ to S0 in End(M) is just the curve (u t , ξ) Sp(n)Sp(1) , where u t is the horizontal lift of τ to u 0 . The lift u t is contained in P by definition, so that [u t , ξ] is contained in Q. This means that Q is invariant under parallel transport. Therefore, denoting by τ0t the parallel transport along τ from t = 0 to t, there is a matrix (Cab (t)) ∈ SO(3) such that τ0t (Ja ) =

3 

Cab (t)Jb ,

a = 1, 2, 3.

b=1

Differentiating at t = 0 we find that ∇ X Ja =

3 

cab Jb ,

a = 1, 2, 3,

b=1

where X =



d  τ, dt t=0 t

and cab = ∇ X Ja =



d  dt t=0 3 

Cab (t) ∈ so(3). Conversely, suppose that

cab Jb ,

a = 1, 2, 3.

cab ωb ,

a = 1, 2, 3,

b=1

Since ∇g = 0 we have that ∇ X ωa =

3  b=1

from where a straightforward computation shows that ∇ = 0.

1.3 A Geometric Description of Some G-structures

27

The curvature tensor, regarded as an element of 2 so(4 p, 4q), is said to be an algebraic curvature tensor of type sp( p, q) if it sits in 2 sp( p, q). This kind of tensor satisfies the following symmetries: R Ja X Y + R X Ja Y = 0, R X Y Ja Z = Ja R X Y Z ,  = 0.

a = 1, 2, 3, a = 1, 2, 3,

The proof of the following proposition can be found in [4]. Proposition 1.3.5 The curvature tensor field of a pseudo-quaternion Kähler manifold (M, g, Q) decomposes as R = νq R 0 + R sp( p,q) , s , R 0 is four times the curvature tensor field of the pseudowhere νq = 16n(n+2) quaternionic hyperbolic space (of the corresponding signature)

R 0X Y Z W = g(X, Z )g(Y, W ) − g(Y, Z )g(X, W ) +

3 

{g(Ja X, Z )g(Ja Y, W )

a=1

−g(Ja Y, Z )g(Ja X, W ) + 2g(X, Ja Y )g(Z , Ja W )} ,

(1.3.5)

and R sp( p,q) is of type sp( p, q). In particular, (M, g, Q) is Einstein. Let p ∈ M, and let Z ∈ T p M with g(Z , Z ) = 0. We consider the 4-dimensional subspace V (Z ) = Span{Z , J1 Z , J2 Z , J3 Z } ⊂ T p M. Let π ⊂ V (Z ) be a non-degenerate 2-plane and {X, Y } an orthonormal basis of π. If K p (Z )(π) = R X Y X Y is constant for every π, we call K p (Z ) the quaternionic sectional curvature with respect to Z at p. We say that (M, g, Q) has constant quaternionic sectional curvature if K p (Z ) is constant for every Z ∈ T p M and every p ∈ M. It is a well-known result that in this case the curvature tensor field takes the form R=

k 0 R , 4

where R 0 is given by (1.3.5) and k is the value of the quaternionic sectional curvature. One can easily adapt the arguments of the case of definite metrics to prove that two spaces of constant and equal quaternionic sectional curvature are locally isometric preserving their pseudo-quaternion Kähler structures (see for instance [103]). When in addition (M, g, Q) is connected, simply-connected and complete, then it is called a quaternion space form. In that case (M, g, Q) is flat or isometric (preserving the pseudo-quaternion Kähler structures) to the symmetric spaces HPnp (k) if k > 0 or HHnp (k) if k < 0, where

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HPnp (k) =

Sp( p, n + 1 − p) Sp( p + 1, n − p) , HHnp (k) = , Sp( p, n − p)Sp(1) Sp( p, n − p)Sp(1)

(1.3.6)

endowed with a suitable metric such that its quaternionic sectional curvature is constant and equal to k. p

n Remark 1.3.6 There is a diffeomorphism between HHs (k) and HPn− p (−k) (for k < 0), which is an isometry up to a change of sign.

An almost pseudo-hyper-complex structure on M, dimM = 4n, is a 3-rank subbundle of End(M) which admits a global basis {J1 , J2 , J3 } satisfying J12 = J22 = J32 = −Id,

J1 J2 = J3

or, equivalently, a reduction of L(M) to the group GL(n, H), which is seen as the common stabilizer of I0 , J0 , K 0 inside GL(4n, R). An almost pseudo-hyper-Hermitian structure on M consists of an almost pseudo-hyper-complex structure J1 , J2 , J3 and a pseudo-Riemannian metric of signature (4 p, 4q) such that g(Ja X, Y ) + g(X, Ja Y ) = 0,

a = 1, 2, 3,

or equivalently, a reduction to the structure group Sp( p, q). We thus have Definition 1.3.7 An almost pseudo-hyper-Hermitian manifold is called pseudohyper-Kähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) The Sp( p, q)-structure is integrable. (b) ∇ Ja = 0 for a = 1, 2, 3, where ∇ is the Levi-Civita connection of g. (c) The holonomy group of g is contained in Sp( p, q). It is evident that the curvature tensor field of a pseudo-hyper-Kähler manifold is of type sp( p, q).

1.3.4 Para-quaternion Kähler Manifolds denote the set of para-quaternions (also known as split-quaternions). An almost Let H para-quaternionic structure on a manifold M of dimension m = 4n is a reduction Sp(1, R). The group GL(n, H)Sp(1, R) of L(M) to the structure group GL(n, H) can be seen as the stabilizer inside GL(4n, R) of the three dimensional subspace of End(R4n ) generated by ⎛

0 ⎜Id I0 = ⎜ ⎝0 0

−Id 0 0 0

0 0 0 Id

⎞ 0 0 ⎟ ⎟, −Id⎠ 0

⎛

0 ⎜Id J0 = ⎜ ⎝0 0

Id 0 0 0

0 0 0 Id

⎞ 0 0⎟ ⎟, Id⎠ 0

1.3 A Geometric Description of Some G-structures

⎛

−Id ⎜ 0 K0 = ⎜ ⎝ 0 0

0 Id 0 0

29

0 0 −Id 0

⎞ 0 0⎟ ⎟. 0⎠ Id

Note that {I0 , J0 , K 0 } generates an algebra isomorphic to the set of imaginary paraquaternions. For this reason, the existence of an almost para-quaternionic structure on M is equivalent to the existence of a 3-rank subbundle Q ⊂ End(M) such that there is a local basis {J1 , J2 , J3 } satisfying J12 = −Id,

J22 = J32 = Id,

J1 J2 = J3 .

Two local bases {J1 , J2 , J3 } and {J1 , J2 , J3 } are related by Ja = 3b=1 Cab Jb for a certain matrix (Cab ) ∈ SO(1, 2). A pseudo-Riemannian manifold of signature (r, s) is said to be strongly oriented if the bundle of orthonormal frames can be reduced to the connected component SO0 (r, s) (since SO(r, s)/SO0 (r, s) is discrete, there always exists a strongly oriented cover of M). An almost para-quaternion Hermitian structure on a strongly oriented pseudoRiemannian manifold (M, g) of signature (2n, 2n) is an almost para-quaternionic structure Q such that g(Ja X, Y ) + g(X, Ja Y ) = 0,

a = 1, 2, 3,

that is, Q is a subbundle of so(M). This is equivalent to a reduction to the structure group Sp(n, R)Sp(1, R) ⊂ SO0 (2n, 2n). Let ωa = g(·, Ja ·),

a = 1, 2, 3.

The 4-form  = ω1 ∧ ω 1 − ω 2 ∧ ω 2 − ω 3 ∧ ω 3 is readily seen to be globally defined. The group Sp(n, R)Sp(1, R) is the stabilizer inside SO0 (2n, 2n) of a 4-form on R4n constructed from {I0 , J0 , K 0 } and the standard metric of signature (2n, 2n) analogously to how  is constructed from J1 , J2 , J3 and g. Definition 1.3.8 An almost para-quaternion Hermitian manifold (M, g, Q) is called para-quaternion Kähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) (b) (c) (d)

The Sp(n, R)Sp(1, R)-structure is integrable. ∇ = 0, where ∇ is the Levi-Civita connection of g. The holonomy of g is contained in Sp(n, R)Sp(1, R). For every local basis {J1 , J2 , J3 } of Q

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1 G-structures, Holonomy and Homogeneous Spaces

∇ Ja =

3 

cab Jb ,

a = 1, 2, 3,

b=1

with (cab ) a matrix of 1-forms in so(1, 2). The fact that (d) is equivalent to (a), (b) and (c) can be proved analogously to the pseudo-quaternion Kähler case. With regard to the curvature, we will say that an algebraic curvature tensor R is of type sp(n, R) if it sits in 2 sp(n, R) when seen as an element of 2 so(2n, 2n). This kind of tensor satisfies the following symmetries: R Ja X Y + R X Ja Y = 0, R X Y Ja Z = Ja R X Y Z ,  = 0.

a = 1, 2, 3, a = 1, 2, 3,

The proof of the following proposition can be found in [4]. Proposition 1.3.9 A para-quaternion Kähler manifold (M, g, Q) has curvature tensor field R = νq R 0 + R sp(n,R) , s , R 0 is four times the curvature of the para-quaternionic hyperwhere νq = 16n(n+2) bolic space (of the corresponding signature)

R 0X Y Z W = g(X, Z )g(Y, W ) − g(Y, Z )g(X, W )  a {g(Ja X, Z )g(Ja Y, W ) − a

−g(Ja Y, Z )g(Ja X, W ) + 2g(X, Ja Y )g(Z , Ja W )} ,

(1.3.7)

with ( 1 , 2 , 3 ) = (−1, 1, 1), and R sp(n,R) is of type sp(n, R). In particular, (M, g, Q) is Einstein. Let p ∈ M, and let Z ∈ T p M with g(Z , Z ) = 0. We consider the 4-dimensional subspace V (Z ) = Span{Z , J1 Z , J2 Z , J3 Z } ⊂ T p M. Let π ⊂ V (Z ) be a non-degenerate 2-plane and {X, Y } an orthonormal basis of π. If K p (Z )(π) = R X Y X Y is constant for every π, we call K p (Z ) the para-quaternionic sectional curvature with respect to Z at p. We say that (M, g, Q) has constant para-quaternionic sectional curvature if K p (Z ) is constant for every Z ∈ T p M and every p ∈ M. It is a wellknown result that in this case the curvature tensor field takes the form R=

k 0 R , 4

where R 0 is given by (1.3.7) and k is the value of the para-quaternionic sectional curvature. It is straightforward to adapt the arguments from the well-known case of

1.3 A Geometric Description of Some G-structures

31

definite metrics to prove that two spaces of constant and equal para-quaternionic sectional curvature are locally isometric preserving their para-quaternion Kähler structures (see for instance [119]). When in addition (M, g, Q) is connected, simplyconnected and complete, then it is called a para-quaternion space form. In that case (M, g, Q) is flat or isometric (preserving the para-quaternion Kähler structures) to the symmetric space n (k) = Sp(n + 1, R) (1.3.8) HP Sp(n, R)Sp(1, R) endowed with a suitable metric such that its para-quaternionic sectional curvature is constant and equal to k. An almost para-hyper-complex structure on M, dimM = 4n, is a 3-rank subbundle of End(M) which admits a global basis {J1 , J2 , J3 } satisfying J12 = −Id,

J22 = J32 = Id,

J1 J2 = J3 ,

which is seen as the or equivalently a reduction of L(M) to the group GL(n, H), common stabilizer inside GL(4n, R) of I0 , J0 , K 0 . An almost para-hyper-Hermitian structure on M is an almost para-hyper-complex structure J1 , J2 , J3 and a pseudoRiemannian metric of signature (2n, 2n) such that g(Ja X, Y ) + g(X, Ja Y ) = 0,

a = 1, 2, 3,

or equivalently, a reduction to the structure group Sp(n, R). We thus have the following Definition 1.3.10 An almost para-hyper-Hermitian manifold (M, g, J1 , J2 , J3 ) is called para-hyper-Kähler if one of the following equivalent conditions holds (see Proposition 1.2.12): (a) The Sp(n, R)-structure is integrable. (b) ∇ Ja = 0 for a = 1, 2, 3, where ∇ is the Levi-Civita connection of g. (c) The holonomy group of g is contained in Sp(n, R). It is evident that the curvature tensor field of a para-hyper-Kähler manifold is of type sp(n, R).

1.3.5 Sasakian and Cosymplectic Manifolds For a complete introduction to Sasakian and cosymplectic manifolds and for detailed proofs of the listed results we shall refer to [19, 113]. Definition 1.3.11 1. An almost contact structure on a (2n + 1)-dimensional manifold M is a triple (φ, ξ, η), where φ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form, such that

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1 G-structures, Holonomy and Homogeneous Spaces

η(ξ) = 1, η ◦ φ = 0, φ2 = −id + η ⊗ ξ. 2. Given a pseudo-Riemannian metric g on M, the quadruple (φ, ξ, η, g) is called an almost contact metric structure if (φ, ξ, η) is an almost contact structure and g(ξ, ξ) = ε ∈ {±1}, η = εξ  , g(φX, φY ) = g(X, Y ) − εη(X )η(Y ), for any vector fields X, Y . An almost contact metric structure on M is equivalent to a reduction to the structure group U( p, q) × {1}. Here, U( p, q) × {1} is the subgroup of O(2 p + 1, 2q) or O(2 p, 2q + 1) (depending on the value of ε) stabilizing both the vector ξ0 = e2n+1 and the (1,1)-tensor field   J0 0 , φ0 = 0 1 where {e1 , . . . , e2n+1 } is the canonical basis of R2n+1 = R2n ⊕ R, and J0 is the standard complex structure on R2n . The space R2n+1 is assumed to be endowed with the scalar product ·, · = ·, ·R2n + εe2n+1 ⊗ e2n+1 , where ·, ·R2n is the standard scalar product of signature ( p, q) on R2n . We define the fundamental 2-form associated to the almost contact metric structure (φ, ξ, η, g) as  = g(·, φ ·). In addition, in analogy with the Nijenhuis tensor field for complex manifolds, we define (see [19]) [φ, φ](X, Y ) = φ2 [X, Y ] + [φX, φY ] − φ[φX, Y ] − φ[X, φY ]. Definition 1.3.12 An almost contact metric structure (φ, ξ, η, g) is called cosymplectic if one of the following equivalent conditions holds: (a) (b) (c) (d)

The U( p, q) × {1}-structure is integrable. (φ, ξ, η, g) satisfies [φ, φ] = 0, dη = 0, and d = 0. ∇φ = 0 (which implies ∇η = 0 and ∇ξ = 0). The holonomy group of g is contained in U( p, q) × {1}.

Definition 1.3.13 An almost contact metric structure (φ, ξ, η, g) is said to be Sasakian if one of the following equivalent conditions hold: 1. (φ, ξ, η, g) satisfies [φ, φ] + 2η ⊗ ξ = 0, and dη = . 2. (∇ X φ)Y = g(X, Y )ξ − εη(Y )X . Note that Sasakian geometry is not an integrable geometry, in the sense that the holonomy group is not contained in U( p, q) × {1}. In connection with this fact, we say a few words about the intrinsic torsion of an almost contact metric structure (for an excellent introduction to the intrinsic torsion of a G-structure see [60, 107]).

1.3 A Geometric Description of Some G-structures

33

As we have seen above, the U( p, q) × {1}-structure determined by the almost contact metric structure (φ, ξ, η, g) is integrable if and only if ∇φ = 0 (or equivalently if and only if ∇ = 0). We can thus see the tensor field ∇φ (or ∇) as the obstruction for (φ, ξ, η, g) to be cosymplectic. Therefore, one can study the possible non-integrable geometries an almost contact metric manifold can present by studying the possible tensor fields ∇φ (or ∇). This can be done by considering the vector space V = R2n+1 endowed with the standard almost contact metric structure described above. We then take the space C(V ) of tensors of type (0, 3) with the same symmetries as ∇, that is, TX Y Z = −TX Y Z = −TX φY φZ + η(Y )TX ξ Z + η(Z )TX Y ξ . C(V ) is a U( p, q) × {1}-module in a natural way, so that one can decompose it into irreducible submodules. This was achieved in [50] for the Riemannian case, obtaining twelve irreducible submodules. The case of metrics with signature is obtained by a straightforward adaptation. Each of these submodules determines a class of geometric structures. The class corresponding to the so-called α-Sasakian structures is given by the submodule C6 (V ) = {T ∈ C(V )/TX Y Z = αε (X, Y η(Z ) − X, Z η(Y )) , α ∈ R} , from which Sasakian structures correspond to α = 1. We finally recall the notion of φ-sectional curvature. Let D p = {X ∈ T p M, η(X ) = 0}. If X ∈ D p is a unitary vector, then X and φX span a non-degenerate plane π, and hence we can consider the sectional curvature K p (π) = R X φX X φX of that plane. If K p is constant for all unitary vectors X ∈ D p and every p ∈ M, then we say that M is of constant φ-sectional curvature. In that case, the curvature tensor field takes the form 4R(X, Y )Z = (k + 3ε){g(Y, Z )X − g(X, Z )Y } +(εk − 1){η(X )η(Z )Y − η(Y )η(Z )X } +(k − ε) {g(X, Z )η(Y )ξ − g(Y, Z )η(X )ξ + g(φY, Z )φX + g(φZ , X )φY − 2g(φX, Y )φZ } , where k is the constant value of the φ-sectional curvature.

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1 G-structures, Holonomy and Homogeneous Spaces

1.4 Homogeneous Spaces and the Canonical Connection 1.4.1 Definition Let G be a Lie group and H a subgroup. It is a classical problem to try to endow the quotient G/H with a “good” differentiable structure, that is, a differentiable structure such that π : G → G/H is a submersion. This is not always possible, the main problem being the Hausdorff property of the quotient topology. However, it is easy to see that the quotient G/H is Hausdorff if and only if H is a closed subgroup of G. In fact, we have the following well-known result (see [79, Chap. I]). Theorem 1.4.1 Let G be a Lie group and H a closed subgroup of G. Then there is a unique differentiable structure on G/H such that the action of G on G/H is C ∞ , that is, the mapping G × G/H → G/H (a, bH ) → abH is C ∞ . In particular, π : G → G/H is a submersion. In addition, dim(G/H ) = dimG − dim H . Note that for every two points a H, bH ∈ G/H , the differentiable mapping induced by left translation by ab−1 takes the point bH to a H , implying that the orbit of a point a H under the action of G is the whole of G/H . This fact suggests the following equivalent definitions. Definition 1.4.2 (First definition of homogeneous space) A homogeneous space is a quotient G/H of a Lie group G by a closed subgroup H , endowed with the unique differentiable structure making π : G → G/H a submersion. Definition 1.4.3 (Second definition of homogeneous space) A manifold M is homogeneous if there is a Lie group G acting on the left on M such that the action is C ∞ and transitive. We shall denote by L a : M → M the action of an element a ∈ G. We shall also often denote by a · p the action of a ∈ G on a point p. It is clear that the first definition of homogeneous space (Definition 1.4.2) implies the second one (Definition 1.4.3), the group G acting transitively on G/H . For the converse, given a manifold M where a group G acts transitively, the isotropy group at any point p ∈ M: H = {a ∈ G/a · p = p}, is a closed subgroup of G and it is a straightforward calculation to check that the map G/H → M a H → a · p

1.4 Homogeneous Spaces and the Canonical Connection

35

defines a diffeomorphism between G/H and M. Note that G → G/H is a principal bundle with structure group H . A group G is said to act effectively on M if the subgroup N = {a ∈ G/ L a = Id M } only contains the neutral element of G. Since N is a normal subgroup of G, we can always assume that G acts effectively on M replacing G by G/N . Definition 1.4.4 A pseudo-Riemannian manifold (M, g) is called homogeneous if there is a Lie group G of isometries acting transitively on the left on M. If a connected pseudo-Riemannian manifold is homogeneous then the isometry groups Isom(M) and Isom0 (M, g) act transitively on M, where Isom0 (M, g) is the connected component of Isom(M, g) containing the identity. Moreover, G can be identified with a Lie subgroup of Isom(M, g). Remark 1.4.5 It can be proved (see [79, Chap. I]) that a homogeneous manifold always admits a real analytic structure such that the action G × G/H → G/H and the projection G → G/H are real analytic maps. Although most of the time we will only be concerned with C ∞ structures and maps, we will make use of this fact when necessary. As we have seen, homogeneous spaces enjoy a large group of internal symmetries. For that reason they constitute a distinguished class of spaces on which the study of pseudo-Riemannian geometry is especially rich and varied. However, this privileged position is often paid for with their rigidity. Weakening Definition 1.4.3 we can obtain a larger and less rigid class of spaces, which still share most of the desirable properties of homogeneous spaces. Definition 1.4.6 A pseudo-Riemannian manifold (M, g) is called locally homogeneous if the pseudo-group of local isometries acts transitively on (M, g), that is, if for every two points p, q ∈ M there are neighbourhoods U and V of p and q respectively and an isometry f : U → V taking p to q. The following definition will be central for the rest of this book. Definition 1.4.7 A homogeneous space G/H is called reductive if the Lie algebra g of G can be decomposed as g = h ⊕ m, where h is the Lie algebra of H and m is an Ad(H )-invariant subspace, that is, Adh (m) ⊂ m for every h ∈ H . The condition Ad(H )(m) ⊂ m implies [h, m] ⊂ m, and the converse holds if H is connected. It is important to emphasize that only in the case of a reductive homogeneous space can we can identify in a natural way the subspace m with To M, where o = H ∈ G/H is the origin of G/H . In fact, for any X ∈ m, it suffices to consider the corresponding infinitesimal generator X ∗ of the action, as defined by (1.1.1), and identify X with X o∗ .

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1 G-structures, Holonomy and Homogeneous Spaces

Proposition 1.4.8 Every homogeneous Riemannian space (M, g) is reductive. Proof We fix a point o ∈ M as the origin. Let M = G/H , and let g and h denote the Lie algebras of G and H respectively. We can suppose that G acts effectively. For every X ∈ g we consider the corresponding infinitesimal generator X ∗ . It is clear that h consists of those X ∈ g such that X o∗ = 0. Let ∇ be the Levi-Civita connection of g. Since X ∗ is a Killing vector field, the operator A X ∗ = L X ∗ − ∇ X∗ = −∇ X ∗ , also know as the Kostant operator, is skewsymmetric, so that A X ∗ |o ∈ so(To M). Let B denote the Killing form of so(To M). We consider the symmetric bilinear form φ on g defined as φ(X, Y ) = −B(A X ∗ |o , AY ∗ |o ). Since g is positive definite, B is negative definite so that φ(X, X ) = 0 implies that A X ∗ |o = 0. Therefore, if X ∈ h and φ(X, X ) = 0, we have A X ∗ |o and X o∗ = 0, whence X ∗ = 0. Since G acts effectively, we obtain X = 0. This proves that φ is definite on h. In addition, for any h ∈ H and X, Y ∈ g and any orthonormal basis {ek } of To M, we have that   φ(Adh (X ), Adh (Y )) = −B AAdh (X )∗ |o , AAdh (Y )∗ |o   = −B A L h∗ (X o∗ ) , A L h∗ (Yo∗ )    g ∇ek L h∗ (X o∗ ), ∇ek L h∗ (Yo∗ ) =− k

   =− g ∇ek X o∗ , ∇ek Yo∗ k

= φ(X, Y ), since L h∗ is an isometry. This shows that φ is Ad(H )-invariant. Finally, we take the orthogonal complement m = h⊥ of h with respect to φ. By construction, g = h ⊕ m and m is Ad(H )-invariant. 

1.4.2 Invariant Connections We start with the following Lemma 1.4.9 For every vector field X on M there is a unique vector field X on L(M) such that (a) X is invariant under the right action of GL(m, R). (b) L X θ = 0. X u ) = X π(u) for every u ∈ L(M). (c) π∗ ( Moreover, for every X on L(M) satisfying (a) and (b), there is a unique vector field X on M satisfying (c). The vector field X is called the natural lift of X .

1.4 Homogeneous Spaces and the Canonical Connection

37

Proof Let f t be the one-parameter group of local transformations of X . We consider X is defined as the vector the set f t of induced local maps on L(M). The vector field field generating f t . It is straightforward to prove uniqueness and properties (a), (b) and (c).  Let (M, g) be a connected homogeneous pseudo-Riemannian manifold, with M = K /H and K ⊂ Isom(M, g). Note that, for convenience, in the following we have changed the notation used for the group acting transitively. Denote by o ∈ M the coset H , which will be called the origin of M. The map H → Aut(To M) h → (L h )∗o is a group homomorphism, which is called the linear isotropy representation. By definition, the isotropy representation is faithful if and only if the action of K is effective. This is equivalent to the property that the induced action of K ⊂ Isom(M, g) on L(M) is free. Hereafter we suppose that K is connected and the isotropy representation is faithful. We will also assume that there is a G-structure P(M, G) which is invariant under the induced action of K on L(M), that is, for every a ∈ K we have an induced map  L a : P → P. Note that all the subsequent results can be applied to pseudo-Riemannian homogeneous spaces by taking the bundle of orthonormal references O(M) as P(M, G). Let u 0 ∈ P such that π(u 0 ) = o. We say that a linear connection on P(M, G) is invariant under the action of K if it is invariant under  L a for every a ∈ K . Identifying To M with Rm through the isomorphism u 0 : Rm → To M, we can view the linear isotropy representation as the homomorphism λ:H →G h → λ(h) = u −1 0 ◦ (L h )∗o ◦ u 0 .

(1.4.9)

We shall also denote by λ the corresponding homomorphism of Lie algebras λ : h → g. Let X ∈ k, and consider the one-parameter subgroup exp(t X ) of K , which determines a one-parameter group of transformations f t = L exp(t X ) of M.  Here we will also denote by X the vector field associated to f t , that is X p = dtd t=0 f t ( p), p ∈ K /H . Theorem 1.4.10 (Wang) Let P(M, G) be an invariant G-structure on a reductive homogeneous space K /H with reductive decomposition k = h ⊕ m. Then, there is a one to one correspondence between invariant connections on P and linear maps m : m → g such that m (Adh (X )) = Adλ(h) ( m (X )), X ∈ m, h ∈ H. The correspondence can be read from

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1 G-structures, Holonomy and Homogeneous Spaces

ωu 0 ( X) =



m (X ) X ∈ m λ(X ) X ∈ h

where ω is the connection form. In particular, the linear map m = 0 satisfies the condition in Theorem 1.4.10 above. The corresponding invariant connection thus enjoys a distinguished position among invariant connections and will be of great importance in the following. Definition 1.4.11 The invariant connection corresponding to the linear map m = 0 is called the canonical connection associated to the reductive decomposition k = h ⊕ m. The canonical connection associated to a certain reductive decomposition admits the following geometric characterization (see [79, Chap. X, Corollary 2.5]), which is often very useful. Proposition 1.4.12 Let P(M, G) be a K -invariant G-structure on a homogeneous space K /H . The canonical connection associated to a reductive decomposition k = h ⊕ m is the unique K -invariant connection on P with the following property: let f t = exp(t X ) be the one-parameter subgroup of K generated by X ∈ m, and let f t (u 0 ) is horizontal, where f t be the induced transformations of P. Then, the orbit u 0 ∈ π −1 (o). Corollary 1.4.13 Let X ∈ m, and consider the curve γt = L exp(t X ) (o) in K /H . The parallel transport along γt from o to γs coincides with the differential of f s = L exp(s X ) . Proposition 1.4.14 The torsion and the curvature tensor fields of the canonical connection associated to a reductive decomposition k = h ⊕ m are given by: (X, Y )o = −[X, Y ]m , T ( R(X, Y )Z )o = −[[X, Y ]h , Z ],

X, Y ∈ m, X, Y, Z ∈ m,

where the subindices h and m indicate projection on the corresponding subspace denote the covariant with respect to the reductive decomposition. In addition, let ∇ derivative with respect to the canonical connection. Then, = 0, R ∇

= 0. T ∇

Proof For a detailed proof, we refer to [79, Chap. X, Theorem 2.6], although one directly obtains the first two formulas from the definition of the torsion and curvature of the connection in Theorem 1.4.10 with m = 0, together with the fact that the linear isotropy representation coincides with the restriction to m of the adjoint representation of H in k. The last two formulas are obtained from the invariance of and R by the action of K and Proposition 1.4.15. T 

1.4 Homogeneous Spaces and the Canonical Connection

39

Proposition 1.4.15 If a tensor field on M is invariant under the action of K , then it is parallel with respect to the canonical connection. Proof Let J be a K -invariant tensor field of type (r, l) on M. Let X ∈ To M. Under the identification of To M and m we consider f t = L exp(t X ) and the curve γt = f t (o). f t (u 0 ). Let By Proposition 1.4.12, the horizontal lift of γt to u 0 ∈ π −1 (o) is γ¯ t =     E = P ×G ⊗r (Rm )∗ ⊗ ⊗l Rm be the associated bundle of tensor fields of type (r, l), and Jo = (u 0 , J0 )G . The f t (u 0 ), J0 )G . But since J is K horizontal lift of γt to (u 0 , J0 )G in E is given by ( invariant, one has ( f t (u 0 ), J0 )G = Jγt . We obtain by the definition of the covariant is invariant J )o = 0. Since the action of K is transitive and ∇ derivative on E that (∇ J = 0. we deduce that ∇  The converse of the previous proposition also holds and it is part of Kiriˇcenko’s Theorem, which will be treated in the next chapter. We end this section by reporting Theorem 2.10 of [79], which describes the natural torsion-free connection of a reductive homogeneous space. Theorem 1.4.16 Let P(M, G) be an invariant G-structure on a reductive homogeneous space K /H with reductive decomposition k = h ⊕ m. Then, there is a unique torsion-free K -invariant connection having the same geodesics as the canonical connection. The corresponding linear map m : m → g ⊂ gl(R, m) is defined as m (X )Y =

1 [X, Y ]m , 2

∀X, Y ∈ m,

where we identify m  To M  Rm by means of the frame u o .

Chapter 2

Ambrose–Singer Connections and Homogeneous Spaces

Homogeneous and locally homogeneous spaces are among the most important objects of study in Differential Geometry. They have been extensively investigated using several methods and techniques. When considering a homogeneous space, many geometric properties translate into algebraic properties. However, a difficulty arises, due to the fact that the same pseudo-Riemannian manifold (M, g) can admit several different descriptions as a coset space G/H . It is surprising how little is understood about this problem for many well-known examples of homogeneous pseudo-Riemannian manifolds. One of the most fruitful approaches to the study of homogeneous spaces started in 1958 with the pioneering work of Ambrose and Singer [7], who extended to homogeneous manifolds Cartan’s definition of symmetric spaces as manifolds of parallel curvature, which we describe in Sect. 2.1. Specifically, the Ambrose–Singer Theorem characterizes connected, simplyconnected and complete homogeneous Riemannian manifolds as Riemannian man such that ifolds (M, g) admitting a linear connection ∇,  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

where ∇ is the Levi-Civita connection of g, R is the curvature tensor field of g and  S = ∇ − ∇. In [65], the Ambrose–Singer Theorem was adapted to pseudo-Riemannian metrics of arbitrary signature. Homogeneous Riemannian manifolds are necessarily reductive (Proposition 1.4.8), but this property is no longer true if the metric is not definite. Indeed, the pseudo-Riemannian version of the Ambrose–Singer Theorem states that the above equations characterize reductive homogeneous pseudo-Riemannian manifolds under suitable conditions on the topology and completeness. In Sect. 2.2 we describe the Ambrose–Singer Theorem in its pseudo-Riemannian version. Section 2.3 deals with Kiriˇcenko’s Theorem [78], which extends the Ambrose– Singer result to the case when a homogeneous manifold is endowed with a geometric structure, defined by some set of tensor fields. © Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_2

41

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2 Ambrose–Singer Connections and Homogeneous Spaces

The Ambrose–Singer Theorem is not just a “nice” characterization of homogeneous spaces; it introduces a new tool for the study of the geometry of this kind of  or, equivalently, the so-called manifold, namely, the Ambrose–Singer connection ∇ homogeneous structure tensor S (or homogeneous structure for short). As we explain in Sect. 2.4, given a homogeneous structure S on a connected, simply-connected and complete pseudo-Riemannian homogeneous manifold (M, g), the Ambrose–Singer Theorem also provides a method to construct a description of (M, g) as a coset space G/H .

2.1 Symmetric Spaces and Cartan’s Theorem We first recall the definition and some properties of symmetric and locally symmetric spaces. We refer to [70] for an extensive study on this topic. Let (M, g) be a connected pseudo-Riemannian manifold, and ∇ its Levi-Civita connection. For every x ∈ M we consider the symmetry sx , which inverts geodesics of ∇ through x. Then, sx is a diffeomorphism from a neighbourhood U of x onto itself, such that sx ◦ sx is the identity transformation (that is, sx is involutive), and has x as an isolated fixed point. Definition 2.1.1 (M, g) is called locally symmetric if sx is a local isometry for every x ∈ M. Moreover, if any of such sx extends to a global isometry of M, then (M, g) is called a globally symmetric space, or simply a symmetric space. It is a well-known result (see [79, Chap. XI]) that by composing symmetries sx along broken geodesics, the isometry group of a symmetric space (M, g) acts transitively on M. Henceforth, a symmetric space is a homogeneous space. In addition, we have the following fact. Let G denote the identity component of the group of isometries of a connected symmetric space M = G/H , where H is the isotropy group at a point o ∈ M. Let so : M → M denote the symmetry at o ∈ M. We consider the map σ : G → G, u → so ◦ u ◦ so−1 , which is obviously an involutive automorphism of G. Let G σ denote the fixed point set of σ and G σ0 its identity component. Then, G σ0 ⊂ H ⊂ G σ . The differential σ∗ : g → g is an involutive automorphism of Lie algebras, so that it has eigenvalues ±1. A Lie algebra with such an involutive automorphism is called a symmetric Lie algebra. It is not hard to see that h is the eigenspace associated to +1. Denoting by m the eigenspace associated to −1, we have that g = h ⊕ m and [h, h] ⊂ h,

[h, m] ⊂ m,

[m, m] ⊂ h.

In particular, G/H is reductive. Via the identification To M m, the pseudoRiemannian metric g induces a non-degenerate symmetric bilinear form B on m, which is necessarily Ad(H )-invariant.

2.1 Symmetric Spaces and Cartan’s Theorem

43

Conversely, let G be a Lie group with an involutive automorphism σ, and let H be a closed subgroup such that G σ0 ⊂ H ⊂ G σ . Let B be an Ad(H )-invariant non-degenerate symmetric bilinear form on m (which is the (−1)-eigenspace of σ∗ ). Then, G/H can be endowed with a G-invariant pseudo-Riemannian metric g, making (G/H, g) a symmetric space. In fact, under the identification m To (G/H ), g is the G-invariant metric on G/H with go = B. In addition, let π : G → G/H denote the canonical projection. The map so defined by so ◦ π = π ◦ σ determines an isometric involution with fixed point o, which can be extended by the action of G, so obtaining isometric involutions at every point of G/H . Theorem 2.1.2 (É. Cartan) A pseudo-Riemannian manifold (M, g) is locally symmetric if and only if ∇ R = 0, where ∇ is the Levi-Civita connection of g and R its curvature tensor field. If M is connected, simply-connected and complete, then (M, g) is symmetric if and only if ∇ R = 0. Proof The above result is well known. For a detailed proof, we refer for instance to Chap. XI in [79]. However, we highlight below its basic arguments so that one can compare them with those used to prove the Ambrose–Singer Theorem, which we present in the next section. Starting from a symmetric space (G/H, σ, g), we consider the canonical connection associated to the reductive decomposition g = h ⊕ m, where h and m are the eigenspaces of σ corresponding to ±1 as before. One proves that this connection has vanishing torsion. Moreover, and since g is G-invariant, such a connection is also metric and so, it coincides with the Levi-Civita connection of g. This implies that ∇ R = 0 by the properties of the canonical connection. Conversely, let (M, g) be a pseudo-Riemannian manifold such that ∇ R = 0. Fixing a point x ∈ M, one considers g = hol ⊕ Tx M, where hol is the Lie algebra of the holonomy group Hol of g. Note that since ∇ R = 0, the curvature tensor Rx is invariant under the action of Hol and hence also under the action of hol. Therefore, the brackets [A, B] = AB − B A, [A, X ] = A · X, [X, Y ] = R X Y ,

A, B ∈ hol, A ∈ hol, X ∈ Tx M, X, Y ∈ Tx M,

endow g with a Lie algebra structure such that σ∗ : g → g, where σ|Tx M = −id and σ|hol = id, is an involutive automorphism. Hence, (g, σ∗ ) is a symmetric Lie algebra. Consider the simply-connected Lie group G with Lie algebra g, and its connected Lie subgroup H with Lie group hol. Then, σ∗ induces an involutive automorphism σ : G → G such that H is the connected component of G σ . The subgroup H is closed in G and we can consider the homogeneous space G/H . Finally, the G-invariant pseudo-Riemannian metric g¯ inherited from g at

44

2 Ambrose–Singer Connections and Homogeneous Spaces

Tx M makes (G/H, σ, g) ¯ a symmetric space, which is moreover locally isometric to (M, g). 

2.2 The Ambrose–Singer Theorem As we have seen, Cartan’s Theorem characterizes locally symmetric spaces as pseudo-Riemannian manifolds whose curvature tensor field is covariantly constant. Moreover, when the manifold is globally symmetric, it provides a coset representation of the manifold. In this section we present the Ambrose–Singer Theorem, which generalizes Cartan’s Theorem to the more general framework of homogeneous spaces. Under suitable global conditions, this result characterizes homogeneous spaces by the existence of  with respect to which the curvature tensor field of the metric a metric connection ∇,  have vanishing covariant derivative. Furthermore, it provides a and the torsion of ∇ method to recover a coset representation of the manifold, which, at least at the Lie algebra level, is achieved in terms of an elementary construction. This topic will be considered in Sect. 2.4. The following result was proved by Ambrose and Singer [7] in the Riemannian case, and later generalized in [65] to metrics of arbitrary signature. Theorem 2.2.1 (Ambrose–Singer) Let (M, g) be a connected, simply-connected and complete pseudo-Riemannian manifold. The following properties are equivalent: (a) (M, g) is a reductive homogeneous pseudo-Riemannian manifold.  satisfying (b) (M, g) admits a linear connection ∇  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

(2.2.1)

 ∇ is the Levi-Civita connection of g, and R its curvature where S = ∇ − ∇, tensor field. Proof The proof of Theorem 2.2.1 can be obtained as a special case of the proof of Kiriˇcenko’s Theorem 2.3.2 (which we give in full detail in the next section), in the absence of the additional structure defined by tensor fields P1 , . . . , Pk (that is, setting P1 = . . . = Pk = 0). For this reason, we only describe below the essential steps of the proof, referring to [65] for more details. Assume first that (M, g) is a reductive homogeneous pseudo-Riemannian manifold, and consider a coset realization M = G/H of (M, g), where G and H denote a group of isometries acting transitively and effectively on (M, g) and the isotropy group at an arbitrary point o ∈ M, respectively. As is well known, the fact that M = G/H is reductive is equivalent to requiring that g = h ⊕ m and m is stable under Ad(H ) (see Definition 1.4.7). Let α belong to the Lie algebra g of G and α∗ be the corresponding infinitesimal generator of the action, as defined by (1.1.1). The Lie algebra of the isotropy group H is then given by h = {α ∈ g : αo∗ = 0}.

2.2 The Ambrose–Singer Theorem

45

The canonical connection ∇˜ associated to the reductive decomposition g = h ⊕ m is uniquely determined by (∇˜ α∗ β ∗ )o = [α∗ , β ∗ ]o = −[α, β]∗o ,  satisfies Eqs. (2.2.1). for all α, β ∈ g. Then, it is easy to check that ∇ Conversely (see the proof of Kirichenko’s Theorem in the next section), if there exists a linear connection ∇˜ = ∇ − S on M satisfying (2.2.1), one then proves that  is complete and ensures the existence, given two points p, q ∈ M, of a global ∇ isometry mapping p to q. Therefore, there exists a group G of isometries acting transitively on M such that M = G/H is reductive and ∇˜ is exactly the canonical connection associated to this reductive decomposition.  The following result describes a set of equations equivalent to the Ambrose–Singer equations (2.2.1). Proposition 2.2.2 Equations (2.2.1) are equivalent to  = 0, ∇g

 = 0, R ∇

 = 0, T ∇

(2.2.2)

 and T  are the curvature and torsion tensor fields of ∇  respectively. where R Proof The equivalence is obtained by direct calculation, taking into account the fact that for any smooth vector fields X, Y tangent to M, one has the relations X Y = SY X − S X Y, T X Y = R X Y + [S X , SY ] + ST Y . R X  Remark 2.2.3 Since the tensor field S is invariant under the action of a Lie group G acting transitively on (M, g), it is completely determined by its value at the origin o ∈ M. In fact, by the properties of the canonical connection α∗ β ∗ )o = −[α, β]∗o , (∇ we see that

α, β ∈ g,

(Sα∗ β ∗ )o = (∇β ∗ α∗ )o = −(Aα∗ β ∗ )o ,

where A denotes the Kostant operator [81], given by A X Y = [X, Y ] − ∇ X Y = −∇Y X, for any tangent vector fields X, Y on M. Therefore, (So ) X Y = −(Aα∗ )o Y,

X, Y ∈ To M,

where α is the unique element in m such that αo∗ = X .

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2 Ambrose–Singer Connections and Homogeneous Spaces

2.3 Kiriˇcenko’s Theorem Let (M, g) denote a homogeneous pseudo-Riemannian manifold, and suppose that M is endowed with some further geometric structure(s), described by means of a finite set of tensor fields P1 , . . . , Pk . It is natural to ask how to characterize the invariance of these tensor fields (and hence, of the additional structure they define) under the  action of the group of isometries G, by means of properties of the connection ∇ (equivalently, the tensor field S) occurring in Ambrose–Singer Theorem. The answer to this question is given by Kiriˇcenko’s Theorem [78], which completes the Ambrose–Singer Theorem by extending it to manifolds with a geometric structure. Before enouncing and proving Kiriˇcenko’s Theorem, we need the following technical lemma, for which we refer to [116]. Lemma 2.3.1 Let M be a connected and simply-connected manifold of dimension m. Let X 1 , . . . , X m be vector fields on M such that (a) X 1 , . . . , X m are complete; are linearly independent at every point p ∈ M; (b) X 1 , . . . , X m k k (c) [X i , X j ] = m k=1 ci j X k , with ci j constant. Then, fixing a point p ∈ M, M has the unique structure of a Lie group such that p is the neutral element and {X 1 , . . . , X m } is a basis of left-invariant vector fields. Theorem 2.3.2 (Kiriˇcenko) Let (M, g) be a connected, simply-connected and complete pseudo-Riemannian manifold with a geometric structure defined by a set of tensor fields P1 , . . . , Pk . Then, the following properties are equivalent: (a) The manifold is reductive homogeneous M = G/H , with G ⊆ Isom(M, g), such that P1 , . . . , Pk are invariant under the action of G.  satisfying (b) (M, g) admits a linear connection ∇  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 Pi = 0, ∇

(2.3.3)

 ∇ is the Levi-Civita connection of g, and for i = 1, . . . , k, where S = ∇ − ∇, R its curvature tensor field. Kiriˇcenko’s Theorem appeared for the first time in [78], although Sekigawa in [109] had already obtained this result for almost Hermitian manifolds (that is, where the tensor P1 = J is an almost complex structure compatible with g). Nevertheless, the proof appearing in the original paper by Kiriˇcenko is incomplete. Moreover, to our knowledge, despite the fact that this result has been extensively used, there is no complete proof of it in the literature. For this reason, we present here a detailed proof. Many of the arguments we follow are inherited from those used in [116] to prove the Ambrose–Singer Theorem. Proof of Kiricenko’s Theorem Let (M, g) denote a reductive homogeneous pseudoRiemannian manifold with invariant tensor fields P1 , . . . , Pk . Let G denote a Lie

2.3 Kiriˇcenko’s Theorem

47

group acting transitively by isometries on (M, g) and preserving P1 , . . . , Pk , and H  the canonical the isotropy group at some point p ∈ M. As usual, we denote by ∇ connection with respect to a reductive decomposition g = h ⊕ m.  = 0 and ∇  = 0, where R  and  satisfies ∇ R T By Proposition 2.2.2, we have that ∇  are the curvature and torsion tensor fields of ∇,  respectively. As we explained in T  R = 0 and ∇  S = 0. Proposition 2.2.2, those equations imply that ∇ Finally, making use of Proposition 1.4.15, since g and P1 , . . . , Pk are invariant  = 0 and ∇  Pi = 0, i = 1, . . . , k. under the action of G, we conclude that ∇g  satisfying (2.3.3). Conversely, suppose that (M, g) admits a linear connection ∇ Let π : O(M) → M be the bundle of orthonormal frames of (M, g). We fix a point x ∈ M and a reference u 0 ∈ O(M) with π(u 0 ) = x. For all indices i = 1, . . . , k, let Hi denote the stabilizer of u ∗0 (Pi )x inside O( p, q) and H = ∩i Hi . The tensor fields Pi determine a reduction Q of O(M) to the group H such that u 0 ∈ Q.  0 ) → M of ∇  at u 0 , which is a reduction We consider the holonomy bundle π : P(u  ∇  Pi = 0 for all indices i = 1, . . . , k. We denote of Q to the group Hol ⊂ H , since ∇   by h and hol∇ the Lie algebras of H and Hol∇ respectively.  Let {A1 , . . . , Ar } be a basis of hol∇ and consider the associated fundamental vector ∗ ∗ fields A1 , . . . , Ar , which are complete in Q. Let {e1 , . . . , em } be the canonical basis of Rm and take the associated standard vector fields B1 = B(e1 ), . . . , Bm = B(em ).  is complete, these vector fields are also complete in Q (see [79, Vol. I, Since ∇ p. 140]). Moreover, it is easy to see that the vector fields A1 , . . . , Ar , B1 , . . . , Bm ,  0 ), are also complete and determine an absolute parallelism on restricted to P(u  P(u 0 ).  on the principal bundle Q → M, and let θ Let ω be the connection 1-form of ∇  be the contact form. We denote by  and  the curvature and torsion forms of ∇. Then, for every i, j = 1, . . . , m, we have (Bi , B j ) = dθ(Bi , B j ) = Bi (θ(B j )) − B j (θ(Bi )) − θ([Bi , B j ]) = −θ([Bi , B j ]), and (Bi , B j ) = dω(Bi , B j ) = Bi (ω(B j )) − B j (ω(Bi )) − ω([Bi , B j ]) = −ω([Bi , B j ]), whence the vertical and  horizontal  parts of [Bi , B j ] are respectively given by −(Bi , B j )∗ and −B (Bi , B j ) . Thus, we can write   [Bi , B j ] = −B (Bi , B j ) − (Bi , B j )∗ .

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2 Ambrose–Singer Connections and Homogeneous Spaces

In addition, by the properties of the fundamental vector fields (recall that the action of H on Q is on the right), we have that [A∗k , Al∗ ] = [Ak , Al ]∗ ,

[Ak , Bi ] = B(Ak (ei )),

for all indices k, l = 1, . . . , r and i = 1, . . . , m.  and T  denote the curvature and torsion tensor fields of On the other hand, let R  Then, for any u ∈ Q and horizontal vector X¯ in Tu Q, we have ∇.       (X i , X j ) = 0, X T X¯ (Bi , B j ) = u −1 ∇       (X i , X j ) = 0, X R X¯ (Bi , B j ) = u −1 ∇ where X, X i , X j ∈ Tπ(u) M are the projections of X¯ , Bi , B j , respectively. Hence, for every i, j = 1, . . . , m, the functions (Bi , B j ) and (Bi , B j ), with values in Rm  0 ). Therefore, the brackets of the vector fields and h respectively, are constant on P(u A∗k , Bi , i = 1, . . . , m, k = 1, . . . , r , have constant coefficients with respect to the basis {A∗k , Bi , i = 1, . . . , m, k = 1, . . . , r } and so, they span a finite-dimensional Lie subalgebra of X(Q).  0 ). On G, ¯ we consider the vector fields Let G¯ be the universal covering of P(u A¯ ∗k , B¯ i , i = 1, . . . , m, k = 1, . . . , r , defined by ρ∗ ( A¯ ∗k ) = A∗k ,

ρ∗ ( B¯ i ) = Bi ,

 0 ) is the covering map. These vector fields are complete and where ρ : G¯ → P(u determine an absolute parallelism. Moreover, their brackets have constant coefficients. Hence, by Lemma 2.3.1, G¯ can be endowed with a Lie group structure with e¯ ∈ ρ−1 (u 0 ) as the neutral element, and with { A¯ ∗k , B¯ i , i = 1, . . . , m, k = 1, . . . , r } spanning its Lie algebra g¯ . Note that { A¯ ∗k , k = 1, . . . , r } itself spans a subalgebra g¯ 0 ⊂ g¯ , whose corresponding connected Lie subgroup of G¯ is denoted by G¯ 0 . ¯ G¯ 0 . Lemma 2.3.3 M is diffeomorphic to G/ Proof The map π1 = π ◦ ρ : G¯ → M determines a fibration on M. Taking its exact homotopy sequence π1∗

∂∗

π1∗

¯ → 0 (G, ¯ b) ¯ → 0 (M, g) . . . → 1 (M, y) → 0 (π1−1 (y), b) || || || 0 0 0 i∗

¯ = 0, that is, the fibers of π1 are connected. Since π1 is we deduce that 0 (π1−1 (y), b) continuous, its fibers are also closed. In addition, π1∗ ( A¯ ∗k ) = 0 for k = 1, . . . , r , hence the fibers are tangent to g¯ 0 . We conclude that the fibers are the integral submanifolds

2.3 Kiriˇcenko’s Theorem

49

of the involutive distribution g¯ 0 , and thus can be represented as classes a¯ G¯ 0 , where ¯ a¯ ∈ G. Therefore, we have a smooth map ¯ G¯ 0 → M, π2 : G/ a¯ G¯ 0 → π1 (a). ¯ The map π2 is then bijective, and its differential is an isomorphism at every point.  Therefore, π2 is a diffeomorphism.  0 ). Then, we can write Let us now consider y ∈ M and v = (y; v1 , . . . , vm ) ∈ P(u   v = b¯ G¯ 0 ; (π1∗ )b¯ ( B¯ 1b¯ ), . . . , (π1∗ )b¯ ( B¯ m b¯ ) , ¯ = v. Thus, we have the following result. where ρ(b) Lemma 2.3.4 The map

L a¯ : M → M b¯ G¯ 0 → a¯ b¯ G¯ 0

 0 ) → P(u  0 ), for every a¯ ∈ G. ¯ induces a transformation  L a¯ : P(u ¯ Then, L a¯ ◦ π1 = π1 ◦ La¯ . Let Proof Let La¯ denote left multiplication by a¯ in G.  ¯ ¯ ¯ ¯ y = b G 0 and v = ρ(b) ∈ P(u 0 ). Since Bi is left-invariant, for all i = 1, . . . , m, we have ¯ ¯ (L a∗ ¯ ) y ◦ (π1∗ )b¯ ( Bi b¯ ) = (π1∗ )a¯ b¯ ◦ (La∗ ¯ ) y ( Bi b¯ ) = (π1∗ )a¯ b¯ ( B¯ i a¯ b¯ ). Taking into account that {b¯ G¯ 0 ; (π1∗ )b¯ ( B¯ 1b¯ ), . . . , (π1∗ )b¯ ( B¯ m b¯ )} = {y; v1 , . . . , vm }  0 ) for every point of M, we conclude that  L a¯ is a transformation is a reference in P(u  ¯  of P(u 0 ) for every a¯ ∈ G.  0 ). Moreover, this action is transitive, So far we have proved that G¯ acts on P(u because    L b¯1 b¯2−1 b¯2 G¯ 0 ; (π1∗ )b¯2 ( B¯ 1b¯2 ), . . . , (π1∗ )b¯2 ( B¯ m b¯2 ) L b¯1 b¯2−1 (ρ(b¯2 )) =    = b¯1 G¯ 0 ; (π1∗ )b¯ ( B¯ m b¯ ), . . . , (π1∗ )b¯ ( B¯ m b¯ ) 1

1

1

1

= ρ(b¯1 ).  0 ) ⊂ Q, so that the maps L a¯ act as isometries of (M, g) preserving Note that P(u the tensor fields P j , j = 1, . . . , k. Let now a¯ be an element in the isotropy group K¯

50

2 Ambrose–Singer Connections and Homogeneous Spaces

 0 ), that is,  of the reference u 0 ∈ P(u L a¯ (u 0 ) = u 0 . Then, L a¯ (π(u 0 )) = π(u 0 ) and

¯ ¯ L a∗ ¯ ◦ (π1∗ )e¯ ( Bi e¯ ) = Bi e¯ ,

i = 1, . . . , m.

Thus, L a¯ is an isometry of M fixing the point x = π(u 0 ) such that its differential at x is the identity, whence L a¯ is the identity transformation of M. Therefore, K¯ is the kernel of the group homomorphism a¯ → L a¯ . As such, it is a normal subgroup  0 ) = G/ ¯ K¯ is a Lie group acting transitively on M by ¯ We thus obtain that P(u of G. isometries preserving P1 , . . . , Pk .  0 ) of G = P(u  0 ) can be decomposed as g = Finally, the Lie algebra g = Te¯ P(u   hol∇ ⊕ m, where hol∇ is spanned by {(A∗k )u 0 , k = 1, . . . , r } and m is spanned by {(Bi )u 0 , i = 1, . . . , m}. We show that this decomposition is reductive. Consider the following maps: ρ : G¯ → G, ¯ K¯ , a¯ → a¯ K¯ , p : G¯ → G/ ¯ ¯ I : G/ K → G, a¯ K¯ →  L a¯ (u 0 ). It is easy to check that I is a diffeomorphism. Let K be the isotropy group of x with respect to the action of G. We have L a¯ (x) = L a¯ (e¯ G¯ 0 ) = a¯ G¯ 0 , so that a¯ ∈ G¯ 0 if and only if L a¯ ∈ K . We thus have that the diffeomorphism I ◦ p identifies G¯ 0 with K . In particular, K is connected. On the other hand, the following diagram is commutative: p

G −→ G/K ρ↓I  (u) P 

so that ρ∗ = I∗ ◦ p∗ . This means that hol∇ = ρ∗ (¯g0 ) = (I ◦ p)∗ (¯g0 ) is contained in  the Lie algebra of K , and counting dimensions we conclude that hol∇ equals the  Lie algebra of K . It is obvious that [hol∇ , m] ⊂ m and, since K is connected, we conclude that m is Ad(K )-invariant.  Definition 2.3.5 Let (M, g) be a pseudo-Riemannian manifold.  on (M, g) satisfying (2.2.1) will be called an Ambrose– (i) A linear connection ∇ Singer connection, or AS-connection for short. (ii) If (M, g) is also endowed with a geometric structure defined by tensor fields  satisfying (2.3.3) will be called an Ambrose– P1 , . . . , Pk , a linear connection ∇ Singer–Kiriˇcenko connection, or ASK-connection for short.

2.4 Homogeneous Pseudo-Riemannian Structures

51

2.4 Homogeneous Pseudo-Riemannian Structures Definition 2.4.1 Let (M, g) be a pseudo-Riemannian manifold with Levi-Civita  be an AS-connection on (M, g). The tensor field S = ∇ − ∇  connection ∇. Let ∇ is called a homogeneous pseudo-Riemannian structure tensor, or a homogeneous structure for short. In the previous Sect. 2.2 we have seen that a homogeneous pseudo-Riemannian manifold admits an AS-connection, and thus a homogeneous structure, whenever it is reductive. Conversely, a pseudo-Riemannian manifold admitting a homogeneous structure is a reductive homogeneous pseudo-Riemannian space under suitable conditions. Dropping any of these assumptions, one only obtains that (M, g) is locally homogeneous. An analogous situation holds when a geometric structure is present. This result and its converse will be treated in the following chapter. It is worth noting that, under the aforementioned topological conditions, the proof of the Ambrose–Singer Theorem (or Kiriˇcenko’s Theorem) provides a method to construct a description of a homogeneous pseudo-Riemannian manifold as a coset G/H starting from an AS-connection (equivalently, from a homogeneous structure S). We now describe this construction in detail. Let V denote a vector space equipped with a non-degenerate bilinear form  ·, · and K : V ∧ V → End(V ), T : V → End(V ), linear morphisms. Definition 2.4.2 The pair (K , T ) is called an infinitesimal model if the following properties are satisfied: TX Y + TY X = 0, K X Y Z + K Y X Z = 0,

(2.4.4) (2.4.5)

K X Y Z , W  + K W Z X, Y  = 0, K X Y · T = 0, K X Y · K = 0, (K X Y Z + TTX Y Z ) = 0,

(2.4.6) (2.4.7) (2.4.8) (2.4.9)

S XYZ

SK

TX Y Z

= 0,

(2.4.10)

XYZ

where K X Y is acting as a derivation on the tensor algebra of V in (2.4.7) and (2.4.8). When a tensor P defining a geometric structure on V is present, we shall say that (K , T, P) is an infinitesimal model if besides (2.4.4)–(2.4.10), (K , T ) satisfies the additional condition K X Y · P = 0.

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 be an AS-connection on (M, g) with associated homogeneous structure S. Let ∇  deterFixing a point x ∈ M, the Ambrose–Singer equations (2.2.1) ensure that ∇ mines an infinitesimal model by setting V = Tx M and TX Y = (Sx )Y X − (Sx ) X Y, x ) X Y . K XY = (R If a geometric structure defined by a tensor field P is present on (M, g), then one  P = 0 implies K X Y · Px = 0. takes Px , and thus ∇ Now, from every infinitesimal model (K , T ), one can construct a Lie algebra via the so-called Nomizu’s construction. For that, one considers the vector space g0 = V ⊕ h 0 , with h0 = {A ∈ so(V )/ A · K = 0, A · T = 0} and the brackets [A, B] = AB − B A, [A, X ] = A · X, [X, Y ] = −TX Y + K X Y ,

A, B ∈ h0 , A ∈ h0 , X ∈ V, X, Y ∈ V.

Conditions (2.4.4)–(2.4.10) imply that g0 has a Lie algebra structure. In the presence of an additional geometric structure defined by a tensor P, one has to take h0 = {A ∈ so(V )/ A · K = 0, A · T = 0, A · P = 0}. Unfortunately, obtaining Nomizu’s construction g0 from the infinitesimal model (K , T ) is not always an easy task, since the computations required to find h0 can become very demanding. Another possibility is to consider the so-called transvection algebra (see [82]). This algebra is defined as g¯ 0 = V ⊕ h¯ 0 , where h¯ 0 is the Lie algebra generated by the endomorphisms K X Y for all X, Y ∈ V (the same definition is valid when a geometric structure is present). In general, g¯ 0 is a proper subalgebra of g0 . If (K , T ) is the  then h¯ 0 coincides with the infinitesimal model associated to an AS-connection ∇,  holonomy algebra of ∇. We now consider the abstract simply-connected Lie group G 0 with Lie algebra g0 , and its connected Lie subgroup H0 with Lie algebra h0 . We also consider the simply-connected Lie group G¯ 0 with Lie algebra g¯ 0 , and its connected Lie subgroup H¯ 0 with Lie algebra h¯ 0 . Definition 2.4.3 An infinitesimal model (K , T ) (or (K , T, P)) is said to be regular if H0 is closed in G 0 . On the other hand, the transvection algebra (¯g0 , h¯ 0 ) is said to be regular if H¯ 0 is closed in G¯ 0 .

2.4 Homogeneous Pseudo-Riemannian Structures

53

In the case when the infinitesimal model (respectively, the transvection algebra) is regular, one can take the homogeneous space G 0 /H0 (respectively, G¯ 0 / H¯ 0 ), which will be called the associated homogeneous model (or the homogeneous model asso or S if the infinitesimal model or the transvection algebra comes from an ciated to ∇  with homogeneous structure S). AS-connection ∇ We recall that in the proof of the Ambrose–Singer Theorem (or Kiriˇcenko’s Theorem), starting from a homogeneous pseudo-Riemannian manifold (M, g), every  0 ) acting transi gives a Lie group G¯ = P(u AS-connection (or ASK-connection) ∇  0 ) is the holonomy bundle of ∇  through tively by isometries on (M, g), where P(u ¯ so that M is diffeomorphic to G/ ¯ H¯ . The u 0 . Its isotropy subgroup H¯ is closed in G, ¯ so that H¯ 0 is closed in Lie group G¯ 0 is then nothing but the universal covering of G, G¯ 0 and M is diffeomorphic to G¯ 0 / H¯ 0 . An interesting feature of these constructions is that different AS-connections on (M, g) might give different representations of M as a coset G/H . For a better understanding of this phenomenon, we need the following results and definitions. Definition 2.4.4 Two homogeneous structures S and S  on pseudo-Riemannian manifolds (M, g) and (M  , g  ) are said to be isomorphic if there exists an isometry ϕ : M → M  such that ϕ∗ S  = S. Note that an isomorphism ϕ between two homogeneous structures S and S  is an  = ∇ − S and ∇  = ∇  − S  . Let now g0 = V ⊕ h0 affine transformation between ∇    and g0 = V ⊕ h0 be the Nomizu constructions associated to S and S  respectively. Then we have the following result. Theorem 2.4.5 (see [116]) If S and S  are isomorphic, then there exists a Lie algebra isomorphism ψ : g0 → g0 such that ψ(V ) = V  and ψ(h0 ) = h0 . Moreover, the restriction of ψ to V is an isometry with respect to the scalar products inherited by V and V  from g0 and g0 , respectively. Proof Let ϕ be an isomorphism between S and S  . We define ψ|V = ϕ∗ and ψ(A) = ϕ∗ A for A ∈ h0 . Then, a straightforward calculation yields that ψ satisfies the statement.  The converse of the above result holds under suitable topological conditions. Theorem 2.4.6 Let (M, g) and (M  , g  ) be two connected, simply-connected and complete pseudo-Riemannian manifolds with homogeneous structures S and S  , respectively. If there exists a Lie algebra isomorphism ψ : g0 → g0 such that ψ(V ) = V  , ψ(h0 ) = h0 and ψ|V is an isometry, then S and S  are isomorphic. Proof Since ψ is a Lie algebra isomorphism, we have that ψ|V is an isometry between  V = Tx M and V  = Tx  M  preserving the curvature and torsion tensor fields of ∇     and ∇ . This implies that there are neighbourhoods U and U of x and x respectively,  and ∇  taking x to x  , and whose and an affine transformation ϕ : U → U  of ∇ differential at x coincides with ψ|V (see [79, Vol. I, Chap. VI]).

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 and ∇  are metric connections and the differential of ϕ at x is an isomSince ∇ etry, ϕ is an isometry. In addition, since (M, g) and (M  , g  ) are connected, simplyconnected and complete, ϕ can be extended to a global isometry (see again [79, Vol. I, Chap. VI]).  Remark 2.4.7 Under the hypotheses of the previous theorem, we conclude that (M, g) and (M  , g  ) are homogeneous pseudo-Riemannian manifolds whose simplyconnected isometry groups G 0 and G 0 constructed from S and S  respectively are isomorphic. Their corresponding isotropy groups H0 and H0 are also isomorphic. We shall now make some comments on the existence of different solutions to the Ambrose–Singer equations and its consequences on the homogeneity of M. From above, we know that isomorphic classes for homogeneous structures provide the same description M = G/H with isomorphic reductive decomposition. In fact, these classes may contain many solutions S to the Ambrose–Singer equations (all S  = ϕ∗ S, for some isometry ϕ) or a single element. For example, this last case occurs in the Heisenberg group, where ϕ∗ S = S for every isometry ϕ (see [116]). With respect to non-isomorphic classes, we have two different situations: (i) There may exist two non-isomorphic homogeneous structures S1 and S2 on (M, g), giving rise to the same Lie algebra g0 but with two different decompositions g0 = V1 ⊕ h1 = V2 ⊕ h2 . This means that there is no isomorphism g0 → g0 such that ψ(V1 ) = V2 , ψ(h1 ) = h2 and ψ|V1 is an isometry. For example, this occurs for the Heisenberg group [116]. (ii) There may exist two homogeneous structures S1 and S2 on (M, g) with nonisomorphic Lie algebras g1 and g2 . They correspond to two different representations of M as a coset, namely, G 1 /H1 and G 2 /H2 . An example of this situation can be found for instance in the standard six-dimensional Riemannian sphere: S6 = SO(7)/SO(6) = G2 /SU(3). We end this section with some first examples of homogeneous structures. Example 2.4.8 (Homogeneous Riemannian structures on the standard three-sphere) We identify in the standard way the sphere S3 with the Lie group SU (2), by the map  (z, w) ∈ S3 ⊂ C2 →

z w w¯ z¯

∈ SU (2).

We consider the basis of the Lie algebra su(2) of SU (2) given by  X1 =

i 0 0 −i



 ,

X2 =

0 1 , −1 0

 X3 =

0i , i 0

2.4 Homogeneous Pseudo-Riemannian Structures

55

for which [X 1 , X 2 ] = 2X 3 , [X 2 , X 3 ] = 2X 1 , [X 3 , X 1 ] = 2X 2 . Observe that {X 1 , X 2 , X 3 } is an orthonormal basis for the canonical metric g on S3 . The Koszul formula (which characterizes the Levi-Civita connection ∇ of g) applied to left-invariant vector fields reads 2g(∇ X Y, Z ) = g([X, Y ], Z ) − g([Y, Z ], X ) + g([Z , X ], Y ),

(2.4.11)

for all X, Y, Z ∈ su(2). Explicitly, with respect to {X 1 , X 2 , X 3 } we have ∇ X 1 X 1 = 0, ∇ X 2 X 1 = −X 3 , ∇X3 X 1 = X 2,

∇X1 X 2 = X 3, ∇ X 2 X 2 = 0, ∇ X 3 X 2 = −X 1 ,

∇ X 1 X 3 = −X 2 , ∇X2 X 3 = X 1, ∇ X 3 X 3 = 0.

Therefore, with respect to {X 1 , X 2 , X 3 }, the curvature tensor is completely determined by R X 1 X 2 X 1 = X 2 , R X 1 X 2 X 3 = 0, R X 1 X 3 X 1 = X 3 , R X 2 X 3 X 1 = 0, R X 2 X 3 X 2 = X 3 . R X 1 X 3 X 2 = 0, Suppose now that S is a homogeneous Riemannian structure on S3 ≡ SU (2), that is, a tensor satisfying conditions (2.2.1). We shall denote again by S the tensor of type (0, 3) corresponding to the homogeneous structure by g. Let {θ1 , θ2 , θ3 } be the basis of left-invariant 1-forms dual to {X 1 , X 2 , X 3 }. Then, for the (0, 3)-tensor S we have S = ρ ⊗ θ1 ∧ θ2 + σ ⊗ θ1 ∧ θ3 + τ ⊗ θ2 ∧ θ3 , for some suitable differential 1-forms ρ, σ, τ on S3 , determined by ρ(Z ) = S Z X 1 X 2 , σ(Z ) = S Z X 1 X 3 , τ (Z ) = S Z X 2 X 3 , for all vector fields Z ∈ S3 . The first equation in (2.2.1) yields S X Y Z + S X Z Y = 0,

X, Y, Z ∈ su(2),

 R = 0) holds automatically. while the second equation (namely, ∇  The connection forms ω˜ i j of ∇ are defined by Z X j = ∇

3

ω˜ i j (Z )X i ,

j = 1, 2, 3.

i=1

Then, ω˜ ji = −ω˜ i j for all indices i, j, and a standard calculation gives

(2.4.12)

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2 Ambrose–Singer Connections and Homogeneous Spaces

ω˜ 12 = ρ − θ3 ,

ω˜ 13 = σ + θ2 ,

ω˜ 23 = τ − θ1 .

Thus, taking into account that S Z X X = 0, the above equations yield Z S)V X 1 X 2 = (∇ Z ρ) − (σ + θ2 )(Z )τ (V ) + (τ − θ1 )(Z )σ(V ), (∇ Z σ) + (ρ − θ3 )(Z )τ (V ) − (τ − θ1 )(Z )ρ(V ), Z S)V X 1 X 3 = (∇ (∇ Z τ ) − (ρ − θ3 )(Z )σ(V ) + (σ + θ2 )(Z )ρ(V ) Z S)V X 2 X 3 = (∇ (∇ and by the third equation in (2.2.1) we conclude that homogeneous structures on the standard three-sphere S3 are in one to one correspondence with the triples (ρ, σ, τ ) of differential 1-forms on S3 satisfying  = (σ + θ2 ) ⊗ τ − (τ − θ1 ) ⊗ σ, (∇ρ)  = −(ρ − θ3 ) ⊗ τ + (τ − θ1 ) ⊗ ρ, (∇σ)  ) = (ρ − θ3 ) ⊗ σ − (σ + θ2 ) ⊗ ρ. (∇τ For more information and a more general study of homogeneous structures on Berger 3-spheres, we refer to [66]. Example 2.4.9 (Natural homogeneous structures on a Lie group) Let G be a Lie group endowed with a left invariant pseudo-Riemannian metric g. This space is obviously homogeneous with G = G/{e} and has trivial reductive decomposition g = h + m = 0 + g. We denote by X ∗ the infinitesimal generator of the action, as defined by Eq. (1.1.1), that is, X ∗p

 d  = exp(t X ) · p, dt t=0

p ∈ G.

Recall that [X, Y ]∗ = −[X ∗ , Y ∗ ], for X, Y ∈ g. For the trivial reductive decomposi is obviously given by tion, the canonical connection ∇ X ∗ Y ∗ = −[X, Y ]∗ , ∇

X, Y ∈ g,

whereas the Levi-Civita connection of g satisfies 2g(∇ X ∗ Y ∗ , Z ∗ ) = −g([X, Y ]∗ , Z ∗ ) + g([Y, Z ]∗ , X ∗ ) + g([X, Z ]∗ , Y ∗ ), (2.4.13) for X, Y, Z ∈ g, which is again the Koszul formula (2.4.11), but rewritten here for  satright-invariant vector fields X ∗ . Hence, the homogeneous structure S = ∇ − ∇ isfies  1 g([X, Y ]∗ , Z ∗ ) + g([Y, Z ]∗ , X ∗ ) + g([X, Z ]∗ , Y ∗ ) . 2 (2.4.14)  and S are invariThe Ambrose–Singer equations associated to S are satisfied as g, R ant under the action of the group (see Proposition 1.4.15). However, in this case, g(S X ∗ Y ∗ , Z ∗ ) =

2.4 Homogeneous Pseudo-Riemannian Structures

57

we can easily check the equations directly. Indeed, skew symmetry of expression  = 0. On the other hand, the curvature ten(2.4.14) in Y and Z is equivalent to ∇g  ∗ , Y ∗ )Z ∗ = ∇ X ∗ ∇ Y ∗ ∇ [X ∗ ,Y ∗ ] Z ∗ identically vanishes as a Y ∗ Z ∗ − ∇ X ∗ Z ∗ − ∇ sor R(X  = 0. Finally, the condiR direct consequence of the Jacobi identity. In particular, ∇  S = 0 is obtained after taking derivatives in (2.4.14) with respect to any vector tion ∇  is metric and some easy algebraic field U ∗ with U ∈ g, and using the fact that ∇ manipulations. There is another simple way of getting G as a homogeneous space. We consider K = G × G acting on G as (a1 , a2 ) · a = a1 a2−1 a, a ∈ G, (a1 , a2 ) ∈ K . It is obvious that K acts transitively and that the isotropy of any point is the diagonal H =  = {(a, a) : a ∈ G}, whose Lie algebra is h = {(X, X ) : X ∈ g}. Among all the possible reductive decompositions k = h + m, there are three that are rather simple: m+ = {(0, X ) : X ∈ g}, m− = {(X, 0) : X ∈ g}, m0 = {(X, −X ) : X ∈ g}. − , ∇ 0 associated to these decompositions are (cf. + , ∇ The canonical connections ∇ [79, Chap. X, §3]) X+∗ Y ∗ = [X ∗ , Y ∗ ], ∇

X−∗ Y ∗ = 0, ∇

X0 ∗ Y ∗ = 1 [X ∗ , Y ∗ ]. ∇ 2

These connections are called Cartan–Schouten connections of G and the notations come from the expressions of their respective torsions: T ± (X ∗ , Y ∗ ) = ±[X ∗ , Y ∗ ],

T 0 (X ∗ , Y ∗ ) = 0,

X, Y ∈ g.

Observe that + is the one described • the homogeneous structure S + associated to the connection ∇ in the first part of this example. • As T − (X ∗ , Y ∗ ) = −[X ∗ , Y ∗ ], the homogeneous structure S − is the canonical connection of the homogeneous space (G × G)/G (see Definition 1.4.11). • The homogeneous structure S 0 is the natural torsion-free connection of the homogeneous space (G × G)/G (see Theorem 1.4.16). − , we have With respect to ∇

S X−∗ Y ∗ = ∇ X ∗ Y ∗

(2.4.15)

0 we have as in Eq. (2.4.13), and for ∇ g(S X0 ∗ Y ∗ , Z ∗ ) =

 1 g([Y, Z ]∗ , X ∗ ) + g([X, Z ]∗ , Y ∗ ) . 2

In the particular case where the metric g is bi-invariant, we get [Z , Y ], X  + Y, [Z , X ] = 0,

X, Y, Z ∈ g,

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where ·, · denotes the restriction of the metric g to Te G = g. This implies that S 0 identically vanishes. Consequently, a Lie group endowed with a bi-invariant metric is locally symmetric (see [77]). The existence of a homogeneous structure with the special behaviour described in (2.4.15) can be used to characterize pseudo-Riemannian Lie groups among all (reductive) homogeneous pseudo-Riemannian manifolds (see also [25, 116]). In fact, we already illustrated above that if G is a Lie group equipped with a left-invariant pseudo-Riemannian metric, then the canonical connection associated to the homogeneous space (G × G)/G satisfies S X−∗ Y ∗ = ∇ X ∗ Y ∗ for all X, Y ∈ g. In particular, from a basis {e1 , . . . , en } of g we obtain a basis {X i = ei∗ } of complete vector fields such that S X−i X j = ∇ X i X j for all indices i, j. Conversely, suppose that (M = G/H, g) is a (connected, simply-connected, complete) reductive homogeneous pseudo-Riemannian manifold. Suppose that (M, g) admits a homogeneous pseudo-Riemannian structure S such that, with respect to the reductive decomposition g = m + h of the Lie algebra g of G, one has S X ∗ Y ∗ = ∇ X ∗ Y ∗ (equivalently, the corresponding canonical connection is given by ∇˜ X ∗ Y ∗ = 0), for all X, Y ∈ g. We shall now prove that h = 0. In fact, consider X ∈ h. Then, by definition, X o∗ = 0. Moreover, the Kostant operator A X ∗ = −∇ X ∗ (see Sect. 1.4.1) then satisfies

0 = X ∗ , Y ∗ o = ∇˜ X o∗ Y ∗ − ∇˜ Yo∗ X ∗ = A X ∗ |o (Y ∗ ), for all Y ∈ g. Therefore, as X o∗ = 0 and A X ∗ |o = 0, we conclude that X ∗ = 0 and so, X = 0. Thus, the isotropy subalgebra is trivial and M = G is a Lie group. Observe that by Lemma 2.3.1, a connected and simply-connected manifold has a Lie group structure if it admits a basis {X i } of complete vector fields such that [X i , X j ] = k cikj X k , with cikj constant for all indices i, j, k. If g is a pseudo-Riemannian metric on M and the above basis {X i } is orthonormal, then clearlyg is left-invariant. Moreover, the Koszul formula yields  at once that [X i , X j ] = k cikj X k for some constants cikj if and only if ∇ei e j = k ε j bikj ek for some constants bikj , where εi = g(ei , ei ). In such a case, the special pseudoRiemannian homogeneous structure S on M corresponding to its Lie group structure is explicitly determined by SXi =

1 k b e j ∧ ek , 2 j,k i j

where e j ∧ ek (·) = g(e j , ·)ek − g(ek , ·)e j .

Chapter 3

Locally Homogeneous Pseudo-Riemannian Manifolds

The conditions on a pseudo-Riemannian manifold (M, g) appearing in the statement of the Ambrose–Singer Theorem (Theorem 2.2.1 in Chap. 2) can be classified into two different types. On the one hand, we have a set of partial differential equations  = 0, ∇  R = 0, ∇  S = 0, ∇g =∇−S to be satisfied by a (1, 2)-tensor S (a homogeneous structure), where ∇ is the corresponding AS-connection and ∇ is the Levi-Civita connection. On the other hand, the geodesic completeness, together with the topological conditions of connectedness and simply-connectedness, are not related to S. The goal of this chapter is to study the results that a homogeneous structure still provides when the second type of conditions (with particular regard to the geodesic completeness) are not satisfied. In the case of positive definite metrics, since geodesic completeness is equivalent to metric completeness, the second group of conditions is entirely of a topological nature. In this Riemannian setting, we have the following known result ([115, Theorem 2.1]). Theorem 3.0.1 A Riemannian manifold (M, g) is locally homogeneous if and only if it admits a homogeneous structure S. The above equivalence sheds light on the role played by the topological conditions: they are connected to the global nature of the isometries of (M, g). When the manifold is not simply-connected but complete, its universal cover is homogeneous and the manifold is, at least, locally isometric to a global homogeneous manifold. The situation is more complicated when the manifold is not complete. There exist locally homogeneous manifolds that are not locally isometric to any global homogeneous manifold (see [83]). A non-complete manifold with a homogeneous structure may belong to this set of purely locally homogeneous manifolds. Globally homogeneous and locally homogeneous manifolds are more than global and particular © Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_3

59

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

cases of the same category, and hence, they may require different perspectives. All these considerations also apply when the manifold is endowed with an additional geometric structure defined by a set of tensors P1 , . . . , Pk . The panorama becomes more sophisticated for non-definite metrics. First, geodesic completeness is not equivalent to metric completeness and goes beyond topology. This situation opens a complex field of research that will not be tackled in this book. We will just explore the information that one can obtain from a homogeneous structure for non-geodesically complete manifolds. Secondly, Theorem 3.0.1 above is no longer true if g is a metric with signature. For example, a non-reductive homogeneous pseudo-Riemannian manifold is, in particular, locally homogeneous, but it does not possess any homogeneous structure. All the proofs of Theorem 3.0.1 known by the authors use the “canonical” ASconnection constructed by Kowalski [85]. The construction of this AS-connection relies strongly on the fact that the Killing form of so(T p M) is definite, which is true if the metric g is definite. Recall that in the non-definite case (see again Theorem 2.2.1, or the original result in [65]), a globally homogeneous pseudo-Riemannian manifold (G/H, g) admitting an AS-connection is reductive. This suggests that the generalization of Theorem 3.0.1 in the pseudo-Riemannian case will require an additional condition, playing the same role as the reductivity condition for globally homogeneous spaces. We provide that condition in Sect. 3.1, with or without additional invariant geometric structure. Furthermore, in Sect. 3.2 we show, under suitable conditions, how to adapt the construction of the “canonical” AS-connection made by Kowalski to metrics with signature. As a consequence we will see that, under those suitable conditions, a locally homogeneous pseudo-Riemannian manifold can be recovered from the curvature and its covariant derivatives at some point up to finite order. This order is known as the Singer homogeneous index in the Riemannian case (for example, see [97]). In this sense, we provide a generalization of this index for pseudo-Riemannian manifolds, with or without additional invariant geometric structure. Several results of this chapter develop the arguments introduced in [88].

3.1 Reductive Locally Homogeneous Manifolds 3.1.1 Pseudo-Riemannian Manifolds In this section we will need some basic notions on pseudo-groups and transitive Lie algebras. The reader is referred to [112] and its references for deeper results and examples in this framework. Here, we will just provide the following essential definitions. Definition 3.1.1 A (smooth) pseudo-group on a smooth manifold M is a set G of diffeomorphisms φ : U → M, defined on open sets U ⊂ M, such that

3.1 Reductive Locally Homogeneous Manifolds

61

Id M ∈ G, if φ ∈ G then φ−1 ∈ G, if φ : U → M belongs to G and V ⊂ U is open, then φ|V ∈ G, if φ : U → M, with U = ∪Uk a union of open sets, such that φ|Uk ∈ G, then φ ∈ G, 5. if φ : U → M and ψ : V → M belong to G and φ(U ) ⊂ V , then ψ ◦ φ ∈ G.

1. 2. 3. 4.

Definition 3.1.2 An (abstract) transitive Lie algebra is a pair (L , L 0 ), where L is a Lie algebra and L 0 is a proper subalgebra of finite codimension such that the only ideal of L contained in L 0 is {0}. Let (M, g) be a locally homogeneous pseudo-Riemannian manifold. We denote by I the pseudo-group of all local isometries which acts transitively on (M, g). All the elements of I satisfy the system of PDEs f ∗ g = g, turning I into a so-called Lie pseudo-group. We refer the reader again to [112] for a formal definition of this concept. For our purposes, all the Lie pseudo-groups will be pseudo-groups of local isometries and there will be no need to deal with a complete definition of this notion. The corresponding system of Lie equations is given by L X g = 0,

(3.1.1)

for vector fields X defined on open neighbourhoods, that is, local Killing vector fields. We fix a point p ∈ M and a coordinate system (x 1 , ..., x m ) around p. We denote by ei = (∂/∂x i ) p , i = 1, ..., m, and (e1 , . . . , em ) the associated dual basis in T p∗ M. We consider then a transitive Lie algebra (i, i0 ) associated with the system (3.1.1) as follows: • The Lie algebra i is the set of vector-valued formal power series in T p M ξ=

∞ 

m 

ξ ij1 ... jr ei ⊗ e j1  . . .  e jr ,

ξ ij1 ... jr ∈ R,

r =0 i, j1 ,..., jr =1

such that ξ ij1 ... jr solve (3.1.1) and all its derivatives. This means that, making the formal identification   ∂r ξi i , (3.1.2) ξ j1 ... jr = ∂x j1 · · · ∂x jr p the local expression of the Eq. (3.1.1) given by ξi

∂g jk ∂ξ i ∂ξ i + gik j + g ji k = 0, i ∂x ∂x ∂x

j, k = 1, . . . , m,

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

together with all possible derivatives with respect to any finite collection of ∂/∂x 1 , . . . , ∂/∂x m , are satisfied at p. • The subalgebra i0 is formed by all the elements of i such that the terms of order zero ξ i vanish. • For the Lie bracket [ξ, η] in i, we consider again the formal identification (3.1.2) and define ∂r (ξ j η ij − η j ξ ij ). [ξ, η]ij1 ,..., jr = ∂x j1 · · · ∂x jr It is now easy to check that i0 does not contain non-trivial ideals. According to [112], we have the following result. Proposition 3.1.3 An element ξ ∈ i is completely determined by the terms of order 0 and 1. Furthermore, the matrix (ξ ij ) belongs to so(T p M) with respect to the metric gp. Definition 3.1.4 A Killing generator at p is a pair (X, A) ∈ T p M × so(T p M) satisfying i ≥ 0, A · ∇ i R p + i X ∇ i+1 R p = 0, where ∇ is the Levi-Civita connection of g. The set kill of Killing generators at p has a Lie algebra structure with bracket [(X, A), (Y, B)] = (AX − BY, (R p ) X Y + [A, B]). We define kill0 = {(X, A) ∈ kill/ X = 0}. Lemma 3.1.5 ([112]) The pair (kill, kill0 ) is a transitive Lie algebra isomorphic to (i, i0 ). Proof Let (x 1 , . . . , x m ) be a set of normal coordinates around p. We consider the map i → kill (ξ i , ξ ij ) → (ξ i (∂/∂x i ) p , ξ ij (∂/∂x i ⊗ d x j ) p ), where (ξ i , ξ ij ) are the terms of order 0 and 1 characterizing an element ξ ∈ i. A straightforward calculation shows that this defines a Lie algebra isomorphism.  We now consider a vector field (which, for the sake of simplicity, will also be denoted by ξ) defined on a connected and simply-connected neighbourhood U of p. We also consider the (1, 1)-tensor field Aξ = Lξ − ∇ξ = −∇ξ. We can replace the Killing condition Lξ g = 0 on U by the set of conditions at p (Lξ g) p = 0, (Lξ ∇ i R) p = 0,

i ≥ 0.

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63

These conditions are equivalent to A · g p = 0, A · ∇ i R p + i X ∇ i+1 R p = 0,

i ≥ 0,

for X = ξ p and A = Aξ | p . Hence, (ξ p , Aξ | p ) is a Killing generator. Corollary 3.1.6 Every formal solution ξ ∈ i is realized by the germ of a local Killing vector field. Proof Adapting the arguments used by Nomizu in [98] to metrics with signature, we can see that if the dimension of the Lie algebra of Killing generators is constant on M, then for every Killing generator (X, A) at a point p there exist a local Killing vector field ξ with (X, A) = (ξ p , Aξ | p ).  The Lie algebra isomorphism exhibited in the proof of Lemma 3.1.5 can be seen as

i → kill [ξ] → (ξ p , Aξ | p ),

where [ξ] denotes the germ of the local vector field ξ at p. We consider a Lie pseudo-group of isometries G ⊂ I acting transitively on (M, g) and the Lie subalgebra g ⊂ i of germs of local Killing vector fields with 1-parameter group contained in G. The Lie algebra k formed by those [ξ] ∈ g vanishing at p is thus a Lie subalgebra of i0 , and the pair (g, k) is a transitive Lie algebra. Definition 3.1.7 Let G be a Lie pseudo-group of isometries acting transitively on (M, g). The isotropy pseudo-group at a point p ∈ M is H p = { f ∈ G/ f ( p) = p} ⊂ G. Since f ( p) = p is not a differential equation, H p is not a Lie pseudo-group in general. For this reason it is more convenient to work with the linear isotropy group. Definition 3.1.8 The linear isotropy group of G at p ∈ M is H p = {F : T p M → T p M/ F = f ∗ , f ∈ H p }. Since every f ∈ H p is an isometry, H p is a Lie subgroup of O(T p M). Lemma 3.1.9 The Lie algebra h p of H p is isomorphic to k. Proof We define the map

k → hp [ξ] → dtd t=0 ( f t )∗ ,

where f t ⊂ H p is the 1-parameter group generated by ξ. A simple inspection shows that this map is a Lie algebra isomorphism. 

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

Note that the previous isomorphism between k and h p can be read as k → hp [ξ] → Aξ | p . There is a natural action of H p on g given by Ad : H p × g → g (F, [ξ]) → [η] , with

 d  f ◦ ϕt ◦ f −1 (q), ηq = dt t=0

for every q in a neighbourhood of p, where ϕt is the 1-parameter group generated by [ξ], and F = f ∗ . When identifying k with h p , the restriction of this action to k is just the usual adjoint action of H p on its Lie algebra. The notation Ad is thus consistent. Definition 3.1.10 Let (M, g) be a pseudo-Riemannian manifold, and let G be a Lie pseudo-group of isometries acting transitively on (M, g). We will say that the triple (M, g, G) is reductive at a point p ∈ M if the transitive Lie algebra (g, k) associated to G at p can be decomposed as g = m ⊕ k, where m is Ad(H p )-invariant. The definition of reductivity does not depend on the point p, as the following result proves. Proposition 3.1.11 If (M, g, G) is reductive at a point p ∈ M, then it is reductive at every point q ∈ M. Proof Let q be another point in M. We denote by (g p , k p ) and (gq , kq ) the transitive Lie algebras associated to G at p and q respectively. Let h ∈ G be a local isometry with h( p) = q. The map h induces isomorphisms hˆ : g p → gq [ξ] → [h ∗ (ξ)]

and hˇ : H p → Hq F → h ∗ ◦ F ◦ h −1 ∗ .

ˆ p ) ⊂ gq . It is Let g p = m p ⊕ k p with m p Ad(H p )-invariant, we define mq = h(m ˆ obvious that gq = mq ⊕ kq , since h is an isomorphism and takes k p to kq . We now show that mq is Ad(Hq )-invariant and independent of the local isometry h. Given F ∈ Hq , consider f ∈ Hq with F = f ∗ . Let [η] ∈ mq . There is an element [ξ] ∈ m p with η = h ∗ (ξ). The 1-parameter group generated by η is thus φt = h ◦ ϕt ◦ h −1 , where ϕt is the 1-parameter group generated by ξ. Therefore,

3.1 Reductive Locally Homogeneous Manifolds

65

  d  −1 f ◦ φ ◦ f t dt    t=0 d  −1 −1 = f ◦ h ◦ ϕ ◦ h ◦ f t dt    t=0 d  −1 −1 −1 −1 h ◦ h ◦ f ◦ h ◦ ϕ ◦ h ◦ f ◦ h ◦ h = t dt t=0

= h ∗ Adhˇ −1 (F) ([ξ]) . 

Ad F ([η]) =

Since hˇ −1 (F) ∈ H p , we have Ad F ([η]) ∈ mq . On the other hand, in order to prove the independence of h, it is enough to prove that for other h  ∈ G with h  ( p) = q we  have that h −1 ∗ ◦ h ∗ ([ξ]) ∈ m p . But h −1 ∗

◦

h ∗ ([ξ])

 =

  d  −1  h ◦ h ◦ ϕt = Ad(h −1 ◦h  )∗ ([ξ]). dt t=0

 Since h −1 ◦ h  ∈ H p and m p is Ad(H p )-invariant, we conclude that h −1 ∗ ◦ h ∗ ([ξ]) ∈  mp.

Remark 3.1.12 Note that being reductive is a property of the triple (M, g, G) rather than a property of the pseudo-Riemannian manifold (M, g) itself. In Sect. 3.4 we will observe that the same locally homogeneous pseudo-Riemannian manifold can be reductive for the action of a certain Lie pseudo-group G, whereas it is non-reductive for the action of another Lie pseudo-group G  . Remark 3.1.13 If the metric g is definite, any pseudo-group of isometries G acting transitively is reductive. Indeed, we endow T p M × so(T p M) with a so(T p M)invariant inner product as the sum of g p in the factor T p M and any adjoint invariant inner product in so(T p M) (the Killing metric if dim(T p M) > 2 or any inner product if dim(T p M) ≤ 2). Then, one can take m as the orthogonal complement k⊥ with respect to the restriction of that inner product to k ⊂ i  T p M × so(T p M). This construction cannot always be done in the indefinite case, because the restriction of the inner product may be degenerate. The following two theorems characterize locally homogeneous pseudoRiemannian manifolds admitting an AS-connection (see Definition 2.3.5). Theorem 3.1.14 Let (M, g, G) be a reductive locally homogeneous pseudoRiemannian manifold. Then, (M, g) admits an AS-connection. Proof Let (r, s) be the signature of g, and let O(M) be the bundle of orthonormal references of M. We fix a point p ∈ M and a reference u 0 ∈ O(M) in the fiber of p. We shall interpret an orthonormal reference u at q ∈ M as an isometry u : (Rm , ·, ·) → (Tq M, gq ), where ·, · is the standard metric of Rm with signature (r, s). Consider the set

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

Q = {u ∈ O(M)/ u =  h(u 0 ), h ∈ G},

(3.1.3)

where  h is the map induced on O(M) by a local isometry h. Q determines a reduction of O(M) with structure group H¯ = {B ∈ O(r, s)/ uˆ 0 (B) = f ∗ , f ∈ H p }, ˆ 0 gives an where uˆ 0 : O(r, s) → O(T p M), B → u 0 ◦ B ◦ u −1 0 . It is obvious that u isomorphism between H¯ and the linear isotropy group H p . The right action of an element B ∈ H¯ on a reference u ∈ Q at q is given by R B (u) = u ◦ B : Rm → Rm → Tq M. Let F = uˆ 0 (B) ∈ H p and f ∈ H p with F = f ∗ . Taking h ∈ G such that u =  h(u 0 ), we can write R B (u) = u ◦ B = u ◦ u 0 ◦ F ◦ u 0 = h ∗ ◦ u 0 ◦ u −1 0 ◦ F ◦ u 0 = h ∗ ◦ f∗ ◦ u 0 = h◦  f (u 0 ). We now consider the map  : g → Tu 0 Q ξu 0 = [ξ] → 



d  dt t=0

ϕ t (u 0 ),

where ϕt is the 1-parameter group of ξ. Recall that  ξ is the natural lift of ξ, as defined in Lemma 1.4.9.  is injective because {ϕt } ⊂ G and the action of G on Q is free. Moreover, dimg = dimT p M + dimk = dimT p M + dimVu 0 Q = dimTu 0 Q, whence  is a linear isomorphism. Let g = m ⊕ k be a reductive decomposition. We define the horizontal subspace at u 0 as Hu 0 Q = (m), and making use of G we define a horizontal distribution on Q as h ∗ (Hu 0 ), Hu Q = 

u = h(u 0 ).

This horizontal distribution is C ∞ and invariant under G. In order to see that H Q  on M, we just have to show that it is equivariant under defines a linear connection ∇ the right action of H¯ .

3.1 Reductive Locally Homogeneous Manifolds

67

For B ∈ H¯ , we take F = uˆ 0 (B), and f ∈ H p with F = f ∗ . Let X u ∈ Hu Q. Then, h ∗ (X u 0 ) for some X u 0 ∈ Hu 0 Q and some h such that u =  h(u 0 ). by definition X u =  h ∗ (([ξ])) for some [ξ] ∈ m. Let ϕt be the 1-parameter group This means that X u =  generated by ξ. We thus have h ∗ ◦ ([ξ]) (R B )∗ (X u ) = (R B )∗ ◦    d   d   RB ◦ h ◦ ϕ t (u 0 ) = f (u 0 ) h◦ϕ t ◦  = dt t=0 dt t=0  d    −1 = t ◦  f (u 0 ) h◦ f ◦ f ◦ϕ dt t=0    d  −1     = (h ◦ f )∗ t ◦ f (u 0 ) f ◦ϕ dt t=0 = ( h◦  f )∗ ((Ad F −1 ([ξ]))) . Since Ad F −1 ([ξ]) ∈ m, we have (Ad F −1 ([ξ])) ∈ Hu 0 Q, whence (R B )∗ (X u ) ∈ h◦  f (u 0 ). H R B (u) Q since R B (u) =   We now study the properties of the connection ∇.  is metric, that is, First of all, since Q is a reduction of O(M), the connection ∇  = 0. On the other hand, the connection ∇  can be characterized in the following ∇g way. Let p, q ∈ M, and let γ be a path in M with γ(0) = p and γ(1) = q. We denote  by γ¯ the horizontal lift of γ to u 0 ∈ Q with respect to ∇. The parallel transport along γ with respect to this connection is thus the linear ¯ But since u = isometry γ : T p M → Tq M given by γ = u ◦ u −1 0 , where u = γ(1).  h(u 0 ) = h ∗ ◦ u 0 for some h ∈ G, we have that the linear isometry γ is exactly h ∗ .  and curvature R  are invariant  implies that its torsion T This characterization of ∇  = 0 and ∇  = 0.  is invariant under G, that is, ∇ T R under parallel transport, since ∇ As usual, these two equations are equivalent to  R = 0, ∇

 S = 0, ∇

 with ∇ the Levi-Civita connection of where R is the curvature of g, and S = ∇ − ∇  is an AS-connection. g. Therefore, ∇  Theorem 3.1.15 Let (M, g) be a pseudo-Riemannian manifold admitting an AS Then there is a Lie pseudo-group of isometries G such that (M, g, G) connection ∇. is reductive locally homogeneous.  is an ASProof Consider two points p, q ∈ M and a path γ from p to q. Since ∇  connection, the parallel transport γ : T p M → Tq M with respect to ∇ is a linear  Therefore, there exist neighisometry preserving the torsion and curvature of ∇. p q bourhoods U and U and an affine transformation f γ : U p → U q with respect to  such that its differential at p coincides with the parallel transport along γ (see [79, ∇ Vol. I, Chap. VI]).

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

 is metric, we have that f γ is an isometry. We consider the set Since ∇ G = { f γ / γ is a path from p to q}. Then, G is a pseudo-group of local isometries of (M, g), which acts transitively on (M, g), so that (M, g) is locally homogeneous. In addition, G coincides with  which consists of all local affine maps of ∇  the so-called transvection group of ∇,    f (P ∇ ) ⊂ P ∇ . This gives G preserving its holonomy bundle P ∇ , that is, such that  the structure of a Lie pseudo-group. We just have to show that it is reductive. For a fixed point p ∈ M, the isotropy pseudo-group is 1:1

H p = { f γ / f γ ( p) = p} ↔ {loops based at p}. The linear isotropy group is thus 

H p = { f ∗γ : T p M → T p M/ f γ ∈ H p } = Hol∇ . Therefore, denoting by (g, k) the transitive Lie algebra associated to G, we have  k  hol∇ . Fix an orthonormal reference u 0 at p and consider the bundle Q defined as in  at u 0 and so, the (3.1.3). It is obvious that Q is exactly the holonomy bundle of ∇  reduces to Q and determines a horizontal distribution H Q, which is connection ∇ invariant under the right action of H p and the left action of G on Q (recall that all  We again take the linear map the elements of G are affine maps with respect to ∇).  : g → Tu 0 Q [ξ] →  ξu 0 . As seen before,  is a linear isomorphism. We consider the subspace m =  −1 (Hu 0 Q) ⊂ g. Obviously, g = m ⊕ k, as (k) = Vu 0 Q. In addition, let [ξ] ∈ m with 1-parameter  group ϕt , and let F = f ∗ ∈ H p . Recall that Ad F ([ξ]) = [η] with ηq = dtd t=0 f ◦ ϕt ◦ f −1 (q) for every q in a neighbourhood of p. Hence,   d   f −1 (u 0 ) =  f ∗ (R F −1 )∗ ( ξu 0 ) . (Ad F ([ξ])) = f ◦ ϕt ◦   dt t=0 Since [ξ] ∈ m, we have that  ξu 0 ∈ Hu 0 Q, whence by the invariance and the equivariance of the horizontal distribution, we find    ξu 0 ) ∈  f ∗ H R F −1 (u 0 ) Q = Hu 0 Q. f ∗ (R F −1 )∗ (

3.1 Reductive Locally Homogeneous Manifolds

69

Therefore, m is Ad(H p )-invariant, showing that (M, g) is reductive.



Let (M, g) be a globally homogeneous pseudo-Riemannian manifold with M = G/H . Obviously, G = G can also be seen as a pseudo-group of isometries acting transitively on M. We now prove that the notion of local reductivity that we have defined is compatible with the standard definition of reductive homogeneous space. We consider the isotropy group H = H p at a point p ∈ M and denote by g and h the Lie algebras of G and H respectively. Recall that (M, g, G) is said to be reductive if g = m ⊕ h for some Ad(H p )-invariant subspace m ⊂ g (see Definition 1.4.7). We denote by (g , k ) the transitive Lie algebra associated to G, viewed as a Lie pseudo-group of local isometries, i.e., the set of germs of local infinitesimal transformations of G. The linear isotropy group as defined in Definition 3.1.7 is just the image of H p under the linear isotropy representation λ (see (1.4.9)). We also recall the definition of an infinitesimal generator of the action, as introduced in (1.1.1): given α ∈ g, we define the vector field α∗ on M as αq∗

 d  = L exp(tα) (q), dt t=0

q ∈ M,

where L a denotes the left action of a ∈ G on M. We consider the map φ : g → g α → [α∗ ] . Note that φ is not a Lie algebra homomorphism, because [α, β]∗ = −[α∗ , β ∗ ]. Nevertheless, we show that it is a linear isomorphism. Consider α ∈ g such that [α∗ ] = 0. Then, α∗ = 0 in a neighbourhood around p. In particular, α∗p = 0 and Aα∗ | p = 0, so that α∗ = 0. This implies α = 0, that is, φ is injective. On the other hand, consider [ξ] ∈ g and  the 1-parameter group of ξ, which determines a curve ϕt ⊂ G. Taking α = dtd t=0 ϕt , we have φ(α) = [ξ], and so, φ is surjective. In addition, if h ∈ H p and h ∗ ∈ λ(H p ), the following diagram is commutative: g

φ



Adh

g

φ

g Adh ∗

g

In fact, if α ∈ g, then Adh ∗ (α∗ ) = [η] with 

d  ∗ ∗ −1 = (L h )∗ ηq = α L ◦ L ◦ L h exp tα h L h −1 (q) = (Ad h (α))q . dt t=0

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

We conclude that, making use of φ, one can transform reductive complements of (g, h) to reductive complements of (g , k ) and vice versa. Henceforth, the notions of reductivity from both the global and the local points of view agree.

3.1.2 Locally Homogeneous Pseudo-Riemannian Manifolds with Invariant Geometric Structures We now consider a pseudo-Riemannian manifold (M, g) endowed with a geometric structure given by a set of tensor fields P = (P1 , . . . , Pk ). We say that the geometric structure given by P is invariant if the Lie pseudo-group of isometries J preserving P, that is J = { f ∈ I/ f ∗ P = P}, acts transitively on M. The corresponding Lie equation is L X P = 0, so that the infinitesimal transformations of J are Killing vector fields which are infinitesimal automorphisms of the geometric structure. A vector field ξ satisfying both Lξ g = 0 and Lξ P = 0 will be called a geometric Killing vector field. We consider the Lie algebra j ⊂ i, which consists of germs of geometric Killing vector fields. The Lie subalgebra j0 ⊂ i0 is defined as the set of elements of j vanishing at p, so that (j, j0 ) is a transitive Lie algebra. Let gkill be the subalgebra of kill containing all the Killing generators (X, A) satisfying j ≥ 0. A · ∇ j Pp + i X ∇ j+1 Pp = 0, Setting gkill0 = kill0 ∩ gkill, we have the following result. Lemma 3.1.16 Let ξ be a Killing vector field and ω a tensor field. If Lξ ω = 0 then Lξ (∇ω) = 0. Proof For the sake of simplicity we illustrate the proof for the case when ω is a 1-form. The generalization for tensor fields of arbitrary type is straightforward. By direct calculation, we have   Lξ (∇ω)(X, Y ) = −ξ · (ω(∇ X Y )) + ω ∇Lξ X Y + ω ∇ X Lξ Y . Making use of Lξ ω = 0, we obtain   Lξ (∇ω)(X, Y ) = ω (Lξ ∇)(X, Y ) = ω Rξ X Y + ∇ X2 Y ξ . But Rξ + ∇ 2 ξ = 0, since it is just the affine Jacobi equation applied to a Killing vector field ξ. 

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71

Proposition 3.1.17 The transitive Lie algebra (gkill, gkill0 ) is isomorphic to (j, j0 ). Proof Let ξ be a geometric Killing vector field and (X, A) = (ξ p , Aξ | p ). By definition we have A · ∇ j P = Lξ (∇ j P) p − ∇ξ ∇ j Pp = Lξ (∇ j P) p − i X ∇ j+1 Pp , and applying Lemma 3.1.16 we obtain that (ξ p , Aξ | p ) ∈ gkill. Making use of Lemma 3.1.5 and Corollary 3.1.6, we see that the map j → gkill [ξ] → (ξ p , Aξ | p ) is a Lie algebra isomorphism taking j0 to gkill0 .



We now consider a Lie pseudo-group G ⊂ J acting transitively on M. We associate to G the Lie algebra g ⊂ j consisting of germs of local geometric Killing vector fields with 1-parameter group contained in G. The Lie algebra k consisting of those [ξ] ∈ g vanishing at p is thus a Lie subalgebra of j0 , and the pair (g, k) is a transitive Lie algebra. We take the isotropy pseudo-group H p and the linear isotropy group H p associated to G. As before, we have that H p is a Lie subgroup of the stabilizer of Pp inside O(T p M) and k  h p . Recall that we have the action Ad of H p on g. Definition 3.1.18 Let (M, g, P) be a pseudo-Riemannian manifold endowed with a geometric structure defined by a tensor field P. Let G be a Lie pseudo-group of isometries acting transitively on (M, g, P) and preserving P. We will say that (M, g, P, G) is reductive if the transitive Lie algebra (g, k) associated to G can be decomposed as g = m ⊕ k, where m is Ad(H p )-invariant. Theorem 3.1.19 Let (M, g, P, G) be a reductive locally homogeneous pseudoRiemannian manifold with P invariant. Then, (M, g, P) admits an ASK-connection. Proof Let (M, g, P, G) be a reductive locally homogeneous pseudo-Riemannian  manifold with P invariant, by Theorem 3.1.14, (M, g) admits an AS-connection ∇.  P = 0. However, recall that ∇  is characterized as the We just have to show that ∇ linear connection whose parallel transport coincides with the differential h ∗ for some h ∈ G. Since G preserves P, we have that P is invariant under parallel transport with  whence ∇  P = 0. respect to ∇,  Theorem 3.1.20 Let (M, g, P) be a pseudo-Riemannian manifold admitting a ASK Then, there is a Lie pseudo-group of isometries G acting transitively on connection ∇. (M, g, P) and preserving P, such that (M, g, P, G) is reductive locally homogeneous with P invariant.

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Proof As in the proof of Theorem 3.1.15, we consider the Lie pseudo-group G = { f γ / γ is a path from p to q}.  and ∇  P = 0, we have that P is Since the local maps f γ are affine maps of ∇, invariant under G. The same arguments used in the proof of Theorem 3.1.15 show that (M, g, P) is reductive locally homogeneous with P invariant. 

3.1.3 Local and Global Homogeneity As we mentioned in the introduction of this chapter, there are locally homogeneous manifolds that are not locally isometric to global homogeneous manifolds (see for example [83]). Roughly speaking, these locally homogeneous manifolds possess a different local geometry from the one of homogeneous spaces. It is natural to ask which conditions locally homogeneous manifolds must satisfy in order to have these kind of isometries. Here we provide a sufficient condition for this problem within the framework of reductive locally homogeneous pseudoRiemannian manifolds introduced in the previous paragraphs. This approach follows the works of [82, 112] for the Riemannian case, as well as their generalization given in [88] for the indefinite case. Proposition 3.1.21 Let (M, g, G) be a reductive locally homogeneous pseudo If the Riemannian manifold endowed with an associated AS-connection ∇.  is regular (see Definition 2.4.3), then (M, g) is infinitesimal model (K , T ) of ∇ locally isometric to a reductive globally homogeneous pseudo-Riemannian mani fold. The same holds if the transvection algebra (g0 , hol∇ ) is regular. Proof Consider a point p ∈ M and the Nomizu construction g0 = T p M ⊕ h0 associated to (K , T ). Let G 0 be the simply-connected Lie group with Lie algebra g0 , and H0 its connected subgroup with Lie algebra h0 . If (K , T ) is regular, then H0 is closed in G 0 , so that we can consider the homogeneous space G 0 /H0 . Moreover, G 0 /H0 is reductive as g0 = T p M ⊕ h0 is a reductive decomposition, and the tangent space of G 0 /H0 at the origin o is identified with T p M through a linear isomorphism F : T p M → T0 (G 0 /H0 ). This homogeneous space is thus endowed with a G 0 -invariant pseudo-Riemannian metric inherited from g at p. can associated to this reductive decomWe consider the canonical connection ∇ position (see Definition 1.4.11). Under the identification F, the curvature and tor coincide with K and T respectively. Thus, there is a linear isometry sion of ∇  and ∇ can . ThereF : T p M → T0 (G 0 /H0 ) preserving the curvature and torsion of ∇ fore, there are open neighbourhoods U and V of p and o, and an affine transformation  and ∇ can taking p to o (see [79, Vol. I, Chap. VI]). f : U → V with respect to ∇ Since both connections are metric we have that f is an isometry. The same arguments can be applied substituting the Nomizu construction with the transvection algebra. 

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73

Remark 3.1.22 Proposition 3.1.21 can be adapted in a straightforward way to the case when (M, g) is endowed with an invariant geometric structure.

3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds Let (M, g) be a pseudo-Riemannian manifold endowed with a geometric structure defined by a tensor field P, and let p be a point in M. For every pair of non-negative integers r, s ≥ 0 we consider the Lie algebras g( p, r ) ⊂ so(T p M) and p( p, s) ⊂ so(T p M) defined as  

g( p, r ) = A ∈ so(T p M), A · ∇ i R p = 0, i = 0, . . . , r ,  

p( p, s) = A ∈ so(T p M), A · ∇ j Pp = 0, j = 0, . . . , s , where A acts as a derivation on the tensor algebra of T p M. We thus have filtrations so(T p M) ⊃ g( p, 0) ⊃ . . . ⊃ g( p, r ) ⊃ . . . so(T p M) ⊃ p( p, 0) ⊃ . . . ⊃ p( p, s) ⊃ . . . Let k( p) and l( p) be the first integers such that g( p, k( p)) = g( p, k( p) + 1) and p( p, l( p)) = p( p, l( p) + 1), and let h( p, r, s) = g( p, r ) ∩ p( p, s). We consider the complex of filtrations so(T p M) ∪ p( p, 0) ∪ .. . ∪ p( p, l( p)) || p( p, l( p) + 1)

⊃

g( p, 0) ∪ ⊃ h( p, 0, 0) ∪ .. . ∪ ⊃ h( p, 0, l( p)) || ⊃ h( p, 0, l( p) + 1)

⊃ ... ⊃

g( p, k( p)) ∪ ⊃ ... ⊃ h( p, k( p), 0) ∪ .. . ∪ ⊃ . . . ⊃ h( p, k( p), l( p)) || ⊃ . . . ⊃ h( p, k( p), l( p) + 1)

=

g( p, k( p) + 1) ∪ = h( p, k( p) + 1, 0) ∪ .. . ∪ = h( p, k( p) + 1, l( p)) || = h( p, k( p) + 1, l( p) + 1).

For the sake of notational completeness, we define g( p, −1) = so(T p M),

p( p, −1) = so(T p M),

so that h( p, −1, s) = p( p, s) and h( p, r, −1) = g( p, r ). We shall call a pair (r ( p), s( p)) of integers in the set N ∪ {0, −1} a stabilizing pair at p ∈ M if r ( p) ≤ k( p), s( p) ≤ l( p) and

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

h( p, r ( p), s( p)) = h( p, r ( p) + 1, s( p)) || || h( p, r ( p), s( p) + 1) = h( p, r ( p) + 1, s( p) + 1). Note that (k( p), l( p)) is a stabilizing pair. Remark 3.2.1 In Sect. 3.4 we exhibit an example of a manifold with a stabilizing pair distinct from (k( p), l( p)). The following definition generalizes the definition of infinitesimal homogeneous space given by Singer ([97]). Definition 3.2.2 Consider a pair of integers (r, s) ∈ (N ∪ {0, −1})2 . We say that (M, g, P) is (r, s)-infinitesimally P-homogeneous if for every p, q ∈ M there is a linear isometry F : T p M → Tq M such that F ∗ (∇ i Rq ) = ∇ i R p ,

i = 0, . . . , r + 1,

F ∗ (∇ j Pq ) = ∇ j Pp ,

j = 0, . . . , s + 1.

If (r ( p), s( p)) is a stabilizing pair at p and (M, g, P) is (r ( p), s( p)) -infinitesimally P-homogeneous, then (r ( p), s( p)) is a stabilizing pair at any point q ∈ M (so that we can omit the point p). In fact, any isometry F : T p M → Tq M with F ∗ (∇ i Rq ) = ∇ i R p and F ∗ (∇ j Pq ) = j ∇ Pp for i = 0, . . . , r ( p) + 1 and j = 0, . . . , s( p) + 1 induces isomorphisms between h( p, i, j) and h(q, i, j) for i ≤ r ( p) and j ≤ s( p). Let H ( p, r, s) be the stabilizing group of the tensors ∇ i R p , and ∇ j Pp , 0 ≤ i ≤ r + 1, 0 ≤ j ≤ s + 1, inside O(T p M). It is evident that h( p, r, s) is the Lie algebra of H ( p, r, s). Obviously, a locally homogeneous pseudo-Riemannian manifold with P invariant is in particular (r, s)-infinitesimally P-homogeneous for every pair (r, s). We shall see that the converse is also true. Definition 3.2.3 Let (r, s) be a stabilizing pair at p ∈ M. We say that (M, g, P) is (r, s)-strongly reductive at p if there is an Ad(H ( p, r, s))-invariant subspace n( p, r, s) ⊂ so(T p M) such that so(T p M) = h( p, r, s) ⊕ n( p, r, s). Lemma 3.2.4 Let (M, g, P) be an (r, s)-infinitesimally P-homogeneous manifold. If (M, g, P) is (r, s)-strongly reductive at p ∈ M, then it is (r, s)-strongly reductive at every point q ∈ M. Proof Let q ∈ M be another point distinct from p. Recall that (r, s) is also a stabilizing pair at q. Let F : T p M → Tq M be a linear isometry such that F ∗ (∇ i Rq ) = ∇ i R p and F ∗ (∇ j Pq ) = ∇ j Pp for i = 0, . . . , r + 1 and j = 0, . . . , s + 1. F induces a lin : so(T p M) → so(Tq M) given by A → F ◦ A ◦ F −1 . ear isomorphism F

3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds

75

 p, r, s)) = h(q, r, s). Let n( p, r, s) be an By construction it is obvious that F(h( Ad(H ( p, r, s))-invariant complement to h( p, r, s). We define  p, r, s)) ⊂ so(Tq M). n(q, r, s) = F(n( This subspace is independent of the isometry F. Indeed, let G : T p M → Tq M be another linear isometry with G ∗ (∇ i Rq ) = ∇ i R p and G ∗ (∇ j Pq ) = ∇ j Pp for i = 0, . . . , r + 1 and j = 0, . . . , s + 1. The composition G −1 ◦ F is an element of O(T p M). Moreover, G −1 ◦ F stabilizes R p , . . . , ∇ r +1 R p and Pp , . . . , ∇ s+1 Pp , so that it is an element of H ( p, r, s). Hence, for any A ∈ n( p, r, s) we have  −1 ◦ F(A) = Ad G −1 ◦F (A) ∈ n( p, r, s), G  p, r, s)) does not depend on the linear isometry F. We finally showing that F(n( show that n(q, r, s) is Ad(H (q, r, s))-invariant. Given B ∈ n(q, r, s), there is an  A ∈ n( p, r, s) with B = F(A). Let b ∈ H (q, r, s) and take a = F −1 ◦ b ◦ F ∈ H ( p, r, s). Then  Adb (B) = b ◦ B ◦ b−1 = F ◦ a ◦ A ◦ a −1 ◦ F −1 = F(Ad a (A)), which belongs to n(q, r, s), since Ada (A) ∈ n( p, r, s).



By virtue of the previous lemma, we will say that an (r, s)-infinitesimally P-homogeneous manifold (M, g, P) is (r, s)-strongly reductive if it is (r, s)-strongly reductive at some point of M. The term “strongly reductive” is motivated by Proposition 3.2.12 and Example 3.4.2, which show that strong reductivity implies reductivity but the converse is not true. Remark 3.2.5 In the case when g is Riemannian, the Killing form of so(T p M) is definite, so that the strong reductivity condition is automatically satisfied choosing for n( p, r, s) the orthogonal complement of h( p, r, s) inside so(T p M) with respect to the Killing form. When the presence of an extra geometric structure is not taken into account, the integer k( p) stabilizing the filtration so(T p M) ⊃ g( p, 0) ⊃ . . . ⊃ g( p, r ) ⊃ . . . is a pseudo-Riemannian invariant of (M, g) known as the Singer invariant. In this case, the choice of g( p, k( p))⊥ as complement of g( p, k( p)) leads to the canonical AS-connection constructed by Kowalski in [85] in a similar way to the proof of Theorem 3.2.9 below. Let π : O(M) → M be the bundle of orthonormal frames with structure group O(ν, n − ν), where ν is the index of the metric. Let u 0 ∈ O(M) with π(u 0 ) = p, and P0 = u ∗0 (Pp ). Let T be the space of tensors to which P0 belongs. For any pair of integers (r, s) ∈ (N ∪ {0, −1})2 consider the following O(ν, n − ν)-equivariant map:

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

k+1 r +4 n ∗ s+1  j n ∗ (r,s) : O(M) → i=0 (R ) ⊕ j=0 (R ) ⊗ T u → u ∗ (Rπ(u) , . . . , ∇ r +1 Rπ(u) , Pπ(u) , . . . , ∇ s+1 Pπ(u) ). Lemma 3.2.6 Let (M, g, P) be (r, s)-infinitesimally P-homogeneous. Then (r,s) (O(M)) is a single O(ν, n − ν)-orbit. Proof Let u ∈ O(M) and define  = (r,s) . If π(u 0 ) = π(u), then u 0 and u are in the same O(ν, n − ν)-orbit, and since  is O(ν, n − ν)-equivariant, we have that (u 0 ) and (u) are in the same O(ν, n − ν)-orbit. If π(u 0 ) = π(u), let q = π(u). Then, there is a linear isometry F : T p M → Tq M such that F ∗ (∇ i Rq ) = ∇ i R p for = 0, . . . , r + 1, and F ∗ (∇ j Pq ) = ∇ j Pp for j = 0, . . . , s + 1.  : O(M) → O(M) such that  ◦ F  = . Since π(u) = F induces a map F  0 )), we conclude that (u 0 ) and (u) are in the same O(ν, n − ν)π( F(u orbit.  Lemma 3.2.7 If (M, g, P) is an (r, s)-infinitesimally P-homogeneous manifold, then there is a metric connection ∇¯ such that ∇¯ X (∇ i R) = 0 for i = 0, . . . , r + 1, and ∇¯ X (∇ j P) = 0 for j = 0, . . . , s + 1. Proof Let u 0 ∈ P with π(u 0 ) = p and  = (r,s) . By Lemma 3.2.6, the orbit (P) is the homogeneous space O(ν, n − ν)/I0 , where I0 is the isotropy group at (u 0 ). We thus have an equivariant map  : O(M) → O(ν, n − ν)/I0 , so that Q = −1 ((u 0 )) determines a reduction of O(M) with group I0 . Since  restricted to Q is constant, all the tensor fields ∇ i R and ∇ j P, i = 0, . . . , r + 1, j = 0, . . . , s + 1, will be parallel with respect to any connection adapted to Q.  Lemma 3.2.8 If (M,  g, P) is an (r, s)-infinitesimally P-homogeneous manifold, then h(M, r, s) = q∈M h(q, r, s) is a vector subbundle of so(M). Moreover, if (M, g, P) is (r, s)-strongly reductive, then n(M, r, s) =



n(q, r, s)

q∈M

is a vector subbundle of so(M) and so(M) = h(M, r, s) ⊕ n(M, r, s). Proof To prove that h(M, r, s) is a vector subbundle of so(M), we have to find a neighbourhood U around every q ∈ M with local sections {H1 , . . . , Ht } such that {H1 (y), . . . , Ht (y)} is a basis of h(y, r, s) for every y ∈ U . Let ∇¯ be a linear connection as in Lemma 3.2.7. We take a normal neighbourhood ¯ U around q with respect to ∇. Given a basis {H1 (q), . . . , Ht (q)} of h(q, r, s), we extend them by parallel trans¯ port with respect to ∇¯ along radial ∇-geodesics in order to define {H1 (y), . . . , Ht (y)}. i ¯ Since ∇ X (∇ R) = 0 for i = 0, . . . , r + 1, and ∇¯ X (∇ j P) = 0 for j = 0, . . . , s + 1,

3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds

77

the parallel transport from q to y defines a linear isometry F : Tq M → Ty M with F ∗ (∇ i R y ) = ∇ i Rq for i = 0, . . . , r + 1, and F ∗ (∇ j Py ) = ∇ j Pq for j = 0, . . . , s + 1. This implies that Hi (y) ∈ h(y, r, s). If (M, g, P) is (r, s)-strongly reductive, we consider the decomposition so(Tq M) = h(q, r, s) ⊕ n(q, r, s) and take a basis {η1 (q), . . . , ηd (q)} of n(q, r, s). Extending ¯ we obtain local {η1 (q), . . . , ηd (q)} by parallel transport along radial ∇-geodesics, sections η1 , . . . , ηd of so(M) defined on U . As seen in Lemma 3.2.4, the linear isometries F determined by the parallel transport take n(q, r, s) to n(y, r, s) for y ∈ U , so {η1 (y), . . . , ηd (y)} is a basis of n(y, r, s) for every y ∈ U .  Theorem 3.2.9 Let (M, g, P) be an (r, s)-infinitesimally P-homogeneous manifold. If (M, g, P) is (r, s)-strongly reductive with a decomposition so(T p M) = n( p, r, s) ⊕ h( p, r, s) where n(M, r, s) is Ad(H ( p, r, s))-invariant, then there is  such that S = ∇  − ∇ is a section of T ∗ M ⊗ n(M, r, s). a unique ASK-connection ∇ Proof Let h(M) denote h(M, r, s) and let n(M) denote n(M, r, s). Let ∇¯ be a linear ¯ which connection as in Lemma 3.2.7. We consider the tensor field B = ∇ − ∇, defines a section of T ∗ M ⊗ so(M) as ∇¯ is metric. By virtue of Lemma 3.2.8 we have the decomposition B = Bh + Bn, with B h and B n sections of T ∗ M ⊗ h(M) and T ∗ M ⊗ n(M), respectively. We define  = ∇ − S. Since S is a section of T ∗ M ⊗ so(M) we have that S = B n , and take ∇  is metric, so that ∇g  = 0. Moreover, ∇ X (∇ i R) = ∇¯ X (∇ i R) + B Xh · (∇ i R) = 0, ∇ X (∇ j P) = ∇¯ X (∇ j P) + B Xh · (∇ j P) = 0, ∇

i = 0, . . . , r + 1, j = 0, . . . , s + 1,

since (r, s) is a stabilizing pair. Let q ∈ M and consider a normal neighbourhood of  Since q with respect to ∇. X (∇ i R) = i X (∇ i+1 R) − S X · (∇ i R), 0=∇ X (∇ j P) = i X (∇ j+1 P) − S X · (∇ j P), 0=∇  differentiating these formulas along a radial ∇-geodesic γ(t), we find 0=0−

 d  i γ(t) Sγ(t) · (∇ i R)γ(t) = − ∇ ˙ ˙ S · (∇ R)γ(t) , dt

0=0−

 d  j γ(t) Sγ(t) · (∇ j P)γ(t) = − ∇ ˙ ˙ S · (∇ P)γ(t) , dt

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

γ(t) for i = 0, . . . , r and j = 0, . . . , s. Thus, ∇ ˙ S ∈ h(γ(t), r, s). In addition, as a consequence of the ad(h(M))-invariance of n(M), the covariant derivative of a section  γ(t) of n(M) is in n(M), so that ∇ ˙ S ∈ n(γ(t), r, s). We conclude that ∇ S = 0.  and ∇  be as in the hypothesis. Then, S − S  We finally prove uniqueness. Let ∇  i R) = ∇  (∇ i R) = 0 and ∇(∇  j P) = is a section of T ∗ M ⊗ n(M). In addition, ∇(∇  j  (∇ P) = 0 for all i, j. In fact, these equations are easily obtained from the fact ∇  (resp.  (resp. ∇  ) are parallel with respect to ∇ that the torsion and the curvature of ∇   P = ∇  P = 0. Consequently, S − S  is a section of T ∗ M ⊗ h(M)  ), and from ∇ ∇ =∇  .  and then, S = S  and ∇ Corollary 3.2.10 If n( p, r, s) ⊂ n( p, r  , s  ) for stabilizing pairs (r, s) and (r  , s  ),  and ∇  constructed from them are equal. then the ASK-connections ∇  − ∇ is a section of both n(M, r, s) and Proof This is evident, because S = ∇   n(M, r , s ).  As we have seen, a strongly reductive locally homogeneous pseudo-Riemannian manifold (M, g, P) with P invariant admits an ASK-connection, so by Theorem 3.1.15 there is a Lie pseudo-group G (which is not necessarily the full isometry pseudo-group) acting transitively by isometries and preserving P, such that (M, g, P, G) is reductive. Moreover, we will show that strongly reductive locally homogeneous spaces with an invariant geometric structure P are reductive for the action of the full pseudogroup of isometries preserving P. In order to prove this, we will make use of some results contained in Sect. 3.3 together with the following lemma.  be an ASK-connection with curvature K and torsion T . Let Lemma 3.2.11 Let ∇ p ∈ M and A ∈ so(T p M) be such that A · K p = 0, A · T p = 0 and A · Pp = 0. Then, A · ∇ i R p = 0 and A · ∇ j Pp = 0 for all i, j ≥ 0.  are related to R and S by Proof The curvature and torsion of ∇ TX Y = SY X − S X Y,

K X Y = R X Y + [S X , SY ] + STX Y .

 R = 0 and ∇  S = 0, an inductive Making use of these formulas in conjunction with ∇  j P) =  i R) = 0 for all i ≥ 0. A similar calculation gives ∇(∇ argument gives that ∇(∇ 0 for all j ≥ 0. Hence, i X ∇ i+1 R = S X · ∇ i R,

i X ∇ j+1 P = S X · ∇ j P,

for all i, j ≥ 0. Let now A ∈ so(T p M) be such that A · K p = 0, A · T p = 0 and A · Pp = 0. By Corollary 3.3.5, A · S p = 0, hence A · R p = 0. A simple calculation thus leads to (A · ∇ i+1 R p ) X = (A · S p ) X · ∇ i R p + (S p ) X · (A · ∇ i R p ),

i ≥ 0,

3.2 Strongly Reductive Locally Homogeneous Pseudo-Riemannian Manifolds

(A · ∇ j+1 Pp ) X = (A · S p ) X · ∇ j Pp + (S p ) X · (A · ∇ j Pp ),

79

j ≥ 0.

Therefore, by induction on i and j, we obtain that A · ∇ i R p = 0 and A · ∇ j Pp = 0 for all i, j ≥ 0.  Proposition 3.2.12 If (M, g, P) is (r, s)-strongly reductive, then (M, g, J ) is reductive, where J is the full Lie pseudo-group of local isometries preserving P.  be the associated ASKProof Let so(T p M) = n( p, r, s) ⊕ h( p, r, s), and let ∇  respecconnection. Let K and T be the curvature and the torsion tensor fields of ∇ tively. The pair (K , T ) defines an infinitesimal model (see Proposition 3.3.6), and we can consider the associated Nomizu construction, that is, we define the Lie algebra g0 = T p M ⊕ h0 with the usual brackets, where h0 = {A ∈ so(T p M)/ A · K p = 0, A · T p = 0, A · Pp = 0}. By Proposition 3.3.7, the Lie algebra h0 is equal to h( p, r, s). On the other hand, h0 ⊂ gkill0 by Lemma 3.2.11, and gkill0 ⊂ h by definition, whence gkill0 ⊂ h = h0 . We thus define the following Lie algebra isomorphism :

g0 → gkill X + A → (X, (S0 ) X + A).

The image of T p M defines a complement m of gkill0 . Making use of Lemma 3.3.4, we have that Ad B (S X ) = S B X for all B in H ( p, r, s) and all X ∈ T p M. Since the linear isotropy group H p is contained in H ( p, r, s), we conclude that m is Ad(H p )invariant. 

3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces We first prove a uniqueness result satisfied by strongly reductive locally homogeneous manifolds. Proposition 3.3.1 Let (M, g, P) and (M  , g  , P  ) be pseudo-Riemannian manifolds endowed with tensor fields P and P  . Suppose (M  , g  , P  ) is locally homogeneous with P  invariant. Suppose furthermore that (M  , g  , P  ) is (r, s)-strongly reductive for some stabilizing pair (r, s). If for each point p ∈ M there is a linear isometry F : T p M → To M  , for a certain fixed point o ∈ M  , such that F ∗ (∇ i Ro ) = ∇ i R p for i = 0, . . . , r + 1, and F ∗ (∇  j Po ) = ∇ j Pp for j = 0, . . . , s + 1, then (M, g, P) is locally homogeneous with P invariant and locally isometric to (M  , g  , P  ), with an isometry preserving P and P  .

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Proof First of all, note that (M, g, P) is (r, s)-infinitesimally P-homogeneous and (r, s)-strongly reductive, so that (M, g, P) is locally homogeneous with P invariant.  and ∇  be connections on M and M  respectively as in Theorem 3.2.9. Let Let ∇  and S  = ∇  − ∇  , and let F : T p M → To M  as in the hypothesis. It is S =∇ −∇ ∗  obvious that F (So ) − S p ∈ T p∗ M ⊗ n( p, r, s). In addition, F ∗ (So ) X − S p X · (∇ i R p ) = i X ∇ i+1 R p − i X ∇ i+1 R p = 0,

i = 0, . . . , r,

F ∗ (So ) X − S p X · (∇ j Pp ) = i X ∇ j+1 Pp − i X ∇ j+1 Pp = 0,

j = 0, . . . , s,

 

so that F ∗ (So ) X − S p X ∈ h( p, r, s). We conclude that F ∗ (So ) = S p .  is given by SY X − S X Y and a similar formula holds for Since the torsion of ∇   the torsion of ∇ , as a simple inspection shows, F preserves the curvature and the  Therefore, there are  and ∇  , which are parallel with respect to ∇. torsion of ∇ neighbourhoods U and V around p and o respectively, and an affine transformation  and ∇  are metric  and ∇  (see [79, Chap. 7]). Since ∇ f : U → V with respect to ∇      and ∇ P = ∇ P = 0, we have that f is an isometry preserving P and P  . Theorem 3.2.9 and Proposition 3.3.1 suggest the possibility of reconstructing a strongly reductive locally homogeneous manifold (M, g, P) with P invariant from the knowledge of the curvature tensor field, the tensor field P, and their covariant derivatives up to some finite order, at a point p. In order to prove this result we must first examine the algebraic properties of the curvature tensor field, P and their covariant derivatives. Let (M, g, P) be a locally homogeneous manifold with P invariant. We fix a point p ∈ M and set V = T p M. Consider the tensors R i = ∇ i R p and P j = ∇ j Pp for i, j ≥ 0. One has R 0X Y Z W = −RY0 X Z W = R 0Z W X Y ,

S

R 0X Y Z W

= 0,

(3.3.4) (3.3.5)

XYZ

R 1X Y Z V W = −R 1X Z Y V W = R 1X V W Y Z ,

S

R 1X Y Z V W

(3.3.6)

= 0,

(3.3.7)

= 0,

(3.3.8)

YZV

SR

1 XY ZV W

XYZ i+2 0 i RYi+2 X − RXY = RXY · R , j+2 j+2 PY X − PX Y = R 0X Y · P j ,

(3.3.9) (3.3.10)

for i, j ≥ 0, where R 0X Y is acting as a derivation on the tensor algebra. In addition,  be an ASK-connection and S = ∇  − ∇. We have that let ∇ i X R i+1 = S X · R i ,

i X P j+1 = S X · P j ,

(3.3.11)

3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces

81

for 0 ≤ i ≤ r + 1, 0 ≤ j ≤ s + 1, where (r, s) is a stabilizing pair at p. We thus consider the following linear maps μi, j : so(V ) → Wi, j A → (A · R 0 , . . . , A · R i , A · P 0 , . . . , A · P j ), and

ν : V → Wr +1,s+1 X → (i X R 1 , . . . , i X R r +2 , i X P 1 , . . . , i X P s+2 ),

with Wi, j

⎤  ⎡ j  i   ⊗α+4 V ∗ ⊗ ⎣ (⊗β V ∗ ) ⊗ P ⎦ , = α=0

β=0

where P is the space of tensors to which P 0 belongs. Then, ker(μr,s ) = ker(μr +1,s ) = ker(μr,s+1 ) = ker(μr +1,s+1 ).

(3.3.12)

Moreover, Eq. (3.3.11) gives ν(V ) ⊂ μr +1,s+1 (so(V )).

(3.3.13)

Finally, let H (r, s) be the stabilizer of R 0 , . . . , R r +1 and P 0 , . . . , P s+1 inside O(V ). In view of Theorem 3.2.9, in order to assure the existence of an ASK-connection, we need that (3.3.14) so(V ) = ker(μr,s ) ⊕ n for an Ad(H (r, s))-invariant subspace n. We shall prove the following result. Theorem 3.3.2 Let V be a vector space endowed with a (non-necessarily definite) inner product ·, ·. Let R 0 , . . . , R r +2 , P 0 , . . . , P s+2 be tensors on V satisfying the conditions (3.3.4)–(3.3.10) for 0 ≤ i ≤ r and 0 ≤ j ≤ s, and such that (3.3.13), (3.3.12), and (3.3.14) hold. Then, 1. There is an (r, s)-strongly reductive locally homogeneous pseudo-Riemannian manifold (M, g, P) with P invariant, whose curvature tensor field, P, and their covariant derivatives coincide with R 0 , . . . , R r +2 , P 0 , . . . , P s+2 at a point p ∈ M. Moreover, (M, g, P) is unique up to local isometries preserving P. 2. If the set of infinitesimal data R 0 , . . . , R r +2 , P 0 , . . . , P s+2 is regular (see Definitions 2.4.3 and 3.3.8), then there is an (r, s)-strongly reductive globally homogeneous pseudo-Riemannian space (G 0 /H0 , g, P) whose curvature tensor field, P, and its covariant derivatives coincide with R 0 , . . . , R r +2 , P 0 , . . . , P s+2 at a point p ∈ M. Moreover, the space (G 0 /H0 , g, P) is unique up to local isometries preserving P.

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

Corollary 3.3.3 An (r, s)-strongly reductive locally homogeneous Riemannian manifold (M, g, P) with P invariant can be reconstructed (up to local isometries) from the data R p , . . . , ∇ r +2 R p , Pp , . . . , ∇ s+2 Pp , where (r, s) is a stabilizing pair at p ∈ M. Before proving Theorem 3.3.2, we recall the definition of an infinitesimal model (see Definition 2.4.3) and show that an infinitesimal model can be associated to every suitable infinitesimal data R 0 , . . . , R s+2 , P 0 , . . . , P r +2 satisfying the hypotheses of Theorem 3.3.2. We define h = ker(μr +1,s+1 ), and consider an Ad(H (r, s))-invariant complement n of h inside so(V ). From (3.3.13) we have that for every X ∈ V there is an endomorphism A(X ) ∈ so(V ) such that i X R i+1 = A(X ) · R i ,

0 ≤ i ≤ r + 1,

= A(X ) · P ,

0 ≤ j ≤ s + 1.

iX P

j+1

j

Since so(V ) = h ⊕ n, we decompose A(X ) = A1 (X ) + A2 (X ), where A1 (X ) ∈ h and A2 (X ) ∈ n. Note that A(X ) is uniquely determined up to an h-component, so that we can take the uniquely defined map S:V →n X → S X = A2 (X ). By the definition of h it is evident that i X R i+1 = S X · R i ,

0 ≤ i ≤ r + 1,

(3.3.15)

= SX · P ,

0 ≤ j ≤ s + 1.

(3.3.16)

iX P

j+1

j

Moreover, by the same arguments used in [97] one sees that S is a linear map. Lemma 3.3.4 If B ∈ H (r, s), then Ad B (S X ) = S B X for every X ∈ V . Proof By the definition of H (r, s), (3.3.15) and (3.3.16), we have for all indices 0 ≤ i ≤ r and 0 ≤ j ≤ s i+1 ) X Z 1 ...Z i+4 R i+1 X Z 1 ...Z i+4 = (B · R

= R i+1 B −1 X B −1 Z 1 ...B −1 Z i+4  = S B −1 X · R i B −1 Z 1 ...B −1 Z i+4  =− R iB −1 Z 1 ...S −1 B −1 Z α ...B −1 Z i+4 α

=−

 α

=−

 α

B

X

R iB −1 Z 1 ...B −1 B S

B −1 X

B −1 Z α ...B −1 Z i+4

(B · R i ) Z 1 ...Ad B (SB −1 X )Z α ...Z i+4

3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces

=−

 α

83

R iZ 1 ...Ad B (SB −1 X )Z α ...Z i+4

 = Ad B (S B −1 X ) · R i Z 1 ...Z i+4 . On the other hand, i X R i+1 = S X · R i , so that Ad B (S B −1 X ) · R i − S X belongs to h. Since S X belongs to n and n is Ad(H (r, s))-invariant, we also have that Ad B (S B −1 X ) ·  R i − S X belongs to n. Hence, Ad B (S B −1 X ) · R i − S X = 0. Corollary 3.3.5 If A ∈ h, then A · S = 0. We take TX Y = SY X − S X Y, K X Y = R 0X Y + [S X , SY ] + STX Y , P = P0 and obtain the following result. Proposition 3.3.6 The triple (T, K , P) is an infinitesimal model. Proof We have to prove that (T, K , P) satisfies conditions (2.4.4)–(2.4.10). With regard to (2.4.4)–(2.4.6), (2.4.9) and (2.4.10), one applies exactly the same arguments used in [97]. For the remaining conditions, we observe that i  i+2 R i+2 X Y − RY X = [S X , SY ] + STX Y · R ,  j+2 j+2 PX Y − PY X = [S X , SY ] + STX Y · P j ,

0 ≤ i ≤ r, 0 ≤ j ≤ s.

In fact, by (3.3.15), i+2 )Y Z 1 ...Z i = (S X · R i+1 ) X Y Z 1 ...Z i+4 R i+2 X Y Z 1 ...Z i+4 = (i X R

= −R i+1 S X Y Z 1 ...Z i+4 −

i+4  α=1

RYi+1 Z 1 ...S X Z α ...Z i+4

i+4    i Y R i+1 Z 1 ...SX Z α ...Z i+4 = − i SX Y R i+1 Z 1 ...Z i+4 − α=1



= − SSX Y · R =

i+4  α=1

R iZ 1 ...SS

i

XY

Z 1 ...Z i+4

Z α ...Z i+4

−

i+4   α=1

+

i+4 

SY · R i

Z 1 ...S X Z α ...Z i+4

R iZ 1 ...SX Z α ...SY Z β ...Z i+4 ,

α,β=1 j+2

and by (3.3.16) a similar argument holds for PX Y . Skew-symmetrizing in X and Y we obtain the desired formulas. Therefore, by (3.3.9) and (3.3.10) and the definition

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

of K , we obtain that K X Y · R i = 0 and K X Y · P j = 0, for 0 ≤ i ≤ r and 0 ≤ j ≤ s. In particular, K X Y · P 0 = 0 and K X Y · R 0 = 0. Making use of (3.3.12), this implies that K X Y ∈ h, whence by Corollary 3.3.5 we find K X Y · S = 0, giving that K X Y · T = 0. Finally, as a straightforward computation shows, for A ∈ h we find (A · K ) X Y = (A · R 0 ) X Y + [(A · S) X , SY ] − [(A · S)Y , S X ] + S(A·T ) X Y , (3.3.17) so that K X Y · K = 0.



Proposition 3.3.7 h = h0 = {A ∈ so(V )/ A · K = 0, A · T = 0, A · P = 0}. Proof For A ∈ h, by Corollary 3.3.5 we have A · S = 0, which implies A · T = 0. In addition, by (3.3.17) we have A · K = 0. Since P = P 0 , by definition we deduce that A ∈ h0 , hence h ⊂ h0 . Conversely, let A ∈ h0 . We have that A · S = 0 since S is recovered from T making use of 2S X Y, Z  = −TX Y, Z  + TY Z , X  − TZ X, Y . On the other hand, by (3.3.17) we obtain A · R 0 = 0, and since P = P 0 we also have A · P 0 = 0. Now, a simple calculation (see Lemma 3.3.4) shows that (A · R i+1 ) X = [A, S X ] · R i − S AX · R i + S X · (A · R i ) = (A · S) X · R i + S X · (A · R i ), (A · P

j+1

0 ≤ i ≤ r + 1,

) X = [A, S X ] · P − S AX · P + S X · (A · P j ) j

j

= (A · S) X · P j + S X · (A · P j ),

0 ≤ j ≤ s + 1.

Using these formulas, by an inductive argument on the indices i and j we obtain that A · R i = 0 and A · P j = 0 for 0 ≤ i ≤ r + 1 and 0 ≤ j ≤ s + 1. Hence A ∈ h,  proving that h0 ⊂ h. Definition 3.3.8 The infinitesimal data R 0 , . . . , R r +2 , P 0 , . . . , P s+2 will be called regular if the associated infinitesimal model (T, K , P) is regular. Remark 3.3.9 R 0 , . . . , R r +2 , P 0 , . . . , P s+2 is recovered from the infinitesimal model (T, K , P) in the following way. As we have seen, S is obtained from T by 2S X Y, Z  = −TX Y, Z  + TY Z , X  − TZ X, Y . With T and S one recovers R 0 using the definition of K . Finally, knowing R 0 and P 0 = P, and using (3.3.15) and (3.3.16), one can subsequently obtain R i and P j for all indices i and j.

3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces

85

We are now in a position to prove Theorem 3.3.2. Proof of Theorem 3.3.2 Suppose that the infinitesimal model (T, K , P) associated to the infinitesimal data R 0 , . . . , R r +2 , P 0 , . . . , P s+2 is regular. We consider the Nomizu construction g0 = h0 ⊕ V , and the Lie groups G 0 and H0 , where G 0 is the connected and simply-connected Lie group with Lie algebra g0 and H0 is its connected Lie subgroup with Lie algebra h0 . Since H0 is closed in G 0 , we consider the homogeneous space G 0 /H0 . It is a reductive homogeneous space with reductive decomposition g0 = h0 ⊕ V , and identifying V = To G 0 /H0 , where o = H0 is the origin of G 0 /H0 , we extend ·, · and P to a G 0 -invariant Riemannian metric g and a G 0 -invariant tensor field P¯ on G 0 /H0 respectively. We consider the canonical connection associated to this reductive decomposition, which is an ASK-connection whose curvature and torsion at the origin o coincide with K and T . Using the properties of the canonical connection, a straightforward calculation shows that R 0 , . . . , R r +2 , P 0 , . . . , P s+2 coincide with the covariant derivatives of the curvature of g and P¯ at the origin o. By the identification of To G 0 /H0 with V , we have that G 0 /H0 is (r, s)-strongly reductive. This proves the second part of the theorem. With regard to the first part of the theorem, we shall prove it adapting to our case the arguments used in [117]. Let (T, K , P) be the infinitesimal model associated to the infinitesimal data R 0 , . . . , R r +2 , P 0 , . . . , P s+2 , which now need not be regular. We consider the corresponding Nomizu construction g0 = h0 ⊕ V . Let G 0 be the connected and simply-connected Lie group with Lie algebra g0 . We choose an orthonormal basis {e1 , . . . , en } of V and denote by {e1 , . . . , en } its dual basis. Let {A1 , . . . , Ad } be a basis of h0 , and {A1 , . . . , Ad } its dual basis. With respect to these bases, we write γ

T = Tαβ eα ⊗ eβ ⊗ eγ , δ eα ⊗ eβ ⊗ eγ ⊗ eδ , K = K αβγ ...βv α1 e ⊗ . . . ⊗ eαu ⊗ eβ1 ⊗ . . . ⊗ eβv P = Pαβ11...α u

and define

ωβα = eα (Aγ (eβ )) ⊗ Aγ ,

where Einstein’s summation convention is used. Note that ωβα ∈ g∗0 , so that ω = ωβα Aα ⊗ Aβ defines a left-invariant 2-form on G 0 with values in h0 ⊂ so(V ). Making use of the brackets defined in g0 , we easily obtain deα =

1 α T − ωβα ∧ eβ , 2 βγ

(3.3.18)

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

1 α γ δ γ dωβα = − K γδβ e e − ωγα ∧ ωβ . 2

(3.3.19)

We now consider a coordinate system φ = (x 1 , . . . , x n , y 1 , . . . , y d ) around the identity element e ∈ G 0 such that d x|eα = e|eα , and take f :

→ U U (a1 , . . . , an ) → φ−1 (a1 , . . . , an , 0, . . . , 0),

 is an open subset of Rn where f where U is the coordinate neighbourhood and U can be defined. It is evident that the map f defines an immersion from an open set W ⊂ Rn containing the origin of Rn into G 0 . α = f ∗ (eα ), since these 1-forms are linearly independent at the origin of Setting E n R , there is an open set M ⊂ W around the origin where they are linearly independent. n } be the dual frame field. We define on M the metric 1 , . . . , E Let { E g=

n 

α ⊗ E α , (α) E

α=1

with (α) = eα , eα  ∈ {−1, +1}, and the tensor fields β ⊗ E γ , α ⊗ E  = Tγ E T αβ δ α  = K αβγ β ⊗ E γ ⊗ E δ , K E ⊗E ...βv α1 αu ⊗ E β1 ⊗ . . . ⊗ E βv .  = Pαβ1...α E ⊗ ... ⊗ E P 1 u

In addition, we consider  ω = f ∗ ω, which is a 1-form on M with values in h0 . Note   that { E 1 , . . . , E n } is an orthonormal frame field defined on the whole of M, so that it is a trivializing section of the bundle of orthonormal frames of M. Hence, this  on O(M). section  ω is the 1-form of a metric connection ∇ By (3.3.18) and (3.3.19), which are nothing but the structure equations for the  and K  are the torsion and curvature of the torsion and curvature of ω, we have that T , K  and P  are  respectively. Since  connection ∇ ω takes values in h0 , we have that T  is  that is, ∇  is an ASK-connection. Therefore, (M, g, P) parallel with respect to ∇,  invariant. locally homogeneous with P Finally, making use of Remark 3.3.9, it is easy to see that the covariant derivatives  and the curvature of g at the origin coincide with R 0 , . . . , R r +2 , P 0 , . . . , P s+2 of P under the identification To M  V . In addition, by this identification M is (r, s)strongly reductive. In both the first and the second part of the theorem, uniqueness (up to local isometry) follows from Proposition 3.3.1.  Note that the strong reductivity condition (3.3.14) is essential in the proof of Theorem 3.3.2, since otherwise we are not able to construct the infinitesimal model (T, K , P) from the infinitesimal data

3.3 Reconstruction of Strongly Reductive Locally Homogeneous Spaces

R 0 , . . . , R r +2 ,

87

P 0 , . . . , P s+2 .

This means that in general a locally homogeneous pseudo-Riemannian manifold, whose metric is not definite, might not be recovered from infinitesimal data.  this problem can be solved if we If the manifold admits an ASK-connection ∇,  − ∇, add to R 0 , . . . , R r +2 , P 0 , . . . , P s+2 the knowledge of either S p , where S = ∇  at p, or the curvature of ∇  at p (these three last items provide the torsion of ∇ equivalent information in view of Remark 3.3.9). In addition, an analogous result to Proposition 3.3.1 can be proved by a straightforward adaptation.

3.4 Examples and the Reductivity Condition It is well known that a globally homogeneous space can be represented by different coset spaces G/H . In the same way, we can consider the action of different Lie pseudo-groups of isometries on the same locally homogeneous pseudo-Riemannian manifold (M, g). Since the notion of reductivity is tied to the action of a Lie pseudo-group in particular, the following question naturally arises: let G and G  be Lie pseudo-groups of isometries acting transitively on (M, g). Is it possible that (M, g, G) is reductive but (M, g, G  ) is non-reductive? We now present some examples which give a positive answer to this question, and explore the possible scenarios when G is a subgroup of G  and vice versa. We will also show that the reductivity condition does not imply the strong reductivity condition. It is worth pointing out that this situation is not a consequence of the freedom obtained by enlarging the (rather rigid) family of globally homogeneous spaces to the family of locally homogeneous spaces. In fact, we can find illustrative examples restricting ourselves to globally homogeneous pseudo-Riemannian manifolds. We will finally give an example of a stabilizing pair distinct from (k( p), l( p)). Example 3.4.1 Consider R5 endowed with the standard metric η of signature (2, 3). We take the 4-dimensional submanifold H41 = {x ∈ R5 / η(x, x) = −1}, equipped with a pseudo-Riemannian metric g inherited from η. (H41 , g) is a Lorentz space of constant sectional curvature, and it is well known that it is the (globally) symmetric space SO0 (2, 3) . H41  SO0 (1, 3) j

Let {e1 , . . . , e5 } be the standard basis of R5 , and let ei denote the endomorphism e j ⊗ ei of R5 . The isotropy algebra of the point p = (0, 1, 0, 0, 0) ∈ H41 is

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3 Locally Homogeneous Pseudo-Riemannian Manifolds

so(1, 3) = Span{e13 + e31 , e14 + e41 , e15 + e51 , e34 − e43 , e35 − e53 , e45 − e54 }. An SO0 (1, 3)-invariant complement is m = Span{e12 − e21 , e23 + e32 , e24 + e42 , e25 + e52 }, and so, (H41 , g, SO0 (2, 3)) is reductive. Consider now the Lie subalgebra g spanned by the following seven elements: e14 + e41 − e23 − e32 ,

1 2 (e − e21 + e13 + e31 + e24 + e42 + e43 − e34 ), 2 1

1 3 1 (e + e31 + e21 − e12 + e24 + e42 + e34 − e43 ) , (e21 − e12 + e24 + e42 + e43 − e34 − e13 − e31 ), 2 1 2

1 1 √ (e15 + e51 + e54 − e45 ) , √ (e35 − e53 − e25 − e52 ) , e14 + e41 + e23 + e32 . 2 2 The isotropy algebra k at p is spanned by the elements 1 2(e14 + e41 + e14 ), e13 + e31 + e34 − e43 , √ (e15 + e51 + e54 − e45 ). 2 Let G be the connected Lie subgroup of SO0 (2, 3) with Lie algebra g, then G acts transitively on H41 . But taking into account [59] (see the Lie algebra A5 of that reference), there is no ad(k)-invariant complement of k. Therefore, (H41 , g, G) is non-reductive. This and the other examples of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds will be analyzed in detail in the last chapter. Example 3.4.2 We consider R4 endowed with the pseudo-Riemannian metric g = 2e y1 cos(y2 )(dy1 dy4 − dy2 dy3 ) − 2e y1 sin(y2 )(dy1 dy3 + dy2 dy4 ) + Le4y1 dy2 dy2 ,  with L ∈ R − {0}. The group G  = SL (2, R)  R2 × R acts transitively by isome4 tries on (R , g) (see Sect. 5 of [59]). The non-trivial brackets of the Lie algebra of G  are [e1 , e2 ] = 2e2 , [e1 , e5 ] = −e5 ,

[e1 , e3 ] = −2e3 , [e2 , e5 ] = e4 ,

[e2 , e3 ] = e1 , [e3 , e4 ] = e5 ,

[e1 , e4 ] = e4 ,

with respect to some basis {e1 , . . . , e6 }, which is the Lie algebra B3 in [59]. In addition, the above is the full isometry algebra of (R4 , g) and can be realized by the complete Killing vector fields

3.4 Examples and the Reductivity Condition

89

Y1 = cos(2y2 )∂ y1 − sin(2y2 )∂ y2 + y3 ∂ y3 − y4 ∂ y4 , 1 Y2 = sin(2y2 )∂ y1 + cos2 (y2 )∂ y2 + y3 ∂ y4 , 2 1 Y3 = sin(2y2 )∂ y1 − sin2 (y2 )∂ y2 + y4 ∂ y3 , 2 Y4 = ∂ y4 , Y5 = −∂ y3 , Y6 = e y1 cos(y2 )∂ y3 + e y1 sin(y2 )∂ y4 . The isotropy algebra at (0, 0, 0, 0) ∈ R4 is Span{e3 , e5 + e6 }. As stated in [59], the homogeneous pseudo-Riemannian manifold (R4 , g, G  ) is non-reductive. We take the subalgebra g = Span{e1 , e2 , e4 , e5 , e6 }. Making use of the distribution generated by the corresponding Killing vector fields, we see that the action of the connected Lie subgroup G ⊂ G  with Lie algebra g is transitive. The isotropy algebra at (0, 0, 0, 0) is k = Span{e5 + e6 }, and m = Span{e1 , e2 , e4 , e5 } is an Ad(K )-invariant complement, where K ⊂ G is the isotropy group with respect to the action of G at (0, 0, 0, 0). Therefore, (R4 , g, G) is reductive. On the other hand, we can check that (R4 , g) is not strongly reductive. In this case, since there is no extra geometric structure, the complex of filtrations reduces to so(T p M) ⊃ g( p, 0) ⊃ g( p, 1) ⊃ . . . A simple calculation shows that the only non-zero component of the curvature is R∂ y1 ∂ y2 ∂ y1 ∂ y2 = −3Le4y1 , and that ∇ R = 0. We take p = (0, 0, 0, 0) and L = 1 for the sake of simplicity, so that the filtration actually is so(T p M) ⊃ g( p, 0) = g( p, 1), where ⎧⎛ ⎫ ⎞ −e 2(b − c) b 0 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ f 2a a c ⎜ ⎟ , / a, b, c, d, e, f ∈ R so(T p M) = ⎝ 2(d − f ) 0 −2a 2(b − c)⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 2(d − f ) d e

 g( p, 0) = A ∈ so(T p M)/ e = 2a, f = d . It is easy to check that g( p, 0) does not admit any complement n invariant under the adjoint action of g( p, 0), hence (R4 , g) cannot be strongly reductive. We finally exhibit an example of a locally homogeneous pseudo-Kähler manifold with a stabilizing pair distinct from (k, l), where as usual (k, l) are the first integers such that g( p, k) = g( p, k + 1) and p( p, l) = p( p, l + 1).

90

3 Locally Homogeneous Pseudo-Riemannian Manifolds

Example 3.4.3 Consider the space C2 with complex coordinates (w, z). We take M = C2 − {||w|| = 0} with the standard complex structure J and the pseudoRiemannian metric g = dw 1 dz 1 + dw 2 dz 2 + b(dw 1 dw 1 + dw 2 dw 2 ), where w = w1 + iw 2 , z = z 1 + i z 2 , and b is a function depending on w 1 and w 2 and R0 satisfying b = ||w|| 4 for some R0  = 0. We will see in Sect. 5.1 that this manifold is locally homogeneous since it admits an ASK-connection. The curvature tensor and its first covariant derivative are R=

1 R0 (dw 1 2 ||w||4

∧ dw 2 ⊗ dw 1 ∧ dw 2 ),

∇ R = 4θ ⊗ R, 1 1 1 2 2 where θ = − ||w|| 2 (w dw + w dw ). We set R0 = 2 and take the point p = (−1, 0, 0, 0), so that R p = dw 1 ∧ dw 2 ⊗ dw 1 ∧ dw 2 , ∇ R p = 4dw 1 ⊗ R p , ∇ 2 R p = (20dw 1 ⊗ dw 1 − 4dw 2 ⊗ dw 2 ) ⊗ R p .

On the other hand, J p is the standard complex structure of C2 and ∇ J p = 0, because the manifold is pseudo-Kähler. A straightforward calculation then shows that the complex of filtrations is so(R4 )6 ⊃ g( p, 0)2 ⊃ g( p, 1)1 = g( p, 2)1 ∪ || || || p( p, 0)4 ⊃ h( p, 0, 0)2 ⊃ h( p, 1, 0)1 = h( p, 2, 0)1 || || || || p( p, 1)4 ⊃ h( p, 0, 1)2 ⊃ h( p, 1, 1)1 = h( p, 2, 1)1 , where superindices indicate dimension. We have that (k, l) = (1, 0), but (r, s) = (1, −1) is a stabilizing pair.

Chapter 4

Classification of Homogeneous Structures

In previous chapters we have seen how AS-connections and ASK-connections can be used to study homogeneous and locally homogeneous spaces. In particular, the presence of an AS-connection characterizes connected, simplyconnected and complete globally homogeneous reductive pseudo-Riemannian manifolds and provides a representation of such a manifold as a coset space. For this reason, not only does a classification of the possible AS-connections or ASK-connections help to understand different coset representations of the same homogeneous space, but also sheds light on the structure of the vast world of homogeneous and locally homogeneous spaces. A very efficient way to tackle this problem is to classify the possible homogeneous structures S, which essentially correspond to the torsion tensor of the corresponding AS or ASK-connections. The advantage of this approach is that the work can be done completely at an algebraic level, in a similar way to how the intrinsic torsion is studied in Riemannian geometry (see for example [60]), from the perspective of representation theory. In this chapter we illustrate a procedure to classify homogeneous structures associated to AS-connections and ASK-connections with integrable underlying geometric structure. We will then apply that procedure to the geometric structures that will be treated in subsequent chapters. In the last part of this chapter we shall use homogeneous structures to obtain a complete characterization of three-dimensional homogeneous (Riemannian and Lorentzian) manifolds.

4.1 The General Procedure Let S be a homogeneous structure on a pseudo-Riemannian manifold (M, g) of  We will use signature (r, s), r + s = dimM, associated to an AS-connection ∇. S to refer indistinctively to either the (1, 2)-tensor field or the metric equivalent (0, 3)-tensor field, defined by © Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_4

91

92

4 Classification of Homogeneous Structures

S X Y Z = g(S X Y, Z ). This convention will be used hereafter. Let ∇ be the Levi-Civita connection of g.  = 0 reads The first Ambrose–Singer equation ∇g  = ∇g − S · g = −S · g = 0, 0 = ∇g that is, for every X ∈ X(M) we have S X · g = 0, where S X acts as a derivation on the tensor algebra. More explicitly, we have S X Y Z + S X Z Y = 0,

X, Y, Z ∈ X(M).

Let x ∈ M. Choosing an orthonormal basis of Tx M, we can consider the vector space V = Rm endowed with the standard symmetric bilinear form ·, · of signature (r, s) as a model of (Tx M, gx ). We take the space of tensors S(V ) ⊂ ⊗3 V ∗ with the same symmetries as the homogeneous structure S, that is, defined by S(V ) = {S ∈ ⊗3 V ∗ / S X Y Z + S X Z Y = 0}. As a vector space, S(V ) is isomorphic to V ∗ ⊗ ∧2 V ∗ , and carries a non-degenerate symmetric bilinear form defined by S, S  =

m 

εi ε j εk Sei e j ek Se i e j ek ,

(4.1.1)

i, j,k=1

where {e1 , . . . , em } is any orthonormal basis of (V, ·, ·) and εi = ei , ei . Furthermore, there is a natural left action of the orthogonal group O(r, s) on S(V ), given by A ∈ O(V ), X, Y, Z ∈ V, (A · S) X Y Z = S A−1 X A−1 Y A−1 Z , turning S(V ) into an O(r, s)-module. Identifying ∧2 V ∗ with so(r, s) (by sending α ∧ β ∈ ∧2 V ∗ to the matrix diag(ε1 , . . . , εm ) · A, where A = (ai j )i,m j=1 , ai j = α(ei )β(e j )), we have that S(V )  V ∗ ⊗ so(r, s), and the action of O(r, s) is seen as the tensor product of the standard representation and the adjoint representation. Suppose now that there is a geometric structure on (M, g), defined by a tensor field P (the case of geometric structures given by more than one tensor field is clearly analogous), and that S is associated to an ASK-connection. Assume that this geometric structure is integrable. Recall that this means that the holonomy of g at x ∈ M can be seen as a subgroup of the stabilizer Hx of Px inside O(Tx M), or  P = 0 now yields equivalently that ∇ P = 0. Thus, the equation ∇ 0 = ∇ P − S · P = −S · P,

4.1 The General Procedure

93

whence (Sx ) X can be seen as an element of the Lie algebra hx of H (x), for every X ∈ Tx M. With the help of an orthonormal basis of Tx M, we consider a tensor field P0 on V as the model of Px , and denote by H and h the stabilizer of P0 inside O(r, s) and its Lie algebra respectively. The space of tensors on V with the same symmetries as S is thus identified with V ∗ ⊗ h ⊂ S(V ). The natural action of H as a subgroup of O(r, s) restricts to V ∗ ⊗ h, as it is just the tensor product of the standard and the adjoint representation. This turns V ∗ ⊗ h into an H -module. Observe that the action of H is orthogonal with respect to the bilinear form ·, · on S(V ) in formula (4.1.1). The H -module V ∗ ⊗ h can thus be decomposed into the direct sum of mutually orthogonal indecomposable H -submodules. As was the case with the holonomy representation (see the paragraph after Definition 1.2.14), if g is definite we can assure that V ∗ ⊗ h is decomposable into the direct sum of irreducible H -submodules. This is also the case when H is semisimple (for instance, all the groups appearing in Berger’s list are semisimple), but not in general pseudoRiemannian settings. Assume now that we have decomposed V ∗ ⊗ h into H -submodules W 1 , . . . , W l , that is, V ∗ ⊗ h = W1 ⊕ . . . ⊕ Wl. For every x ∈ M, this gives a decomposition of the Hx -module Tx∗ M ⊗ hx into Hx -submodules Wx1 , . . . , Wxl , that is, Tx∗ M ⊗ hx = Wx1 ⊕ . . . ⊕ Wxl . Proposition 4.1.1 Let S be a homogeneous structure of (M, g, P). If Sx belongs to the submodule Wxi for some i = 1, . . . , l at a point x ∈ M, then it belongs to the submodule W yi at every point y ∈ M. Proof Since (M, g, P) is (at least locally) homogeneous and P is invariant, there is a linear isometry φ : Tx M → Ty M preserving both P and S. Choosing an orthonormal basis, we can then identify Tx M and Ty M with V , so that φ turns into an element of the group H . This implies that the induced transformation of V ∗ ⊗ h preserves the H -submodules W 1 , . . . , W l . Since S is also invariant under φ, we conclude that  Sx ∈ Wxi if and only if S y ∈ W yi . The previous proposition shows that a certain algebraic decomposition of V ∗ ⊗ h into H -submodules induces a corresponding classification of homogeneous structures. For practical purposes, this decomposition can mainly be carried out in two different (though complementary) ways: using either representation theory, or real tensors. For the cases where we will use the representation theory approach, we will make use of the techniques in [107], which provide a method to decompose tensor products of representations of Lie groups (see also [9, 11, 121]).

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4 Classification of Homogeneous Structures

We shall now illustrate the classification procedure, considering first the general case of homogeneous structures S on a pseudo-Riemannian manifold (M, g), without adding any geometric structure on it. In other words, we assume that S is associated to just an AS-connection so that we only have to take into account Eq. (2.2.1). We need to decompose the O(r, s)-module S(V ) into irreducible submodules. Suppose that m = dimM ≥ 3. From the general representation theory for the orthonormal group (see for example [9, 121]), we have the decomposition V ∗ ⊗ ∧2 V ∗  V ∗ ⊗ W ⊗ ∧3 V ∗ , where W is the irreducible representation of O(r, s) associated to the Young element Id +(12) − (23) − (132). For the explicit expression of the tensors belonging to each of these submodules, we first consider the O(r, s)-equivariant map c12 : S(V ) → V ∗ m i S → c12 (S)(Z ) = i=1 ε Sei ei Z , where {ei } is any orthonormal basis of V and εi = ei , ei . The subspace ker(c12 ) is non-degenerate with respect to the symmetric bilinear form on S(V ), and its orthogonal complement is ker(c12 )⊥ = {S ∈ S(V )/ S X Y Z = X, Y ϕ(Z ) − X, Z ϕ(Y ), ϕ ∈ V ∗ }. On the other hand, we also consider the O(r, s)-equivariant map L : S(V ) → S(V ) S → L(S) = S

XYZ

SX Y Z .

This map satisfies L 2 = 3L. Therefore, it is diagonalizable, with real eigenvalues 0 and 3. The corresponding eigenspaces S 0 (V ) = {S ∈ V/

SS

XY Z

= 0},

XYZ

S 3 (V ) = {S ∈ S(V )/ S X Y Z + SY X Z = 0} are mutually orthogonal and invariant under O(r, s). It is easy to check that S 3 (V ) ⊂ ker(c12 ) and ker(c12 )⊥ ⊂ S 0 (V ). We set S1 (V ) = ker(c12 )⊥ , S2 (V ) = S 0 (V ) ∩ ker(c12 ), and S3 (V ) = S 3 (V ). Then, the following result holds. Proposition 4.1.2 [65, 116] If m ≥ 3, then the space S(V ) decomposes into irreducible and mutually orthogonal O(r, s)-submodules as

4.1 The General Procedure

95

S(V ) = S1 (V ) ⊕ S2 (V ) ⊕ S3 (V ). If m = 2, then S(V ) = S1 (V ). Let S denote the set of homogeneous pseudo-Riemannian structures. Then, the following explicit classification holds:   S1 = S ∈ S / S X Y Z = g(X, Y )ϕ(Z ) − g(X, Z )ϕ(Y ), ϕ ∈ 1 (M) ,   S2 = S ∈ S / S X Y Z = 0, c12 (S) = 0 ,

S



XYZ

 S3 = S ∈ S / S X Y Z + SY X Z = 0 . Observe that dim S1 (V ) = m, dim S2 (V ) =

  m(m − 2)(m + 2) m , dim S3 (V ) = . 3 3

Thus, homogeneous pseudo-Riemannian structures in the class S1 are sections of a vector bundle, whose rank grows linearly with the dimension of the manifold. This fact leads to the notion of a homogeneous structure of linear type, which will be a central object of study in Chap. 5. Definition 4.1.3 A homogeneous pseudo-Riemannian structure is called of linear type if it belongs to the class S1 . It is easy to see that a homogeneous pseudo-Riemannian structure of linear type, thought of as a (1, 2)-tensor field, takes the form S X Y = g(X, Y )ξ − g(Y, ξ)X,

(4.1.2)

for some vector field ξ ∈ X(M), and that the Ambrose–Singer equations are equivalent to  R = 0,  = 0. ∇ ∇ξ Since we are working with metrics with signature, the class of these structures furthers splits as explained in the following. Definition 4.1.4 A homogeneous pseudo-Riemannian structure of linear type S, defined by the vector field ξ, is said to be 1. non-degenerate if g(ξ, ξ) = 0, 2. degenerate if g(ξ, ξ) = 0.

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4 Classification of Homogeneous Structures

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures We shall now describe the classification procedure for some integrable geometric structures.

4.2.1 Homogeneous Pseudo-Kähler Structures Consider homogeneous structures S on a pseudo-Kähler manifold (M, g, J ) (see Sect. 1.3.1). They satisfy the Ambrose–Singer–Kiriˇcenko equations, that is,  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 J = 0. ∇

Such structures will be called homogeneous pseudo-Kähler structures. In this case, the stabilizer of the complex structure inside O(r, s) is the corresponding unitary group U( p, q), where 2 p = r and 2q = s. We thus decompose the U( p, q)-module K(V ) = V ∗ ⊗ u( p, q) ⊂ S(V ) into irreducible submodules. As a representation of U( p, q), the space K(V ) is isomorphic to the tensor product of the standard representation and the adjoint representation of U( p, q). Suppose that m = dimM ≥ 6.

Following [107], we adopt the notation k,l ⊗C = k,l +l,k and [k,k ]⊗C = k,k , where k,l denotes the space of forms of type (k, l) on V ⊗ C with

respect to J . The standard and the adjoint representation can thus be written as 0,1 and R + [1,1 0 ] respectively, where the subindex 0 denotes the primitive part with respect to the symplectic form associated to J . In terms of complex representations, we then have 2,1 1,0 + 2,1 1,0 ⊗ 1,1 0  0 + S0 , 2,1 (see [60]). where S02,1 is the kernel of the anti-symmetrization 1,0 ⊗ 1,1 0 →  Therefore,



0,1 2,1  2,1  0,1 +  + 0 + S0 . V ∗ ⊗ u( p, q)  0,1 ⊗ R + [1,1 0 ]  

We now give the explicit expression of the tensors in these submodules. For that, we consider the equivariant map L : K(V ) → K(V ) S → L(S) X Y Z =

1 2

(SY Z X + S Z X Y + S J Y J Z X + S J Z X J Y ) ,

which is also orthogonal with respect to the symmetric bilinear form inherited from S(V ). Then, a simple computation shows that L 2 = Id, so that L is diagonalizable

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

97

with eigenvalues ±1. The corresponding eigenspaces K±1 (V ) are mutually orthogonal and invariant under U( p, q). Taking the contraction c12 , each eigenspace splits in two mutually orthogonal submodules K±1 (V ) = K±1 (V ) ∩ ker(c12 ) + K±1 (V ) ∩ ker(c12 )⊥ . We set:

K1 (V ) = K1 (V ) ∩ ker(c12 ), K2 (V ) = K1 (V ) ∩ ker(c12 )⊥ , K3 (V ) = K−1 (V ) ∩ ker(c12 ), K4 (V ) = K−1 (V ) ∩ ker(c12 )⊥ .

We then have the following. Proposition 4.2.1 [1, 13] If m ≥ 6, the space K(V ) is decomposed into mutually orthogonal and irreducible U( p, q)-submodules as K(V ) = K1 (V ) ⊕ K2 (V ) ⊕ K3 (V ) ⊕ K4 (V ), where  1 K1 (V ) = S ∈ K(V ) / S X Y Z = (SY Z X + S Z X Y + S J Y J Z X + S J Z X J Y ), 2

 c12 (S) = 0 ,

 K2 (V ) = S ∈ K(V ) / S X Y Z = X, Y θ1 (Z ) − X, Z θ1 (Y ) + X, J Y θ1 (J Z )  − X, J Z θ1 (J Y ) − 2J Y, Z θ1 (J X ), θ1 ∈ V ∗ ,  1 K3 (V ) = S ∈ K(V ) / S X Y Z = − (SY Z X + S Z X Y + S J Y J Z X + S J Z X J Y ), 2  c12 (S) = 0 ,  K4 (V ) = S ∈ K(V ) / S X Y Z = X, Y θ2 (Z ) − X, Z θ2 (Y ) + X, J Y θ2 (J Z )  − X, J Z θ2 (J Y ) + 2J Y, Z θ2 (J X ), θ2 ∈ V ∗ . If m = 4, then K(V ) = K2 (V ) ⊕ K3 (V ) ⊕ K4 (V ). If m = 2, then K(V ) = K4 (V ). Taking into account the fact that dim K1 (V ) = n(n + 1)(n − 2), dim K2 (V ) = dim K4 (V ) = 2n, dim K3 (V ) = n(n − 1)(n + 2),

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4 Classification of Homogeneous Structures

where m = 2n, we note that homogeneous pseudo-Kähler structures in the composed class K2 ⊕ K4 are sections of a vector bundle, whose rank grows linearly with the dimension of the manifold. It is easy to prove that such a homogeneous pseudo-Kähler structure, seen as a (1, 2)-tensor field, takes the form S X Y = g(X, Y )ξ−g(Y, ξ)X −g(X, J Y )J ξ+g(J Y, ξ)J X −2g(J X, ζ)J Y, (4.2.3) for some vector fields ξ, ζ ∈ X(M). In addition, if S takes the form (4.2.3), then the Ambrose–Singer–Kiriˇcenko equations are equivalent to  R = 0, ∇

 = 0, ∇ξ

 = 0. ∇ζ

(4.2.4)

These facts motivate the following definition. Definition 4.2.2 Homogeneous pseudo-Kähler structures belonging to the class K2 ⊕ K4 are called homogeneous pseudo-Kähler structures of linear type. A homogeneous pseudo-Kähler structure of linear type S described as in (4.2.3) in terms of vector fields ξ and ζ is called 1. non-degenerate if g(ξ, ξ) = 0, 2. degenerate if g(ξ, ξ) = 0, 3. strongly degenerate if g(ξ, ξ) = 0 and ζ = 0.

4.2.2 Homogeneous Para-Kähler Structures We proceed in a way similar to the previous section but considering now homogeneous structures S on a para-Kähler manifold (M, g, J ) (see Sect. 1.3.2). In this case, the Ambrose–Singer–Kiriˇcenko equations read  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 J = 0. ∇

Such structures will be called homogeneous para-Kähler structures. The stabilizer of the para-complex structure J inside O(r, s) is the para-unitary group Gl(n, R), where dimM = m = 2n. We thus have to decompose the Gl(n, R)-module PK(V ) = V ∗ ⊗ gl(n, R) ⊂ S(V ) into irreducible submodules. Note that V is an indecomposable but reducible representation of Gl(n, R), since there are two invariant maximal isotropic and complementary subspaces of V . These subspaces are exactly the eigenspaces V+ and V− corresponding to the eigenvalues ±1 of J , and they have the same dimension. Taking V + and V − the dual spaces of V+ and V− respectively, we have

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

99

V ∗ ⊗ gl(n, R)  V + ⊗ gl(n, R) + V − ⊗ gl(n, R). We define k,−l = ∧k V + ⊗∧l V − . Under the identification so(r, s)  ∧2 V ∗ , we have that gl(n, R) is identified with 1,−1 . Considering the symplectic form ω ∈ 1,−1 associated to J , we have the decomposition 1,−1 = Rω + 1,−1 . Hence, 0 . V + ⊗ gl(n, R)  V + ⊗ R + V + ⊗ 1,−1 0 On the other hand,

 V + + 2,−1 + S02,−1 , V + ⊗ 1,−1 0 0

→ 2,−1 , and where S02,−1 is the kernel of the anti-symmetrization V + ⊗ 1,−1 0 2,−1 2,−1 is the primitive part of  with respect to ω. Since the complexification 0 of gl(n, R) is isomorphic to gl(n, C), which coincides with the complexification of u( p, q), the representation theory for gl(n, R) and u( p, q) is analogous. Using this and S02,−1 are irreducible. fact, it is easy to see that the submodules 2,−1 0 − 1,−1 is carried out in a similar way. Summarizing, A decomposition of V ⊗  we obtain + 1,−2 + S02,−1 + S01,−2 . V ∗ ⊗ gl(n, R)  2V + + 2V − + 2,−1 0 0 On the other hand, we consider the equivariant map L : K(V ) → K(V ) S → L(S) X Y Z =

1 2

(SY Z X + S Z X Y − S J Y J Z X − S J Z X J Y ) ,

which is also orthogonal with respect to the symmetric bilinear form inherited from S(V ). It is easy to check that L 2 = Id, so that L is diagonalizable with eigenvalues ±1. The corresponding eigenspaces W ±1 (V ) are mutually orthogonal and invariant under Gl(n, R). Taking now the contraction c12 , each eigenspace splits into two mutually orthogonal submodules W ±1 = W ±1 ∩ ker(c12 ) + W ±1 ∩ ker(c12 )⊥ . We set

U1 U2 U3 U4

= W 1 ∩ ker(c12 ), = W 1 ∩ ker(c12 )⊥ , = W −1 ∩ ker(c12 ), = W −1 ∩ ker(c12 )⊥ .

The submodules U1 , . . . , U4 are indecomposable but reducible. Indeed, the splitting V = V + ⊕ V − induces decompositions Ui = Ui+ ⊕ Ui− for all indices i, where

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4 Classification of Homogeneous Structures

  Ui+ = S ∈ Ui / S X − Y Z = 0, X − ∈ V − ,   Ui− = S ∈ Ui / S X + Y Z = 0, X + ∈ V + . We set PK1 (V ) = U1+ , PK3 (V ) = U3+ , PK5 (V ) = U1− ,

PK2 (V ) = U2+ , PK4 (V ) = U4+ , PK6 (V ) = U2− ,

PK7 (V ) = U3− ,

PK8 (V ) = U4− .

We then have the following. Proposition 4.2.3 [64] If m ≥ 6, the space PK(V ) is decomposed into irreducible Gl(n, R)-submodules as PK(V ) =PK1 (V ) ⊕ PK2 (V ) ⊕ PK3 (V ) ⊕ PK4 (V ) ⊕ PK5 (V ) ⊕ PK6 (V ) ⊕ PK7 (V ) ⊕ PK8 (V ), where  1 PK1 (V ) = S ∈ PK(V ) / S X Y Z = (SY Z X + S Z X Y − S J Y J Z X − S J Z X J Y ), 2 

c12 (S) = 0, S X − Y Z = 0, X − ∈ V − ,

 PK2 (V ) = S ∈ PK(V ) / S X Y Z = X, Y θ1 (Z ) − X, Z θ1 (Y )

 − X, J Y θ1 (J Z ) + X, J Z θ1 (J Y ) + 2J Y, Z θ1 (J X ), θ1 ∈ V + ,  1 PK3 (V ) = S ∈ PK(V ) / S X Y Z = − (SY Z X + S Z X Y − S J Y J Z X − S J Z X J Y ), 2  c12 (S) = 0, S X − Y Z = 0, X − ∈ V − ,  PK4 (V ) = S ∈ PK(V ) / S X Y Z = X, Y θ2 (Z ) − X, Z θ2 (Y )  − X, J Y θ2 (J Z ) + X, J Z θ2 (J Y ) − 2J Y, Z θ2 (J X ), θ2 ∈ V + ,

and PK5 , . . . , PK8 are obtained from PK1 , . . . , PK4 interchanging V + and V − . If m = 4, then PK(V ) = PK2 (V ) ⊕ PK3 (V ) ⊕ PK4 (V ) ⊕ PK6 (V ) ⊕ PK7 (V ) ⊕ PK8 (V ). If m = 2, then K(V ) = PK4 (V ) ⊕ PK8 (V ).

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

101

Taking into account that dim PK1 (V ) = dim PK5 (V ) = n(n+1)(n−2) , 2 n 2 (n+1) dim PK3 (V ) = dim PK7 (V ) = 2 − n, dim PK2 (V ) = dim PK4 (V ) = dim PK6 (V ) = dim PK8 (V ) = n, we note that homogeneous para-Kähler structures in the composed class PK2 ⊕ PK4 ⊕ PK6 ⊕ PK8 are sections of a vector bundle, whose rank grows linearly with the dimension of the manifold. For such a homogeneous para-Kähler structure, one has S X Y = g(X, Y )ξ−g(Y, ξ)X +g(X, J Y )J ξ−g(J Y, ξ)J X −2g(J X, ζ)J Y, (4.2.5) for some vector fields ξ, ζ ∈ X(M), and the Ambrose–Singer–Kiriˇcenko equations are equivalent to  R = 0,  = 0,  = 0. ∇ ∇ξ ∇ζ (4.2.6) This leads to the following definition. Definition 4.2.4 Homogeneous para-Kähler structures belonging to the class PK2 ⊕ PK4 ⊕PK6 ⊕PK8 are called homogeneous para-Kähler structures of linear type. A homogeneous para-Kähler structure of linear type S described as in (4.2.5) in terms of vector fields ξ and ζ is called 1. non-degenerate if g(ξ, ξ) = 0, 2. degenerate if g(ξ, ξ) = 0, 3. strongly degenerate if g(ξ, ξ) = 0 and ζ = 0.

4.2.3 Homogeneous Pseudo-Quaternion Kähler Structures Let (M, g, Q) be a pseudo-quaternion Kähler manifold (see Sect. 1.3.3). Its homogeneous structures S (to which we shall refer as homogeneous pseudo-quaternion Kähler structures) satisfy the Ambrose–Singer–Kiriˇcenko equations  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 = 0. ∇

The stabilizer of the 4-form  inside O(r, s) is Sp( p, q) Sp(1), where 4 p = r and 4q = s. We thus decompose the Sp( p, q) Sp(1)-module QK(V ) = V ∗ ⊗ (sp( p, q) + sp(1)) ⊂ S(V ) into irreducible submodules. Suppose that m = dimM ≥ 8 and set n = p + q. Following [42], we denote by E = C2n and H = C2 the standard representations of Sp( p, q) and Sp(1) respectively.

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4 Classification of Homogeneous Structures

The product of the quaternionic structures on E and H gives real structures on tensor products of E and H , and we denote the real part with respect to these real structures with brackets. For the sake of simplicity, we will omit some tensor products and write E H for E ⊗ H , etc. The standard representation of Sp( p, q) Sp(1) is thus V = [E H ], and the adjoint representation is isomorphic to [S 2 E] + [S 2 H ]. So, we have V ∗ ⊗ (sp( p, q) + sp(1))  [E H ] ⊗ ([S 2 E] + [S 2 H ])  [E H ] ⊗ [S 2 E] + [E H ] ⊗ [S 2 H ].

As complex representations, E and S 2 E have highest weights (1, 0, . . . , 0) and (2, 0, . . . , 0) respectively. Making use of the techniques in [107], we obtain the decomposition (1, 0, . . . , 0) ⊗ (2, 0, . . . , 0)  (2, 1, 0 . . . , 0) + (3, 0, . . . , 0) + (1, 0, . . . , 0)  K + S 3 E + E, where K is the module associated to the irreducible representation with highest weight (2, 1, 0, . . . , 0). In addition, H ⊗ S 2 H  H + S 3 H , where H is identified with the kernel of the symmetrization H ⊗ S 2 H → S 3 H . Summarizing, we have the decomposition into irreducible representations [60] V ∗ ⊗ (sp( p, q) + sp(1)) = [E H ] + [E S 2 H ] + [E H ] + [S 3 E H ] + [K H ].  = 0 is We now give the explicit expression of the tensors. The condition ∇ equivalent to 3  X Ja = dab Jb , a = 1, 2, 3, (4.2.7) ∇ b=1

where (dab ) is a matrix of 1-forms belonging to sp(1). This implies that Ja (S X Y ) − S X (Ja Y ) =

3 

cab (X )Jb Y,

a = 1, 2, 3,

b=1

where (cab ) is a matrix of 1-forms sitting in sp(1). Note that (cab ) can be obtained as the sp(1)-part of S X ∈ sp( p, q) + sp(1). Hence, the symmetries satisfied by a pseudo-quaternion Kähler structure are S X Y Z = −S X Z Y ,

(4.2.8)

S X Ja Y Ja Z − S X,Y,Z = −π (X )g(Jb Y, Ja Z ) + π (X )g(Jc Y, Ja Z ), c

b

(4.2.9)

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

103

for any cyclic permutation (a, b, c) of (1, 2, 3), where π 1 , π 2 , π 3 are local 1-forms on M, where we use the Einstein summation convention. We take the sum of the 1 three equations in (4.2.9), which up to a factor 4 is given by  S = π a ⊗ ωa . 2 Using the left-hand sides of (4.2.9), one sees that  S satisfies (4.2.8) and (4.2.9) for the same 1-forms π 1 , π 2 , π 3 as S. We now consider the tensor TXSY Z

1 = 4

 SX Y Z +

3 

 S X Ja Y Ja Z .

a=1

This tensor satisfies TXS Ja Y Ja Z − TXSY Z = 0, a = 1, 2, 3, and T S +  S = S. We can thus define invariant subspaces of QK(V )   1 Vˇ =  ∈ QK(V )/  X Y Z = π a (X )Ja Y, Z , π a ∈ V ∗ , a = 1, 2, 3 , 2   S ˆ V = T ∈ QK(V )/ TX J Y J Z − TXSY Z = 0, a = 1, 2, 3 , a

so that

a

ˆ QK(V ) = Vˇ + V.

The kernel of the equivariant map c12 restricted to Vˇ gives the space Vˇ 0 = ker(c12 ) ⊂ ˇ on which the symmetric bilinear form inherited from S(V) is non-degenerate. V, Hence, Vˇ = Vˇ 0 + Vˇ 0⊥ , where    ˇ =− Vˇ 0⊥ =  ∈ V/ (θ ◦ Ja ) ⊗ ωa , θ ∈ V ∗ . a

ˆ we consider the equivariant map With regard to the space V, L : Vˆ → Vˆ  3  S → L(S) X Y Z = S Z X Y + SY Z X + a=1 S Ja Y Ja Z X + S Ja Z X Ja Y . This map satisfies L 2 = 8Id−2L. Henceforth, L is diagonalizable with eigenvalues 2 and −4. Denoting by Vˆ 2 and Vˆ −4 the corresponding mutually orthogonal eigenspaces, we have Vˆ = Vˆ 2 + Vˆ −4 . The kernel Vˆ 0 ⊂ Vˆ of the restriction of c12 to Vˆ is non-degenerate with respect to the inherited symmetric bilinear form, so that we can consider its orthogonal ˆ A simple inspection shows that Vˆ −4 ⊂ Vˆ 0 and Vˆ ⊥ ⊂ Vˆ 2 , complement Vˆ 0⊥ ⊂ V. 0 whence we conclude that QK(V ) = Vˇ 0 ⊕ Vˇ 0⊥ ⊕ Vˆ 0⊥ ⊕ (Vˆ 2 ∩ Vˆ 0 ) + Vˆ −4 .

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4 Classification of Homogeneous Structures

We set

QK1 (V ) = Vˇ 0 , QK2 (V ) = Vˇ 0⊥ , QK3 (V ) = Vˆ 0⊥ , QK4 (V ) = Vˆ 2 ∩ Vˆ 0 , QK5 (V ) = Vˆ −4 .

We have the following. Proposition 4.2.5 [13, 42] For m ≥ 8, the space QK(V ) decomposes into irreducible and mutually orthogonal Sp( p, q) Sp(1)-submodules as QK(V ) = QK1 (V ) ⊕ QK2 (V ) ⊕ QK3 (V ) ⊕ QK4 (V ) ⊕ QK5 (V ), where 3    QK1 (V ) = S ∈ QK(V )/ S X Y Z = θ(Ja X )Ja Y, Z , θ ∈ V ∗ ,



QK2 (V ) = S ∈ QK(V )/ S X Y Z =

a=1 3 

θa (X )Ja Y, Z ,

a=1



3 

 θa ◦ Ja = 0, θa ∈ V ∗ ,

a=1

QK3 (V ) = S ∈ QK(V )/ S X Y Z = X, Y θ(Z ) − X, Z θ(Y ) +

3 

 (X, Ja Y θ(Ja Z ) − X, Ja Z θ(Ja Y )), θ ∈ V ∗ ,

a=1



QK4 (V ) = S ∈ QK(V )/ 6S X Y Z =

S

SX Y Z +

XYZ

 QK5 (V ) = S ∈ QK(V )/

3 

S

S X Ja Y Ja Z , a=1X Ja Y Ja Z



 c12 (S) = 0 ,

S SX Y Z = 0 . XYZ

If m = 4, then QK(V ) = QK1 (V ) ⊕ QK2 (V ) ⊕ QK3 (V ) ⊕ QK4 (V ). Making use of the isomorphisms QK1 (V )  QK3 (V )  [E H ], QK2 (V ) = [E S 3 H ], QK4 (V )  [S 3 E H ], QK5 (V )  [K H ],

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

we find

105

dim QK1 (V ) = dim QK3 (V ) = 4n, dim QK2 (V ) = 8n, dim QK4 (V ) = 43 n(n + 1)(2n + 1), dim QK5 (V ) =

16 n(n 2 3

− 1).

In particular, pseudo-quaternion Kähler structures in the composed class QK1 ⊕ QK2 ⊕ QK3 are sections of a vector bundle, whose rank grows linearly with the dimension of the manifold. Such a homogeneous pseudo-quaternion Kähler structure takes the form S X Y = g(X, Y )ξ − g(Y, ξ)X +

3 

(g(Ja Y, ξ)Ja X − g(X, Ja Y )Ja ξ)

a=1

+

3 

g(X, ζ a )Ja Y,

(4.2.10)

a=1

for some vector fields ξ and ζ a , a = 1, 2, 3. These facts lead to the following definition. Definition 4.2.6 Homogeneous pseudo-quaternion Kähler structures belonging to the class QK1 ⊕ QK2 ⊕ QK3 are called homogeneous pseudo-quaternion Kähler structures of linear type. A homogeneous pseudo-quaternion Kähler structure of linear type defined by the vector fields ξ and ζ a , a = 1, 2, 3, is called 1. non-degenerate if g(ξ, ξ) = 0, 2. degenerate if g(ξ, ξ) = 0. A homogeneous pseudo-hyper-Kähler structure is a homogeneous structure on a  Ja = 0, a = 1, 2, 3. pseudo-hyper-Kähler manifold (M, g, J1 , J2 , J3 ) satisfying ∇ This implies that S satisfies (4.2.8) and (4.2.9) with π a = 0 for a = 1, 2, 3. Therefore, the decomposition of HK(V ) = V ∗ ⊗ sp(r, s) can be read in terms of the pseudoquaternion Kähler case. In fact, HK(V ) = HK1 (V ) ⊕ HK2 (V ) ⊕ HK3 (V ), where HK1 (V ), HK2 (V ), HK3 (V ) have the same expressions as QK3 (V ), QK4 (V ) and QK5 (V ) respectively. A homogeneous pseudo-hyper-Kähler structure S is said to be of linear type if it belongs to the class HK1 . In that case, S, seen as a (1, 2)-tensor field, has the form

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4 Classification of Homogeneous Structures

S X Y = g(X, Y )ξ − g(Y, ξ)X +

3 

(g(Ja Y, ξ)Ja X − g(X, Ja Y )Ja ξ) ,

a=1

where ξ = θ . In addition, S ∈ HK1 (V ) is said to be degenerate if ξ is isotropic, and non-degenerate if ξ is non-isotropic.

4.2.4 Homogeneous Para-Quaternion Kähler Structures We now illustrate the classification of homogeneous para-quaternion Kähler structures. Many of the arguments in this section follow their analogues for the pseudoquaternion Kähler case, as described in the previous section. Let (M, g, Q) denote a para-quaternion Kähler manifold (see Sect. 1.3.4). Homogeneous structures S on (M, g, Q) satisfy the Ambrose–Singer–Kiriˇcenko equations  = 0, ∇g

 R = 0, ∇

 S = 0, ∇

 =0 ∇

and will be called homogeneous para-quaternion Kähler structures. In this case, the stabilizer of the 4-form  inside O(r, s) is given by Sp(n, R) Sp(1, R), where dimM = m = 4n ≥ 8. Thus, we need to decompose into irreducible submodules the Sp(n, R) Sp(1, R)-module PQ(V ) = V ∗ ⊗ (sp(n, R) + sp(1, R)) ⊂ S(V ). We denote by E = R2n and H = R2 the standard representations of Sp(n, R) and Sp(1, R), respectively. For the sake of simplicity, we will omit some tensor products and write E H for E ⊗ H and so on. The standard representation of Sp(n, R) Sp(1, R) is thus V = E H , and the adjoint representation is isomorphic to S 2 E + S 2 H . We then have V ∗ ⊗ (sp(n, R) + sp(1, R))  E H ⊗ (S 2 E + S 2 H )  E H ⊗ S 2 E + E H ⊗ S 2 H. The representations E and S 2 E have the highest weights (1, 0, . . . , 0) and (2, 0, . . . , 0) respectively. Making use of the techniques in [107] and taking into account the fact that the complexifications of sp(n, R) and sp(n) coincide, we have the decomposition (1, 0, . . . , 0) ⊗ (2, 0, . . . , 0)  (2, 1, 0 . . . , 0) + (3, 0, . . . , 0) + (1, 0, . . . , 0)  K + S 3 E + E, where K is the module associated to the irreducible representation with highest weight (2, 1, 0, . . . , 0). In addition, H ⊗ S 2 H  H + S 3 H , where H is identified with the kernel of the symmetrization H ⊗ S 2 H → S 3 H . Summarizing, we have the decomposition into irreducible representations

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

107

V ∗ ⊗ (sp(n, R) + sp(1, R)) = E H + E S 2 H + E H + S 3 E H + K H. (4.2.11) With regard to real tensors, we introduce the notation ( 1 , 2 , 3 ) = (−1, 1, 1), so  is equivalent to that Ja2 = a for a = 1, 2, 3. The condition ∇ 3 

X Ja = ∇

dab Jb ,

a = 1, 2, 3,

(4.2.12)

b=1

where (dab ) is a matrix of 1-forms sitting in sp(1, R). Consequently, we find Ja (S X Y ) − S X (Ja Y ) =

3 

cab (X )Jb Y,

a = 1, 2, 3,

b=1

where (cab ) a matrix of 1-forms sitting in sp(1, R). Note that (cab ) can be obtained as the sp(1, R)-part of S X ∈ sp(n, R) + sp(1, R). The symmetries satisfied by a para-quaternion Kähler structure are then given by S X Y Z = −S X Z Y ,

(4.2.13)

S X Ja Y Ja Z − S X,Y,Z = b π c (X )g(Jb Y, Ja Z ) − c π b (X )g(Jc Y, Ja Z ),

(4.2.14)

for any cyclic permutation (a, b, c) of (1, 2, 3), where π 1 , π 2 , π 3 are local 1-forms on M, and the Einstein summation convention is used. We take the sum of the three equations in (4.2.9), but multiplying each one by a . 1 Up to a factor 4, this gives  S = π a ⊗ ωa . 2 From the left-hand sides of (4.2.9), one sees that  S satisfies (4.2.8) and (4.2.9) for the same 1-forms π 1 , π 2 , π 3 as S. We now consider the tensor   3  1 TXSY Z = a S X Ja Y Ja Z . SX Y Z − 4 a=1 This tensor satisfies TXS Ja Y Ja Z + a TXSY Z = 0, a = 1, 2, 3, and T S +  S = S. Hence, we can define the following invariant subspaces of PQ(V ):   1 a a ∗ ˇ V =  ∈ PQ(V )/  X Y Z = π (X )Ja Y, Z , π ∈ V , a = 1, 2, 3 , 2   S Vˆ = T ∈ PQ(V )/ TX J Y J Z + a TXSY Z = 0, a = 1, 2, 3 , a

so that

a

ˆ PQ(V ) = Vˇ + V.

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4 Classification of Homogeneous Structures

The kernel of the equivariant map c12 restricted to Vˇ gives the space Vˇ 0 = ker(c12 ) ⊂ ˇ on which the symmetric bilinear form inherited from S(V) is non-degenerate. V, Hence Vˇ = Vˇ 0 + Vˇ 0⊥ , where    ˇ =− (θ ◦ Ja ) ⊗ ωa , θ ∈ V ∗ . Vˇ 0⊥ =  ∈ V/ a

ˆ we consider the self-adjoint equivariant map With regard to the space V, L : Vˆ → Vˆ   3 S → L(S) X Y Z = S Z X Y + SY Z X − a=1 a S Ja Y Ja Z X + S Ja Z X Ja Y . This map satisfies L 2 = 8 Id −2L, so that it is diagonalizable with eigenvalues 2 and −4. Denoting by Vˆ 2 and Vˆ −4 the corresponding mutually orthogonal eigenspaces, we have Vˆ = Vˆ 2 + Vˆ −4 . The kernel Vˆ 0 ⊂ Vˆ of the restriction of c12 to Vˆ is non-degenerate with respect to the inherited symmetric bilinear form, so that we can consider its orthogonal ˆ A simple inspection shows that Vˆ −4 ⊂ Vˆ 0 and Vˆ ⊥ ⊂ Vˆ 2 , complement Vˆ 0⊥ ⊂ V. 0 whence we conclude that PQ(V ) = Vˇ 0 ⊕ Vˇ 0⊥ ⊕ Vˆ 0⊥ ⊕ (Vˆ 2 ∩ Vˆ 0 ) ⊕ Vˆ −4 . We then set

PQ1 (V ) = Vˇ 0 , PQ2 (V ) = Vˇ 0⊥ , PQ3 (V ) = Vˆ 0⊥ , PQ4 (V ) = Vˆ 2 ∩ Vˆ 0 , PQ5 (V ) = Vˆ −4

and we have the following. Proposition 4.2.7 For m ≥ 8, the space PQ(V ) decomposes into irreducible and mutually orthogonal Sp(n, R) Sp(1, R)-submodules as PQ(V ) = PQ1 (V ) ⊕ PQ2 (V ) ⊕ PQ3 (V ) ⊕ PQ4 (V ) ⊕ PQ5 (V ), where 3    θ(Ja X )Ja Y, Z , θ ∈ V ∗ , PQ1 (V ) = S ∈ PQ(V )/ S X Y Z = a=1



PQ2 (V ) = S ∈ PQ(V )/ S X Y Z =

3  a=1

θa (X )Ja Y, Z ,

3  a=1

θa ◦ Ja = 0,  θa ∈ V ∗ ,

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

109

 PQ3 (V ) = S ∈ PQ(V )/ S X Y Z = X, Y θ(Z ) − X, Z θ(Y ) −

3 

 a (X, Ja Y θ(Ja Z ) − X, Ja Z θ(Ja Y )), θ ∈ V ∗ ,

a=1



PQ4 (V ) = S ∈ PQ(V )/ 6S X Y Z =

SS

XY Z

XYZ

 PQ5 (V ) = S ∈ PQ(V )/

SS

XY Z

−

3  a=1

a

S

S X Ja Y Ja Z , X Ja Y Ja Z

 =0 .

 c12 (S) = 0 ,

XYZ

If m = 4, then PQ(V ) = PQ1 (V ) ⊕ PQ2 (V ) ⊕ PQ3 (V ) ⊕ PQ4 (V ). Proof We adapt to para-quaternion structures the proof of the pseudo-quaternion Kähler case, as illustrated in [42]. We shall identify each submodule PQi (V ) with one appearing in decomposition (4.2.11). First we clearly have PQ1 (V )  E H  PQ3 (V ), with non-zero projections E H ⊗ S 2 H → PQ1 (V ),

E H ⊗ S 2 E → PQ3 (V ).

On the other hand, as dimPQ2 (V ) = 8n and PQ2 (V ) is orthogonal to PQ1 (V ) ˇ we have that PQ2 (V )  E S 3 H . Now, PQ3 (V ) + PQ4 (V ) + PQ5 (V )  inside V, E H + S 3 E H + K H . Observe that ∧2 E ⊗ E = Rω ⊗ E + (Rω)⊥ ⊗ E  E + ∧20 E ⊗ E  2E + K + V (1,1,1,0,...,0) , where V (1,1,1,0,...,0) is the representation with highest weight (1, 1, 1, 0, . . . , 0). Hence, K H is a submodule of both S 2 V ∗ ⊗ V ∗ ⊃ ∧2 E ⊗ E H and V ∗ ⊗ ∧2 E ⊃ E H ⊗ S 2 E. Using Schur’s Lemma, an equivariant map PQ4 (V ) + PQ5 (V ) → ∧2 E ⊗ E H will have kernel isomorphic to S 3 E H and will be non-zero in a submodule isomorphic to K . The module ∧2 E ⊂ S 2 V ∗ can be seen as the space of bilinear forms satisfying b(Ja ·, Ja ·) = a b(·, ·), a = 1, 2, 3, so that we consider the map p : S 2 V ∗ ⊗ V ∗ → ∧2E ⊗ E H  T → 41 TX Y Z − a a TJa X Ja Y Z . The projection π : V ∗ ⊗ ∧2 V ∗ → PQ5 (V ) is given by T → 16 (2 − L)U , where we put    1 UXY Z = a TX Ja Y Ja Z . TX Y Z − 4 a

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4 Classification of Homogeneous Structures

After some long calculations, one can check that the image under the composition of maps sym p π V ∗ ⊗ ∧2 V ∗ → PQ5 (V ) → S 2 V ∗ ⊗ V ∗ → ∧2 E ⊗ E H of an element α ⊗ β ∧ γ, α, β, γ ∈ V ∗ , is never zero if β and γ are linearly  (observe that this fact is made possible independent over the para-quaternions H by the condition n ≥ 2). Consequently, PQ5 (V )  K H and hence, PQ4 (V )   S3 E H . Because of the isomorphisms obtained in the previous proof, we have dim PQ1 (V ) = dim PQ3 (V ) = 4n, dim PQ2 (V ) = 8n, dim PQ4 (V ) = 43 n(n + 1)(2n + 1), dim PQ5 (V ) =

16 n(n 2 3

− 1).

Therefore, para-quaternion Kähler structures in the composed class PQ1 ⊕ PQ2 ⊕ PQ3 are sections of a vector bundle, whose rank grows linearly with the dimension of the manifold. A homogeneous para-quaternion Kähler structure belonging to PQ1 ⊕ PQ2 ⊕ PQ3 takes the form S X Y = g(X, Y )ξ − g(Y, ξ)X −

3 

a (g(Ja Y, ξ)Ja X − g(X, Ja Y )Ja ξ)

a=1

+

3 

g(X, ζ a )Ja Y,

a=1

(4.2.15) for some vector fields ξ and ζ a , a = 1, 2, 3. The facts above motivate the following definition. Definition 4.2.8 Homogeneous para-quaternion Kähler structures belonging to the class PQ1 ⊕ PQ2 ⊕ PQ3 are called homogeneous para-quaternion Kähler structures of linear type. A homogeneous para-quaternion Kähler structure of linear type defined as in (4.2.15) by the vector fields ξ and ζ a , a = 1, 2, 3, is called 1. non-degenerate if g(ξ, ξ) = 0, 2. degenerate if g(ξ, ξ) = 0.

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

111

A homogeneous para-hyper-Kähler structure is a homogeneous structure S on a  Ja = 0, a = 1, 2, 3. para-hyper-Kähler manifold (M, g, J1 , J2 , J3 ) satisfying ∇ In this case, S satisfies (4.2.13) and (4.2.14) with π a = 0 for a = 1, 2, 3. Therefore, the decomposition of PHK(V ) = V ∗ ⊗ sp(n, R) can be read in terms of the para-quaternion Kähler case. In fact, PHK(V ) = PHK1 (V ) ⊕ PHK2 (V ) ⊕ PHK3 (V ), where the expressions of PHK1 (V ), PHK2 (V ), PHK3 (V ) are the same as PQ3 (V ), PQ4 (V ) and PQ5 (V ) respectively. A homogeneous para-hyper-Kähler structure S is said to be of linear type if it belongs to the class PHK1 . In that case, S seen as a (1, 2)-tensor field has the form S X Y = g(X, Y )ξ − g(Y, ξ)X −

3 

a (g(Ja Y, ξ)Ja X − g(X, Ja Y )Ja ξ) ,

a=1

where ξ = θ . In addition, S ∈ PHK1 (V ) is said to be degenerate if ξ is isotropic, and non-degenerate if ξ is non-isotropic.

4.2.5 Homogeneous Sasakian and Cosymplectic Structures Let (M, g, φ, ξ, η) be an almost contact metric manifold (see Sect. 1.3.5). Homogeneous structures S on it satisfy the corresponding Ambrose–Singer–Kiriˇcenko equations  = 0,  R = 0,  S = 0,  =0 ∇g ∇ ∇ ∇φ and will be called homogeneous almost contact metric structures.  = 0 is equivalent to ∇  = 0, and implies ∇η  =0 Remark 4.2.9 The condition ∇φ  and ∇ξ = 0, since X ξ, Y ) = 0. X (ξ, Y ) + (∇ ∇ A classification of homogeneous almost contact metric structures was carried out in [60] with a representation theoretical approach. We recall that classification and obtain the corresponding classes of tensors. Hereafter we suppose dimM ≥ 5. We take V = R2n+1 endowed with its standard almost contact metric structure (·, ·, φ, ξ, η) as the model of the tangent space Tx M at a fixed point x ∈ M. We also consider the space of (0, 3)-tensors on V satisfying the same algebraic symmetries as a homogeneous almost contact metric structure, that is, S(V ) = {S ∈ ⊗3 V / S X Y Z + S X Z Y = 0}.  = 0 reads The condition ∇φ

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4 Classification of Homogeneous Structures

∇ X φ = [S X , φ],

(4.2.16)

where the bracket denotes the usual commutator of endomorphisms. We thus take the subspace S+ (V ) ⊂ S(V ) consisting of tensors such that S X commutes with φ, that is, S+ (V ) = {S ∈ S(V )/ S X φY φZ − S X Y Z = 0}. A simple computation shows that the symmetric bilinear form defined on S(V ) is non-degenerate on S+ (V ), so that we can consider its orthogonal complement S− (V ) = {S ∈ S(V )/ S X φY φZ + S X Y Z = η(Y )S X ξ Z + η(Z )S X Y ξ }. Identifying S(V ) with V ∗ ⊗ (∧2 V ∗ )  V ∗ ⊗ so(2 p + 1, 2q) (or so(2 p, 2q + 1), depending on the value of ε) we have that S+ (V ) is isomorphic to V ∗ ⊗ u( p, q), where u( p, q) is seen as the Lie algebra of

U( p, q) × {1} ⊂ O(2 p + 1, 2q) (or O(2 p, 2q + 1) ). Hence, S− (V ) is identified with V ∗ ⊗ u( p, q)⊥ (note that the Killing forms of so(2 p+1, 2q) and so(2 p, 2q +1) are non-degenerate on u( p, q)). Therefore, S+ (V ) is the space of tensors with the same symmetries as homogeneous almost contact metric structures on a cosymplectic manifold, and S− (V ) is the space of tensors with the same symmetries as ∇φ. In addition, by (4.2.16) we have that ∇φ gives the component of S in S− (V ). It is obvious that S+ (V ) is invariant under the induced U( p, q) × {1} representation on S(V ), hence so is S− (V ). Following [60], and using the same notation as in Sect. 4.2.1, we have for n ≥ 3 the decompositions into irreducible U( p, q) × {1}-modules 

2,1 + [B] , S+ (V ) = R + 2 1,0 + [1,1 0 ] + 0

2,0 2,1  3,0 2,0

S− (V ) = 2R + 2 1,0 + 2[1,1 + 0 +  + σ + [A] . 0 ]+2  

and For n = 2, the same decompositions are valid except that the modules 2,1 0 

3,0

is absent in S+ (V ). As we indicated in  are absent in S− (V ), and 2,1 0 Sect. 1.3, a decomposition of S− (V ) using real tensors is obtained in [50]. The irreducible submodule corresponding to α-Sasakian structures is the one-dimensional space C6 (V ) = {S ∈ S− (V )/ S X Y Z = αε(η(Z )X, Y  − η(Y )X, Z ), α ∈ R}.

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

113

Therefore, for a homogeneous almost contact metric structure S on a Sasakian manifold (i.e., a 1-Sasakian manifold), the component in S− (V ) is given by (S− ) X Y Z = ε(η(Z )X, Y  − η(Y )X, Z ). We finally obtain a decomposition of S+ (V ) into irreducible U( p, q) × {1}-modules using real tensors. Proposition 4.2.10 For n ≥ 3, the space S+ (V ) decomposes into irreducible and mutually orthogonal U( p, q) × {1}-modules as S+ (V ) = CS 1 (V ) ⊕ CS 2 (V ) ⊕ CS 3 (V ) ⊕ CS 4 (V ) ⊕ CS 5 (V ) ⊕ CS 6 (V ), where   1 CS 1 (V ) = S ∈ D2 / S X Y Z = S Z X Y + SY Z X + SφZ X φY + SφY φZ X , 2 

c12 (S) = 0 ,



CS 2 (V ) = S ∈ D2 / S X Y Z = X, Y ψ1 (Z ) − X, Z ψ1 (Y ) + X, φY ψ1 (φZ )  − X, φZ ψ1 (φY ) − 2φY, Z ψ1 (φX ), ψ1 ∈ V¯ ∗ ,   1 CS 3 (V ) = S ∈ D2 / S X Y Z = − S Z X Y + SY Z X + SφZ X φY + SφY φZ X , 2  c12 (S) = 0 ,  CS 4 (V ) = S ∈ D2 / S X Y Z = X, Y ψ2 (Z ) − X, Z ψ2 (Y ) + X, φY ψ2 (φZ )  − X, φZ ψ2 (φY ) + 2φY, Z ψ2 (φX ), ψ2 ∈ V¯ ∗ ,   CS 5 (V ) = S ∈ S+ (V ) / S X Y Z = η(X )ω0 (Y, Z ) ,   CS 6 (V ) = S ∈ S+ (V ) / trace(Sξ ) = 0 .

If n = 2, then S+ (V ) = CS 2 (V ) ⊕ CS 3 (V ) ⊕ CS 4 (V ) ⊕ CS 5 (V ) ⊕ CS 6 (V ). Proof We first decompose V ∗ = Rη ⊕ V¯ ∗ , where V¯ is the orthogonal complement to ξ. This gives the following orthogonal decomposition into U( p, q)-modules V ∗ ⊗ u( p, q) = (Rη ⊕ V¯ ∗ ) ⊗ u( p, q) = Rη ⊗ u( p, q) ⊕ V¯ ∗ ⊗ u( p, q). The first summand is isomorphic to u( p, q) and is identified with D1 = {S ∈ S+ (V )/ S X Y Z = η(X )SξY Z }.

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4 Classification of Homogeneous Structures

Thus, we can further decompose u( p, q) = Rω0 ⊕ su( p, q), where ω0 is the symplectic form on V¯ inherited from . This translates into D1 = CS 5 ⊕ CS 6 . The second summand consists of basic tensors and is identified with D2 {S ∈ S+ (V )/ SξY Z = 0}. Note that D2  V¯ ∗ ⊗ u( p, q) can be seen as the space of tensors with the same symmetries as homogeneous pseudo-Kähler structures on V¯ , so that following Sect. 4.2.1  we obtain the decomposition D2 = CS 1 ⊕ CS 2 ⊕ CS 3 ⊕ CS 4 . It is easy to see that we have the following isomorphisms 

, CS 1 (V )  2,1 0

CS 2 (V )  CS 4 (V )  1,0 , CS 3 (V )  [B] , CS 5 (V )  R, CS 6 (V )  [1,1 0 ], so that the previous are irreducible U( p, q) × {1}-modules. We shall now consider the above classification in the special cases of cosymplectic and Sasakian manifolds. First let (M, g, φ, ξ, η) denote a cosymplectic manifold, and let S be a homogeneous almost contact metric structure on M, which we will simply call a homogeneous cosymplectic structure. Since ∇φ = 0, the S− part of S vanishes, so that S ∈ S+ . We consider the class CS 2 ⊕ CS 4 ⊕ CS 5 , the dimension of which is 4n + 1. Elements of this class are defined by two basic 1-forms together with a one-dimensional vertical term corresponding to the CS 5 part. The expression of the dimension is a polynomial of degree 1 with respect to n and in analogy with the previous cases, the tensors belonging to this class are called homogeneous cosymplectic structures of linear type. The corresponding (1, 2)-vector field takes the form S X Y = g(X, Y )χ − g(χ, Y )X − g(X, φY )φχ + g(χ, J φY )φX − 2g(ζ, φX )φY − αη(X )φY,

(4.2.17)

for some vector fields χ and ζ, and α ∈ R. Next, let (M, g, φ, ξ, η) be a Sasakian manifold, and let S be a homogeneous almost contact metric structure on M, which we will simply call a homogeneous Sasakian structure. Since ∇ X φ = [S X , φ], the S− part of S belongs to C6 with α = 1. We consider homogeneous Sasakian structures whose S+ part belongs to the class CS 2 ⊕ CS 4 ⊕ CS 5 . For the same reason as before, we will call such S a homogeneous

4.2 Classification of Some Special Homogeneous Pseudo-Riemannian Structures

115

Sasakian structure of linear type. The corresponding (1, 2)-tensor field takes the form S X Y = g(X, Y )χ − g(χ, Y )X − g(X, φY )φχ + g(χ, J φY )φX −2g(ζ, φX )φY − αη(X )φY + g(X, Y )ξ − εη(Y )X,

(4.2.18)

for some vector fields χ and ζ, and α ∈ R. Since the metric g restricted to the contact distribution D = Span{ξ}⊥ may have a signature, we have to distinguish between the following cases. Definition 4.2.11 A homogeneous cosymplectic (Sasakian) structure of linear type is called 1. non-degenerate if g(χ, χ) = 0, and 2. degenerate if g(χ, χ) = 0. Remark 4.2.12 The same study can be done replacing the Sasakian condition by the α-Sasakian condition, which implies that the intrinsic torsion belongs to the class C6 with α not necessarily equal to 1.

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds It is a natural problem to classify homogeneous pseudo-Riemannian manifolds (M, g) of a given dimension. The aim is to know all the simply-connected examples, which provide a complete local description of all such spaces. This problem has been intensively studied. As may be expected, the higher the dimension, the bigger the computational difficulties. For this reason, the main results focus on the low-dimensional cases. It is possible to use different approaches to tackle this classification problem. A first possibility is to treat the problem “algebraically”, that is, classifying all the pairs (g, h) formed by a Lie algebra g ⊂ so( p, q) (( p, q) being the signature of the homogeneous pseudo-Riemannian manifold one is considering) and an isotropy subalgebra h, such that dim(g/h) = p+q = dim M. Another, more geometrical way is to understand which kind of geometric properties are true for the (locally) homogeneous pseudo-Riemannian manifolds of a given dimension, in order to understand which properties they share. Three-dimensional homogeneous Riemannian manifolds provide a nice example of these two different approaches. The explicit classification of connected, simplyconnected three-dimensional homogeneous Riemannian manifolds is well known. The possible dimensions of the isometry group g ⊂ so(3) are 6 (real space forms), 4 (essentially, the Bianchi–Cartan–Vranceanu spaces, see for example [118]) or 3 (Riemannian Lie groups [91]).

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4 Classification of Homogeneous Structures

On the other hand, a three-dimensional connected, simply-connected and complete homogeneous Riemannian manifold is either symmetric or isometric to some three-dimensional Lie group equipped with a left-invariant Riemannian metric [108]. Homogeneous pseudo-Riemannian structures play a fundamental role in this study. This is evident in the case of the first, algebraic approach, as each representation of a reductive homogeneous pseudo-Riemannian manifold (M, g) corresponds to a homogeneous pseudo-Riemannian structure and conversely. But homogeneous pseudo-Riemannian structures are also needed in the more geometric approach. In fact, in order to prove that some homogeneous pseudo-Riemannian manifold is isometric to a Lie group equipped with a left-invariant pseudo-Riemannian metric, one proves that it admits the special homogeneous structure which can be used to characterize the existence of a Lie group structure on the homogeneous manifold (see Example 2.4.9). The purpose of this section is to describe the geometric characterization of threedimensional locally homogeneous pseudo-Riemannian manifolds and to outline the state-of-the-art in the study of four-dimensional locally homogeneous pseudoRiemannian manifolds. Up to reversing the metric [101], a three-dimensional pseudo-Riemannian manifold is either Riemannian or Lorentzian. Let (M, g) be a connected three-dimensional pseudo-Riemannian manifold. Its curvature tensor is completely determined by the Ricci tensor , defined as in Eq. (1.2.2), that is, (X, Y ) p =

3 

εi g(R(X, ei )Y, ei ),

i=1

for any point p ∈ M and X, Y ∈ T p M, where {e1 , e2 , e3 } is an orthonormal basis of T p M and εi = g(ei , ei ) = ±1 for all i. When (M, g) is locally homogeneous, the existence of local isometries between neighbourhoods of arbitrary points permits us to construct an orthonormal frame field {ei } such that the components of  and of its derivatives of any order remain constant along M. As we shall see, we only use the constancy of the components of , ∇ and ∇ 2  with respect to {ei }, which expresses the weaker property that the manifold is curvature homogeneous up to order 2 (for a survey on curvature homogeneity properties, we refer to [68]). With respect to such a frame field {ei }, we now put ∇ei e j =



ε j Bi jk ek .

(4.3.19)

k

Clearly, the functions Bi jk completely determine the Levi-Civita connection, and conversely. Observe that from ∇g = 0 it follows at once that Bik j = −Bi jk ,

(4.3.20)

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

117

for all i, j, k. In particular, Bi j j = 0

(4.3.21)

for all indices i and j. Taking into account the constancy of the components of  and ∇, an easy calculation yields   ε j Bi jt tk + εk Bikt t j (4.3.22) ∇i  jk = − t

and ∇ri2  jk = −



(εi Brit ∇t  jk + ε j Br jt ∇i tk + εk Br kt ∇i t j ),

(4.3.23)

t

for all indices i, j, k, r . We want to prove that whenever (M, g) is not locally symmetric, there exists a suitable orthonormal frame field {ei }, as before, for which all Bi jk are constants. As illustrated in Example 2.4.9, this implies the existence of a special homogeneous pseudo-Riemannian structure T satisfying Tei e j = ∇ei e j for all indices i, j, which ensures that (M, g) is locally isometric to a Lie group with a left-invariant metric g. Because of the symmetries of the curvature tensor, the Ricci tensor  is symmetric. Consequently, the Ricci operator Ric, defined by g(Ric(X ), Y ) = (X, Y ), is selfadjoint. As this yields different consequences depending on the signature of the metric tensor, we shall now treat separately the Riemannian and the Lorentzian cases. The arguments below are an adaptation of the results obtained for the two cases respectively in [25, 108].

4.3.1 The Riemannian Case If the metric tensor g is Riemannian, we have εi = 1 for all indices i. However, we shall write formally the equations for this case using εi (1 ≤ i ≤ 3) in order to use them again for the analogous Lorentzian case (where ε1 = ε2 = −ε3 = 1). A well-known linear algebra argument yields that, since Ric is self-adjoint with respect to g, we can pass to an orthonormal basis of Ricci eigenvectors (which we shall denote again by {ei }), for which i j = εi δi j qi , for all i, j, where q1 = a, q2 = b and q3 = c denote the eigenvalues of Ric. Hence, (4.3.22) simplifies as follows: ∇i  jk = −ε j εk (q j − qk )Bi jk .

(4.3.24)

In particular, from (4.3.24) we get that ∇i  j j = 0 for all i, j. Depending on the algebraic multiplicity of the Ricci eigenvalues, we consider separately the following cases. (I-a) a = b = c. Then, (M, g) is an Einstein manifold and so, being threedimensional, it has constant sectional curvature. In particular, it is locally symmetric.

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4 Classification of Homogeneous Structures

(I-b) a = b = c = a. In this case, q j − qk = 0 for all j = k and so, (4.3.24) yields at once that Bi jk is constant for all j = k. Taking into account (4.3.21), all Bi jk are then constant. (I-c) a = b = c. Writing (4.3.24) with ( j, k) = (1, 2) we get ∇i 12 = 0, while for ( j, k) = (1, 3) and ( j, k) = (2, 3) we respectively obtain that Bi13 and Bi23 are constant for all i. We shall prove that, unless (M, g) is locally symmetric, there exists a suitable orthonormal frame field {ei } with respect to which Bi12 are also constant for all i. (M, g) being locally homogeneous, the scalar curvature τ (see Eq. (1.2.2)) is constant. Hence, from the well-known divergence formula dτ = 2 div (see [12, p. 88]), we get  ε j ∇ j i j , (4.3.25) 0 = ei (τ ) = 2 j

for all i = 1, 2, 3. Writing (4.3.25) for i = 1, 2, 3, we have ∇1 13 + ∇2 23 = ∇3 13 = ∇3 23 = 0, that is, by (4.3.24), B113 + B223 = B313 = B323 = 0.

(4.3.26)

Since B313 = B323 = 0, the integral curves of e3 are geodesics. Therefore (see also [108]), we can choose {ei } so that ∇e3 ei = 0, that is, B312 = 0. All the components of , ∇ and ∇ 2  are constant. Moreover, Br 3t is constant for all indices r, t. Thus, writing (4.3.23) for (i, j, k) = (1, 2, 3), we get ∇ri2 23 + ε3 Br 3t ∇i t2 = −εi Brit ∇t 12 − Br 2t ∇i t1 ,

(4.3.27)

where the left-hand side is constant. For i = 1, taking into account (4.3.24) and the constancy of Br 23 , (4.3.27) yields that (∇1 13 − ∇2 23 )Br 12 is constant, for all indices r . Therefore, all Br 12 are constant, unless ∇1 13 = ∇2 23 . By the same argument, from (4.3.27) for i = 2 we get that (∇1 23 + ∇2 13 )Br 12 is constant, leading again to the constancy of all coefficients Br 12 , unless ∇1 23 = −∇2 13 . Hence, we are left with the case when ∇1 13 = ∇2 23 and ∇1 23 = −∇2 13 , that is, by (4.3.24) and (4.3.26), B113 = B223 = 0 and B213 = −B123 . In this remaining case, ∇ is completely determined by the only possibly non-zero components ∇1 23 = −∇2 13 = −ε3 (a − c)α,

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

119

where α = B123 is a real constant. Clearly, if α = 0, then (M, g) is Ricci-parallel and so (being three-dimensional) locally symmetric. Thus, from now on we shall assume α = 0, and consider the system of partial differential equations ⎧ ⎨ e1 η = B112 , e2 η = B212 , ⎩ a . e3 η = − 2α

(4.3.28)

We can compute again the Ricci components using the covariant derivatives ∇ei e j and we get ⎧ 2 2 a = −e1 (B212 ) + e2 (B112 ) − B112 − B212 , ⎪ ⎪ ⎨ 2 c = 2ε3 α , (4.3.29) 0 = 13 = e3 (B212 ) + ε3 αB112 , ⎪ ⎪ ⎩ 0 = −23 = e3 (B112 ) + ε3 αB212 . Using (4.3.28) and (4.3.29), it is easy to check that [ei , e j ]η = ei (e j η) − e j (ei η), for all i, j. Hence, a basic theorem on partial differential equations ensures that (4.3.28) admits a unique solution under an initial condition η0 = η( p), with p ∈ M. For such a solution η of (4.3.28), we can construct a new orthonormal frame {ei∗ }, defined by e1∗ = (cos η)e1 − (sin η)e2 , e2∗ = (sin η)e1 + (cos η)e2 , e3∗ = e3 ,

(4.3.30)

and it is easy to check that the only possibly non-vanishing functions Bi∗jk are given by a ∗ ∗ ∗ ∗ ∗ ∗ B123 = −B132 = −B213 = B231 = α, B312 = −B321 =− , 2α which are all constant. Summarizing, in all the possible cases we found that either (M, g) is locally symmetric, or all connection coefficients Bi jk are constant with respect to a suitable orthonormal basis {ei }. In the latter case, (M, g) then admits the special homogeneous pseudo-Riemannian structure determined by Tei e j = ∇ei e j and so, (M, g) is locally isometric to some Riemannian Lie group, that is, a Lie group equipped with a left-invariant Riemannian metric. Under the hypotheses of (M, g) being simply-connected and complete, one then gets the corresponding global result. Thus, we have the following result (see [108]).

120

4 Classification of Homogeneous Structures

Theorem 4.3.1 A three-dimensional connected, simply-connected, complete homogeneous Riemannian manifold (M, g) is either symmetric or isometric to some threedimensional Riemannian Lie group. If any of the hypotheses of simple connectedness or completeness is missing, then (M, g) is either locally symmetric or locally isometric to some Riemannian Lie group. Taking into account Milnor’s study of three-dimensional Riemannian Lie groups [91], the above Theorem 4.3.1 leads to the complete classification of locally homogeneous Riemannian manifolds in dimension three. With regard to the locally symmetric cases, from the proof of Theorem 4.3.1 we see that they occur exactly for the possibilities listed below in terms of the Ricci eigenvalues a, b, c: (i) a = b = c: then, (M, g) has constant sectional curvature; (ii) a = b = c, and all connection functions Bi jk vanish, except for B112 and B212 . Then, ∇ei e3 = 0 and so, e3 is a parallel vector field. Consequently, (M, g) locally splits into the Riemannian product M 2 × R, where R is spanned by e3 and M 2 has constant (Gaussian) curvature. Thus, we have the following. Theorem 4.3.2 A three-dimensional connected, simply-connected, complete homogeneous Riemannian manifold (M, g) is isometric to one of the following Riemannian spaces: (a) A Riemannian real space form: R3 , S3 (k), H3 (−k); (b) the Riemannian product of a real line and a Riemannian surface of constant Gaussian curvature k = 0: S2 (k) × R, H2 (−k) × R; (c) one of the following Riemannian Lie groups (G, g): (I) If G is unimodular, then its Lie algebra g admits an orthonormal basis {e1 , e2 , e3 } such that [e1 , e2 ] = λ3 e3 , [e2 , e3 ] = λ1 e1 , [e3 , e1 ] = λ2 e2 . More precisely, depending on the sign of λi , G is one of the following Lie groups [91]: G SU (2) (or S O(3)) S L(2, R) (or S O(1, 2)) ˜ E(2) E(1, 1) H3 R⊕R⊕R 3D Riemannian Lie groups

λ1 +

λ2 +

λ3 +

+

+

−

+ + + 0

+ − 0 0

0 0 0 0

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

121

In the table (and throughout the book), S L(2, R) denotes the universal cov˜ ering of S L(2, R), E(2) the universal covering of the group of rigid motions of the Euclidean plane, E(1, 1) the group of rigid motions of the Minkowski space and H3 the Heisenberg group. (II) If G is not unimodular, then its Lie algebra g admits an orthonormal basis {e1 , e2 , e3 } such that [e1 , e2 ] = αe2 + βe3 , [e1 , e3 ] = γe2 + δe3 , [e2 , e3 ] = 0, α + δ  = 0, αγ + βδ = 0.

4.3.2 The Lorentzian Case In Lorentzian settings, four different cases (called Segre types) can occur for a selfadjoint operator like Ric (see [101, p. 261]). More precisely, there exists an orthonormal basis {e1 , e2 , e3 }, with e3 timelike, such that Ric takes one of the following canonical forms: ⎛ ⎞ ⎛ ⎞ a00 a 0 0 Segre type {11, 1} : ⎝ 0 b 0 ⎠ , Segre type {1z z¯ } : ⎝ 0 b c ⎠ , 00c 0 −c b ⎛ ⎞ ⎛ ⎞ a 0 0 b a −a ε ⎠, Segre type {21} : ⎝ 0 b Segre type {3} : ⎝ a b 0 ⎠ . (4.3.31) 0 −ε b − 2ε a0 b When (M, g) is locally homogeneous, the constancy of the Ricci components with respect to a suitable orthonormal basis implies that the Segre type of Ric remains constant along M. We shall now treat separately the different cases, according to the Segre type of the Ricci operator of (M, g). (I) Segre type {11, 1}. In this case, Ric is diagonal, like in the Riemannian case, and Eq. (4.3.24) remains valid. So, ∇i  j j = 0 for all i, j and we have the following subcases to consider, depending on the multiplicity of the Ricci eigenvalues a, b, c. (I-a) a = b = c. As in the Riemannian case, (M, g) is an Einstein manifold and so, being three-dimensional, it has constant sectional curvature. In particular, it is locally symmetric. (I-b) a = b = c = a. Because of (4.3.24), all Bi jk are constant. (I-c) a = b = c. All the equations of the corresponding Riemannian case a = b = c hold, and by exactly the same argument we conclude that either (M, g) is locally symmetric (when the only non-vanishing connection functions are B112 and B212 ), or there exist some special orthonormal basis {ei }, for which all connection functions are constant. (I-d) a = b = c. Although it is similar to case I-c) above, this case must be treated separately, because e3 is a time-like vector field.

122

4 Classification of Homogeneous Structures

Writing (4.3.24) with ( j, k) = (2, 3) we find ∇i 23 = 0, while for ( j, k) = (1, 3) and ( j, k) = (2, 3) it respectively yields that Bi12 and Bi13 are constant for all i. The divergence formula (4.3.25) now gives ∇2 12 − ∇3 13 = ∇1 12 = ∇3 13 = 0, that is, by (4.3.24), B212 + B313 = B112 = B113 = 0.

(4.3.32)

Since B112 = B113 = 0, the integral curves of e1 are geodesic. Therefore, we can choose {ei } so that ∇e1 ei = 0, that is, B123 = 0. All the components of , ∇ and ∇ 2  are constant and Br 1t is constant for all indices r, t. Thus, writing (4.3.23) for (i, j, k) = (3, 1, 2) and for (i, j, k) = (2, 1, 2), we get that (∇2 12 + ∇3 13 )Br 23 , (∇3 12 + ∇2 13 )Br 23 are constant. Hence, all coefficients Br 23 are constant, unless ∇2 12 = −∇3 13 and ∇3 12 = −∇2 13 , that is, by (4.3.24) and (4.3.32), B212 = B313 = 0 and B312 = B213 . In this remaining case, ∇ is completely determined by the only possibly non-zero components ∇2 13 = −∇3 12 = −(a − c)α, where α = B213 is a real constant. If α = 0, then (M, g) is Ricci-parallel and so (being three-dimensional) locally symmetric. So, we now assume α = 0, and consider the system of partial differential equations ⎧ b , ⎨ e1 η = − 2α e2 η = B223 , ⎩ e3 η = B323 .

(4.3.33)

Computing the curvature components in terms of the connection functions, we find ⎧ a = −2α2 , ⎪ ⎪ ⎨ 2 2 − B323 , b = −e2 (B323 ) + e3 (B223 ) + B223 = e (B ) + αB , 0 = − ⎪ 12 1 323 223 ⎪ ⎩ 0 = −13 = e1 (B223 ) + αB323 .

(4.3.34)

Using (4.3.34) it is easy to check that [ei , e j ]η = ei (e j η) − e j (ei η), for all i, j. Hence, (4.3.33) admits a unique solution under an initial condition η0 = η( p), with p ∈ M. If η is such a solution of (4.3.33), we put e1∗ = e1 , e2∗ = (cosh η)e1 − (sinh η)e2 , e3∗ = (sinh η)e1 − (cosh η)e2 . (4.3.35)

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

123

Then, {ei∗ } is again an orthonormal frame (with e3∗ timelike), and the connection functions Bi∗jk are completely determined by ∗ =− B123

b , 2α

∗ ∗ = B312 = −α. B213

Therefore, all Bi∗jk are constant. (II) Segre type {1z z¯ }. In this case, 11 = a, 22 = −33 = b, 12 = 13 = 0 and 23 = c = 0. Writing (4.3.22) for ( j, k) = (2, 2), (1, 2) and (1, 3), we easily get ⎧ ⎨ ∇i 22 = −2cBi23 , ∇i 12 = (a − b)Bi12 − cBi13 , ⎩ ∇i 13 = −cBi12 − (a − b)Bi13 .

(4.3.36)

Since c = 0 and all ∇i  jk are constant, (4.3.36) implies at once that Bi jk are constant whenever j = k. This, together with (4.3.21), implies that all Bi jk are constant. (III) Segre type {21}. In this case, 11 = a, 22 = b, 33 = 2ε − b, 12 = 13 = 0 and 23 = ε, where ε = ±1. When a = b − ε, we can proceed as in the previous case. In fact, we write (4.3.22) for ( j, k) = (2, 2), (1, 2) and (1, 3) and we get ⎧ ⎨ ∇i 22 = −2εBi23 , ∇i 12 = (a − b)Bi12 − εBi13 , ⎩ ∇i 13 = −εBi12 − (a − b + 2ε)Bi13 .

(4.3.37)

If a = b − ε, then (4.3.37) implies at once that Bi jk are constant for j = k and so, for all i, j, taking into account (4.3.21). Thus, we are left with the case when a = b − ε. Writing (4.3.22) for all possible j, k, we easily get ⎧ ⎨ ∇i 11 = 0, ∇i 22 = ∇i 33 = ∇i 23 = −2εBi23 , ⎩ ∇i 12 = ∇i 13 = −ε(Bi12 + Bi13 ),

(4.3.38)

for all i. In particular, Bi23 = constant,

Bi12 + Bi13 = constant,

(4.3.39)

for all i. From the divergence formula (4.3.25), using (4.3.38) we now get B212 + B213 = B312 + B313 ,

B112 + B113 = −2(B223 − B323 ).

Next, we write (4.3.23) for ( j, k) = (1, 2). Using (4.3.38), we obtain ∇ri2 12 = −εi Brit ∇t 12 − ∇i 22 (Br 12 + Br 13 ) − Br 23 ∇i 13 ,

(4.3.40)

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4 Classification of Homogeneous Structures

that is, taking into account (4.3.39), εi Brit ∇t 12 = constant,

(4.3.41)

for all r, i. We write (4.3.41) for (r, i) = (1, 2), (2, 2), (3, 3), (1, 3) and (2, 3). Since Bi23 is constant for all indices i, we obtain that B112 ∇1 12 ,

B212 ∇1 12 ,

B313 ∇1 12 ,

B113 ∇1 12 ,

B213 ∇1 12

are constant. If ∇1 12 = 0, this implies (by (4.3.40)) that all Bi jk are constant. Hence, we are left with the case when ∇1 12 = 0, that is, B112 + B113 = 0 and, by (4.3.40), B223 = B323 . We now calculate (4.3.23) for ( j, k) = (2, 2). Taking into account the constancy of the components of ∇ and ∇ 2  and of Br 23 , we then have εi Brit ∇t 22 − 2Br 12 ∇i 12 = −∇ri2 22 − 2Br 23 ∇i 23 = constant,

(4.3.42)

for all indices r, i. For i = 1, (4.3.42) (taking into account (4.3.38)) yields the constancy of −2εB223 (Br 12 − Br 23 ), for all r . If B223 = 0, then Br 12 − Br 23 is a constant and so, by (4.3.39), we conclude that Br 12 , Br 13 are all constant. Thus, we are left with the case when B223 = 0 (whence, B323 = 0 by (4.3.40)). The only possibly non-vanishing connection functions are now given by B123 = α,

B212 ,

B213 = β − B212 ,

B312 ,

B313 = β − B312 ,

where α, β are real constants. Writing (4.3.42) for i = 2, we find that −(∇1 22 + 2∇2 12 )Br 12 is constant for all r , leading to the constancy of all Br 12 (and so, of all Bi jk ), unless ∇1 22 + 2∇2 12 = 0, which yields β = −α. Finally, the remaining components of ∇ are now given by ∇1 22 = ∇1 33 = ∇1 23 = −2εα, ∇2 12 = ∇2 13 = −∇3 12 = −∇3 13 = εα. Therefore, if α = 0, then (M, g) is (Ricci-parallel and so) locally symmetric. On the other hand, if α = 0, then by (4.3.42) with i = 3, taking into account the above components of ∇ we get the constancy of 2εα(Br 12 − Br 13 ),

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

125

for all indices r . Since α = 0, we conclude that Br 12 − Br 13 is constant, which, by (4.3.39), yields that all Bi jk are constant. (IV) Segre type {3}. In this case, 11 = 22 = −33 = b, 12 = −13 = a = 0 and 23 = 0. We now write (4.3.22) for ( j, k) = (1, 2), (2, 2) and (3, 3) and we get ⎧ ⎨ ∇i 12 = a Bi23 , ∇i 22 = 2a Bi12 , ⎩ ∇i 33 = 2a Bi13 .

(4.3.43)

Therefore, as a = 0, from (4.3.43) it follows at once that Bi jk are constant for all indices j, k. Thus, we have the following Lorentzian counterpart of Theorem 4.3.1. Theorem 4.3.3 A three-dimensional connected, simply-connected, complete homogeneous Lorentzian manifold (M, g) is either symmetric or isometric to some threedimensional Lorentzian Lie group. If any of the hypotheses of simple connectedness or completeness is missing, then (M, g) is either locally symmetric or locally isometric to some Lorentzian Lie group. The above characterization is perfectly analogous to that of the Riemannian case. However, it leads to a different and richer classification result. In fact, while there exists just one standard form for the Lie algebra of three-dimensional unimodular Riemannian Lie groups and another one for the non-unimodular ones [91], • as proved in [106], there exist 4 distinct forms of three-dimensional unimodular Lorentzian Lie algebras, depending on the Segre type of the self-adjoint operator defined, using the Lorentzian cross product ×, by L([ei , e j ]) = ei × e j ; • as proved in [52], there exist 3 distinct forms of three-dimensional Lorentzian non-unimodular Lie algebras, depending on the unimodular kernel being either space-like, time-like or degenerate. Moreover, from the proof of Theorem 4.3.3, we see that locally symmetric Lorentzian examples occur in some cases with diagonal Ricci operator, which (although more numerous) are analogues of the corresponding Riemannian cases, and also in the case of Segre type {21}, which has no Riemannian counterpart. In fact, they occur exactly for the possibilities listed below: (i) Segre type {(11, 1)}, that is, when a = b = c: then (M, g) has constant sectional curvature. (ii) Segre type {(11), 1}, when a = b = c and all connection functions Bi jk vanish, except for B112 and B212 . Then ∇ei e3 = 0 and so, e3 is a parallel vector field. Consequently, (M, g) locally splits into the Lorentzian product M 2 ×R1 , where R1 is spanned by e3 (time-like) and M 2 is a Riemannian surface of constant (Gaussian) curvature. (iii) Segre type {(1(1, 1)}, when a = b = c and all connection functions Bi jk vanish, except for B223 and B323 . Then ∇ei e1 = 0 and so, e1 is a parallel vector field.

126

4 Classification of Homogeneous Structures

Consequently, (M, g) locally splits into the Lorentzian product R × M12 , where R is spanned by e1 (space-like) and M12 is a Lorentzian surface of constant (Gaussian) curvature. (iv) Segre type {(21)}, when a = b − ε and all connection functions Bi jk vanish, except for B112 , B113 = −B112 , B212 , B213 = −B212 , B312 , B313 = −B312 . Put u = e2 − e3 . Then u is a null vector field, and the above equations for the connection functions yield at once that ∇ei u = 0, that is, u is a parallel null vector field. The existence of parallel null vector fields clearly does not have a Riemannian analogue. Three-dimensional strictly Walker manifolds, that is, Lorentzian manifolds admitting a parallel null vector field, were investigated in [47]. A three-dimensional locally symmetric Lorentzian manifold (M, g), having a parallel null vector field, admits local coordinates (t, x, y) such that,   ∂ ∂ ∂ with respect to the local frame field ( ∂t ), ( ∂x ), ( ∂ y ) , the Lorentzian metric g and the Ricci operator are explicitly given by ⎛

⎞

00 1 g = ⎝0 ε 0 ⎠, 10 f

⎛

0 0 − 1ε α

⎜ Ric = ⎜ ⎝0 0 00

⎞

⎟ ⎟ 0 ⎠, 0

(4.3.44)

where ε = ±1, u = ( ∂t∂ ) and f (x, y) = x 2 α + xβ(y) + ξ(y),

(4.3.45)

for any real constant α (= 0, otherwise (M, g) is flat) and any functions β, ξ (see [47, Theorem 6]). The Ricci operator, as described in (4.3.44), is indeed of Segre type {(21)}, having λ = 0 as the unique eigenvalue, associated to a two-dimensional eigenspace. Taking into account the results of [52, 106], we then have the following. Theorem 4.3.4 A three-dimensional connected, simply-connected, complete homogeneous Lorentzian manifold (M, g) is isometric to one of the following Lorentzian spaces: (a) A Lorentzian space form: R3 , S3 (k), H3 (−k); (b) a Lorentzian direct product of a real line and a surface of constant Gaussian curvature k = 0: S2 (k) × R1 , H2 (−k) × R1 , S21 (k) × R, H21 (−k) × R; (c) a three-dimensional locally symmetric strictly Walker manifold, explicitly described by (4.3.44), (4.3.45) in local coordinates; (d) one of the following Lorentzian Lie groups (G, g):

4.3 Three-Dimensional Homogeneous Pseudo-Riemannian Manifolds

127

(I) If G is unimodular, then there exists an orthonormal frame field {e1 , e2 , e3 }, with e3 time-like, such that the Lie algebra of G is one of the following: [e1 , e2 ] = αe1 − βe3 , g1 : [e1 , e3 ] = −αe1 − βe2 , [e2 , e3 ] = βe1 + αe2 + αe3

(4.3.46) α = 0.

If β = 0, then G is S L(2, R), while G = E(1, 1) for β = 0. [e1 , e2 ] = −γe2 − βe3 , g2 : [e1 , e3 ] = −βe2 + γe3 , [e2 , e3 ] = αe1 .

γ = 0,

(4.3.47)

In this case, G = S L(2, R) if α = 0, while G = E(1, 1) if α = 0. g3 : [e1 , e2 ] = −γe3 , [e1 , e3 ] = −βe2 , [e2 , e3 ] = αe1 .

(4.3.48)

Table 4.1 lists all the Lie groups G which admit a Lie algebra g3 , according to the different possibilities for α, β and γ. [e1 , e2 ] = −e2 + (2ε − β)e3 , g4 : [e1 , e3 ] = −βe2 + e3 , [e2 , e3 ] = αe1 .

ε = ±1, (4.3.49)

Table 4.2 describes all Lie groups G admitting a Lie algebra g4 . (II) If G is non-unimodular, then there exists an orthonormal frame field {e1 , e2 , e3 }, with e3 time-like, such that the Lie algebra of G is one of the following: Table 4.1 3D Lorentzian Lie groups with Lie algebra g3 Lie group α β S L(2, R) + +

γ +

S L(2, R) SU (2)  E(2)

+ +

− +

− −

+

+

0

 E(2) E(1, 1) E(1, 1) H3 H3 R⊕R⊕R

+ + + + 0 0

0 − 0 0 0 0

− 0 + 0 − 0

128

4 Classification of Homogeneous Structures

Table 4.2 3D Lorentzian Lie groups with Lie algebra g4 Lie group (ε = 1) S L(2, R) E(1, 1) E(1, 1)  E(2) H3

α

β

= 0 0 0 0

1 1

[e1 , e2 ] = 0, g5 : [e1 , e3 ] = αe1 + βe2 , [e2 , e3 ] = γe1 + δe2 ,

Lie group (ε = −1) S L(2, R) E(1, 1) E(1, 1)  E(2) H3

α

β

= 0 0 >0

= −1 = −1 −1

2 admits a nontrivial homogeneous structure S ∈ S1 if and only if it is isometric to the real hyperbolic space RH (m). (ii) ([63, Theorem 1.1]) A connected, simply-connected and complete irreducible Kähler manifold of dimension 2m > 4 admits a nontrivial homogeneous Kähler structure S ∈ K2 ⊕ K4 if and only if it is holomorphically isometric to the complex hyperbolic space CH (m). (iii) ([42, Theorem 1.1]) A connected, simply-connected and complete quaternionic Kähler manifold of dimension 4m > 8 admits a nontrivial homogeneous quaternionic Kähler structure S ∈ QK1 ⊕ QK2 ⊕ QK3 if and only if it is isometric to the quaternionic hyperbolic space HH (m). In this case, the homogeneous structure is necessarily of type QK3 . From this last proposition, we see that the real, complex and quaternionic hyperbolic spaces become the reference manifolds for the linear classes of their corresponding geometries. It is natural to ask what are the homogeneous descriptions of these spaces associated to homogeneous structures of linear type. The following results provide the answer to this question. Proposition 5.1.2 (i) ([116, p. 55], [43, Sect. 3.1]) The homogeneous Riemannian structures of linear type on RH (m) can be realized by the homogeneous model AN , where AN stands for the solvable part of the Iwasawa decomposition of the full isometry group S O(m, 1). (ii) ([43, Theorem 4.4]) The homogeneous Kähler structures of linear type on CH (m) can be realized by the homogeneous model U (1)AN /U (1), AN standing here for the solvable part of the Iwasawa decomposition of the full isometry group SU (m, 1). The homogeneous model CH (m) = AN can be defined by a tensor in the non-linear class K2 ⊕ K3 ⊕ K4 . (iii) ([42, Theorem 5.4]) The homogeneous quaternionic Kähler structures of linear type on HH (m) can be realized by the homogeneous model Sp(1)AN /Sp(1), where AN now stands for the solvable part of the Iwasawa decomposition of the full isometry group Sp(m, 1). The homogeneous model HH (m) = AN can be defined by a tensor in the non-linear class QK1 ⊕ QK3 ⊕ QK4 .

5.1 Homogeneous Structures in the Class S1

135

It is interesting to note that these hyperbolic spaces may also admit homogeneous structures belonging to other (non-linear) classes. A homogeneous structure of the class S1 has the expression S X Y = g(X, Y )ξ − g(Y, ξ)X, for a certain vector field ξ ∈ X(M). If we want to generalize the above results to metrics with signature, the causal nature of ξ (which turns out to be constant along the manifold) plays an essential role. We recall Definition 4.1.4, where a homogeneous structure S ∈ S1 is called a non-degenerate or degenerate homogeneous structure of linear type according to the vector field ξ being isotropic or non-isotropic. In the non-degenerate case, we have the following results. Proposition 5.1.3 ([65, Proposition 3.2]) A connected pseudo-Riemannian manifold (M, g) admitting a non-degenerate homogeneous structure S ∈ S1 defined by a vector field ξ has nonzero constant sectional curvature −g(ξ, ξ). The previous proposition shows some remarkable differences to the Riemannian case. First, spaces equipped with non-degenerate homogeneous structures in the class S1 may have constant positive curvature. On the other hand, it can be proved that the pseudo-Riemannian sphere and hyperbolic spaces, which can be thought of as model spaces of constant curvature, do not admit globally defined vector fields ξ associated to a homogeneous class S ∈ S1 , that is, if such a vector field ξ exists, it necessarily has a singular locus that prevents the full application of the Ambrose–Singer theorem when all the conditions are considered. In fact, we have the following result. Proposition 5.1.4 ([65, Theorem 3.4]) Let (M, g) be a connected, simply-connected and complete pseudo-Riemannian manifold of signature (k, m − k). Then (M, g) admits a non-degenerate homogeneous structure S ∈ S1 if and only if (M, g), up to a change of sign of g, is a Riemannian manifold (that is, k = m or k = 0) of constant negative curvature. With respect to degenerate homogeneous structures, the panorama becomes more interesting. Proposition 5.1.5 ([93, 94]) Let (M, g) be a connected pseudo-Riemannian manifold of dimension m + 2 admitting a degenerate homogeneous pseudo-Riemannian structure S ∈ S1 . Then (M, g) is locally isometric to Rm+2 with coordinates (x 1 , ..., x m , u, v) and metric g = dudv +

Bab a b 2 m x b du + a=1 a (d x a )2 , u2

(5.1.1)

where the signature of the metric is (1 , . . . , m ) and Bab is a symmetric matrix. Equation (5.1.1) corresponds to the geometric model of a so-called singular scaleinvariant plane wave. The notion of plane waves originates from Einstein equations in

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5 Homogeneous Structures of Linear Type

cosmology, and the proposition above means that degenerate homogeneous structures of linear type are locally characterized by this special class of waves. Conversely, it is easy to check that a singular scale-invariant plane wave admits a degenerate homogeneous structure with ξ = − u1 ∂v . Note again that the structure is not globally defined as it has a singularity in the locus u = 0.

5.2 The Pseudo- and Para-Kähler Cases Since many features in the geometry of pseudo-Kähler and para-Kähler manifolds are very similar, it is very convenient to develop the arguments and the results simultaneously. For this reason, we unify the treatment of these geometries through the notion of -Kähler manifolds. Definition 5.2.1 Let (M, g) be a pseudo-Riemannian manifold. 1. An almost -Hermitian structure on (M, g) is a smooth section J of so(T M) such that J 2 = Id. 2. (M, g) is called -Kähler if it admits a parallel almost -Hermitian structure J with respect to the Levi-Civita connection. In this way, one recovers the corresponding formula or result in the pseudoKähler and the para-Kähler cases by substituting  = −1 and  = 1 respectively. In particular, we can write a homogeneous -Kähler structure of linear type as S X Y = g(X, Y )ξ − g(ξ, Y )X + g(X, J Y )J ξ − g(ξ, J Y )J X − 2g(ζ, J X )J Y, (5.2.2) for some vector fields ξ and ζ. The notions of degenerate and non-degenerate structures remain the same (Definitions 4.2.4 and 4.2.2). We shall also use the terms -complex and -holomorphic, which include the complex and para-complex cases in the obvious way. In addition C will denote the complex and the para-complex numbers for  = ±1 respectively, where i  will stand for the corresponding imaginary unit.

5.2.1 The Non-degenerate Case Lemma 5.2.2 Let (M, g, J ) be a connected -Kähler manifold of dimension 2n  4 admitting a non-degenerate homogeneous -Kähler structure of linear type. Then (M, g, J ) is Einstein.  R = 0 reads Proof Equation ∇ (∇ X R)Y Z W U = −R SX Y Z W U − RY SX Z W U − RY Z SX W U − RY Z W SX U ,

(5.2.3)

5.2 The Pseudo- and Para-Kähler Cases

137

so applying the second Bianchi identity and substituting (5.2.2) we have  2g(X, ξ)RY Z W U + g(X, W )RY Z ξU + g(X, U )RY Z W ξ 0=

S

XYZ

 + 2g(X, J Y )R J ξ Z W U + g(X, J W )RY Z J ξU + g(X, J U )RY Z W J ξ . √ Since g(ξ, ξ) = 0 we can choose an orthonormal basis including ξ/ |g(ξ, ξ)|. Contracting the previous formula with respect to X and W and applying the first Bianchi identity we obtain (2n + 2)R Z Y ξU = −2g(Y, ξ)(Z , U ) + 2g(Z , ξ)(Y, U ) − 2g(Y, J Z )(J ξ, U ) − g(Y, U )(Z , ξ) − g(Y, J U )(Z , J ξ) + g(Z , U )(Y, ξ) + g(Z , J U )(Y, J ξ),

(5.2.4)

where  is the Ricci curvature tensor (Eq. (1.2.2)). In particular, by contracting (5.2.4) with respect to Y and U with the same orthonormal basis as before, we get (Z , ξ) = (τ /2n)g(Z , ξ), where τ denotes the scalar curvature (see again Eq. (1.2.2)). Setting a = 1/(2n + 2) and ν = τ /2n, we can write 1 RξU = 2θ ∧ Ric(U ) − 2νθ(J U )ω + bU  ∧ θ + ν(J U ) ∧ (θ ◦ J ), a

(5.2.5)

where RξU = RξU ξU , Ric is the Ricci operator associated to  through g and ω is the symplectic form associated to g and J . Using the identity RW U J ξ· = Rξ J W U · − Rξ J U W · we can write (5.2.4) as 0 = 2θ ∧ RW U + W  ∧ RξU − U  ∧ RξW − 2ω ∧ (Rξ J U W − Rξ J W U ) 

(5.2.6)



− (J W ) ∧ Rξ J U + (J U ) ∧ Rξ J W . Denoting the right-hand side of (5.2.5) by (U ) and substituting into (5.2.6) we obtain 2 0 = θ ∧ RW U + W  ∧ (U ) − U  ∧ (W ) a − 2ω ∧ (i W (J U ) − iU (J W )) − (J W ) ∧ (J U ) + (J U ) ∧ (J W ). Taking W = ξ, the previous formula transforms into 0 = (2g(ξ, ξ)ω + θ ∧ (θ ◦ J )) ∧ (Ric(J U ) − ν(J U ) ), and contracting first with ξ and then with J ξ we obtain g(ξ, ξ)(Ric(J U ) − ν(J U ) ) = 0, from which, since g(ξ, ξ) = 0, we conclude that the manifold is Einstein.



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5 Homogeneous Structures of Linear Type

Theorem 5.2.3 Let (M, g, J ) be a connected -Kähler manifold of dimension 2n  4 admitting a non-degenerate homogeneous -Kähler structure S of linear type. Then (M, g, J ) has constant -holomorphic sectional curvature c = −4g(ξ, ξ) and ζ = 0. Proof Since by the previous Lemma (M, g, J ) is Einstein, formula (5.2.4) transforms into RY Z ξW = c RY0 Z ξW , where c = s/(4n(n + 1)) and R 0 is the curvature of a manifold with constant holomorphic sectional curvature equal to 4, i.e., R 0X Y Z W = g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) + g(X, J Z )g(Y, J W ) − g(x, J W )g(Y, J Z ) + 2g(X, J Y )g(Z , J W ). This implies that R X J X ξ = c {−2g(J X, ξ)X + 2g(X, ξ)J X − 2g(X, X )J ξ} .

(5.2.7)

 = 0 is equivalent to ∇ X ξ = S X ξ. Using this in On the other hand, ∇ξ R X J X ξ = ∇[X,J X ] ξ − ∇ X ∇ J X ξ + ∇ J X ∇ X ξ, we get

ζ

R X J X ξ = −g(ξ, ξ)R 0X J X ξ +  X J X ξ,

(5.2.8)

where ζ

 X Y ξ = 2g(X, J ζ) {g(Y, J ξ)ξ + g(ξ, ξ)J Y + 2g(ζ, Y )J ξ} − 2g(Y, J ζ) {g(X, J ξ)ξ + g(ξ, ξ)J X + 2g(X, ζ)J ξ} + 2 {g(Y, ζ)g(ξ, J X ) − g(X, ζ)g(ξ, J Y ) + 2g(X, J Y )g(ξ, ζ)} J ξ. Taking Y = X ∈ Span{ζ, J ζ}⊥ and comparing formulas (5.2.7) and (5.2.8), we have ζ that c = −g(ξ, ξ) and g(ξ, ζ) = 0. In addition, this implies that  X J X ξ = 0, whence 2g(ξ, ξ)g(X, ζ) = 0. This together with g(ξ, ζ) = 0 gives ζ = 0. Let now A = R + g(ξ, ξ)R 0 . A direct computation from (5.2.3) gives (∇ X R)Y Z W U = g(Y, ξ)A X Z W U + g(Z , ξ)AY X W U + g(W, ξ)AY Z XU + g(U, ξ)AY Z W X − g(J Y, ξ)A J X Z W U − g(J Z , ξ)AY J X W U − g(J W, ξ)AY Z J XU − g(J U, ξ)AY Z W J X . Since A satisfies the first Bianchi identity, taking a cyclic sum in X, Y, Z we obtain 0 = −2

S XYZ

g(X, ξ)AY Z W U ,

5.2 The Pseudo- and Para-Kähler Cases

139

which is equivalent to θ ∧ A W U = 0. Contracting with ξ and taking into account that AY Z ξW = 0, we have that 0 = g(ξ, ξ)A W U , hence A W U = 0. This proves that (M, g, J ) has constant -holomorphic sectional curvature −4g(ξ, ξ).

Remark 5.2.4 For  = −1, if g(ξ, ξ) > 0 then c = −4g(ξ, ξ) < 0, so that spaces with negative definite metric and constant negative holomorphic sectional curvature cannot admit non-degenerate homogeneous pseudo-Kähler structures of linear type. Similarly, if g(ξ, ξ) < 0 then c > 0, so that spaces with positive definite metric and constant positive holomorphic sectional curvature are also excluded.

5.2.2 The Degenerate Case  R = 0 reads Equation ∇ (∇ X R)Y Z W U = −R SX Y Z W U − RY SX Z W U − RY Z SX W U − RY Z W SX U ,

(5.2.9)

so applying the second Bianchi identity and substituting (5.2.2) we have 0=

S



2g(X, ξ)RY Z W U + g(X, W )RY Z ξU + g(X, U )RY Z W ξ

(5.2.10)

XYZ

 +2g(X, J Y )R J ξ Z W U + g(X, J W )RY Z J ξU + g(X, J U )RY Z W J ξ . Since g(ξ, ξ) = 0, there exists an orthonormal basis {ek } with g(e1 , e1 ) = 1, g(e2 , e2 ) = −1, and ξ = g(ξ, e1 )(e1 + e2 ). Whence, contracting the previous formula with respect to X and W and applying first Bianchi identity, we obtain (2n + 2)R Z Y ξU = −2g(Y, ξ)(Z , U ) + 2g(Z , ξ)(Y, U )

(5.2.11)

−2g(Y, J Z )(J ξ, U ) − g(Y, U )(Z , ξ) −g(Y, J U )(Z , J ξ) + g(Z , U )(Y, ξ) +g(Z , J U )(Y, J ξ). With the same orthonormal basis, contracting the previous expression with respect to Y and U we arrive at (Z , ξ) = (τ /2n)g(Z , ξ). Setting a = 1/(2n + 2) and ν = s/2n, we can write 1 RξU = 2θ ∧ Ric(U ) − 2νθ(J U )ω + νU  ∧ θ − ν(J U ) ∧ (θ ◦ J ), (5.2.12) a

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5 Homogeneous Structures of Linear Type

where ω denotes the symplectic form associated to (g, J ). From the first Bianchi identity we have RW U J ξ· = Rξ J W U · − Rξ J U W · so we can write (5.2.11) as 0 = 2θ ∧ RW U + W  ∧ RξU − U  ∧ RξW

(5.2.13)

−2ω ∧ (Rξ J U W − Rξ J W U ) −(J W ) ∧ Rξ J U + (J U ) ∧ Rξ J W . Denoting by (U ) the right-hand side of (5.2.12) and substituting into (5.2.13) we obtain 0=

2 θ ∧ RW U + W  ∧ (U ) − U  ∧ (W ) a −2ω ∧ (i W (J U ) − iU (J W )) −(J W ) ∧ (J U ) + (J U ) ∧ (J W ).

Then, taking W = ξ in the previous formula 0 = (θ ∧ (θ ◦ J )) ∧ (Ric(J U ) − ν(J U ) ). Now, since U is arbitrary, defining α = Ric − νg, one has θ ∧ (θ ◦ J ) ∧ α(X ) = 0, for any vector field X . This implies that α = λθ + μθ ◦ J, for some 1-forms λ and μ. Note that since (M, g, J ) is -Kähler, α = Ric − νg is symmetric and of type (1, 1). Imposing this on the right-hand side of the previous equation we have that λ = f θ, μ = − f (θ ◦ J ), for some function f , so that we obtain Ric = νg + f (θ ⊗ θ − (θ ◦ J ) ⊗ (θ ◦ J )) .

(5.2.14)

Substituting (5.2.14) into (5.2.12), we find 1 0 RξU = ν RξU + PξU , a

(5.2.15)

where again R 0X Y Z W = g(Y, Z )g(X, W ) − g(X, Z )g(Y, W ) + g(X, J Z )g(Y, J W ) − g(x, J W )g(Y, J Z ) + 2g(X, J Y )g(Z , J W )

5.2 The Pseudo- and Para-Kähler Cases

141

and PξU = −2 f θ(J U )θ ∧ (θ ◦ J ). On the other hand, from ∇ξ = S · ξ and (5.2.2), formula R X Y Z = ∇[X,Y ] Z − [∇ X , ∇Y ]Z gives

ζ

ζ

R X Y ξ = −g(ξ, ξ)R 0X Y ξ +  X Y ξ =  X Y ξ,

(5.2.16)

where ζ

 X Y ξ = − 2g(ζ, J Y )g(X, J ξ)ξ + 2g(ζ, Y )g(X, J ξ)J ξ − 4g(ζ, J Y )g(X, ξ)J ξ + 2g(ζ, J X )g(Y, J ξ)ξ − 2g(ζ, X )g(Y, J ξ)J ξ + 4g(ζ, J X )g(Y, ξ)J ξ + 4g(ξ, ζ)g(Y, J X )J ξ − 4g(ζ, J Y )g(ξ, X )J ξ + 4g(ζ, J X )g(ζ, Y )J ξ.

Taking Y = J X and comparing (5.2.15) and (5.2.16), one finds 2abg(ξ, J X ) = 0,

2abg(ξ, X ) = 0,

for every X , so that ν = 0. Hence, the scalar curvature vanishes. We now choose a basis {ξ, J ξ, q1 , J q1 , X i , J X i } of T p M for every p ∈ M, where g(ξ, q1 ) = 1, g(q1 , q1 ) = 0, and {X i , J X i } is an orthonormal basis of Span{ξ, J ξ, q1 , J q1 }⊥ . Comparing again (5.2.15) and (5.2.16) for X = ξ and Y = J q1 , and for X = J ξ and Y = J q1 we obtain that g(ζ, J ξ) = 0 and g(ζ, ξ) = 0, so that ζ ∈ Span{ξ, J ξ}⊥ . Taking X = X i and Y = J q1 , and X = J X i and Y = J q1 , we also have g(ζ, J X i ) = 0 and g(ζ, X i ) = 0 respectively, so that ζ ∈ Span{ξ, J ξ}. Finally, writing ζ = λξ + μJ ξ for some functions λ and μ, and taking X = q1 and Y = J q1 one finds g(ζ, J q1 ) = 0 and 2a f = −2λ − 4λ2 , so that ζ = λξ,

1 f = − λ( + 2λ). a

 = 0 and ∇ζ  = 0 imply that λ must be constant. This agrees Note that equations ∇ξ with the fact that the Ricci form ρ = f θ ∧ (θ ◦ J ) is closed, as (M, g, J ) is -Kähler. Thus, we have proved the following.

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5 Homogeneous Structures of Linear Type

Proposition 5.2.5 Let (M, g, J ) be an -Kähler manifold admitting a degenerate -Kähler homogeneous structure of linear type given by (5.2.2). Then ζ = λξ for some λ ∈ R and the Ricci curvature is 1  = − λ( + 2λ) (θ ⊗ θ − (θ ◦ J ) ⊗ (θ ◦ J )) , a where a = 1/(dimM + 2) and θ = ξ  . In particular, the scalar curvature vanishes. Next, since ν = 0, formula (5.2.15) becomes R Z Y ξU = a PZ Y ξU = −2a f (θ ∧ (θ ◦ J ) ⊗ (θ ◦ J ))(Z , Y, U ). Looking again at formula (5.2.10), we obtain −

S XYZ

=

S

2g(X, ξ)RY Z W U

(5.2.17)

 2a f (θ ∧ (θ ◦ J )) ⊗ (X  ∧ (θ ◦ J ))(Y, Z , W, U )

XYZ

+ (θ ∧ (θ ◦ J )) ⊗ (J X  ∧ (θ))(Y, Z , W, U ) − 2g(X, J Y )θ ⊗ (θ ∧ (θ ◦ J ))(Z , W, U )} . Substituting this into (5.2.9) and after quite a long computation we get   1 ∇ X R = 4θ(X ) ⊗ (R − ag  ) − 2a (X  ∧ (θ ◦ J )) ρ + (J X  ∧ (θ)) ρ , 2

(5.2.18) where ρ is the Ricci form and  stands for the -complex Kulkarni–Nomizu product (h  k)(X 1 , X 2 , X 3 , X 4 ), defined as h(X 1 , X 3 )k(X 2 , X 4 ) + h(X 2 , X 4 )k(X 1 , X 3 ) −h(X 1 , X 4 )k(X 2 , X 3 ) − h(X 2 , X 3 )k(X 1 , X 4 ) −h(X 1 , J X 3 )k(X 2 , J X 4 ) − h(X 2 , J X 4 )k(X 1 , J X 3 ) +h(X 1 , J X 4 )k(X 2 , J X 3 ) + h(X 2 , J X 3 )k(X 1 , J X 4 ) −2h(X 1 , J X 2 )k(X 3 , J X 4 ) − 2h(X 3 , J X 4 )k(X 1 , J X 2 ), for h and k symmetric (0, 2)-tensors. With the help of (5.2.18) we now compute some terms of the curvature tensor of g. We again choose a basis {ξ, J ξ, q1 , J q1 , X i , J X i }

5.2 The Pseudo- and Para-Kähler Cases

143

of T p M for every p ∈ M. Taking the symmetric sum with respect to X, Y, Z in (5.2.18), we have 0 = 4θ(X ) (RY Z W U − 2ag  Y Z W U )   − 2a (X  ∧ (θ ◦ J )) ρ + (J X  ∧ (θ)) ρ (Y, Z , W, U )   − 2a (Y  ∧ (θ ◦ J )) ρ + (J Y  ∧ (θ)) ρ (Z , X, W, U )   − 2a (Z  ∧ (θ ◦ J )) ρ + (J Z  ∧ (θ)) ρ (X, Y, W, U ). Setting Y, Z ∈ Span {ξ, J ξ}⊥ we obtain RY Z W U = −8ag(Y, J Z )ρ(W, U ),

Y, Z ∈ Span{ξ, J ξ}⊥

(5.2.19)

for every W, U . On the other hand, setting X = q1 , Y = J q1 and Z ∈ Span X i , J X i , we find RY Z W U = a f (g(Z , W )θ(J U ) − g(Z , U )θ(J W ) − g(Z , J W )θ(U ) +g(Z , J U )θ(W )) , for every W, U , so that Rq1 Z W U = a f (g(J Z , U )θ(J W ) − g(J Z , W )θ(J U ) +g(Z , U )θ(W ) − g(Z , W )θ(U ))

(5.2.20)

for Z ∈ Span{X i , J X i } and all W, U . Proposition 5.2.6 (M, g, J ) is Ricci-flat. Proof Let g(q1 , q1 ) = b and suppose for the sake of simplicity that b > 0 (the case b < 0 is analogous). Defining q2 = J q1 , we choose an orthonormal basis   √ q1 √ q2 q1 q2 , b Jξ − , √ , √ , Xi , J Xi b ξ− b b b b of T p M for every p ∈ M, which has signature (−1, , 1, −, εi , −εi ) where g(X i , X i ) = εi ∈ {±1}. We compute the Ricci curvature by contracting the curvature tensor with respect to this orthonormal basis and using (5.2.19) and (5.2.20): √ √ q1

q1

, U, b ξ − (W, U ) = − R W, b ξ − b

b

√ √ q2 q2 + R W, b ξ − , U, b ξ − b b



q1 q2 q1 q2 + R W, √ , U, √ − R W, √ , U, √ b b b b i i + ε R(W, X i , U, X i ) − ε R(W, J X i , U, J X i )

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5 Homogeneous Structures of Linear Type

= 4a f (θ ⊗ θ − (θ ◦ J ) ⊗ (θ ◦ J ))  + 2a f  εi (θ ⊗ θ − (θ ◦ J ) ⊗ (θ ◦ J )) i

= (4a + 2a



εi )(W, U ).

i

We deduce that if (W, U ) = 0 then 4a + 2a

 i

dimM + 2 = 4 + 2a

εi = 1. Therefore,



εi ,

i

whence dimM = 2 + 2a



εi < dimM.

i

Since this is impossible we conclude that  = 0.



Corollary 5.2.7 The only possible values for λ are λ = 0 and λ = − 2 . In the next section we shall study the cases λ = 0 and λ = − 2 separately. Proposition 5.2.8 The curvature tensor of g is given by R = k(θ ∧ (θ ◦ J )) ⊗ (θ ∧ (θ ◦ J )), for some function k. Moreover, if k = 0 the holonomy algebra of g is given by ⎞ i i 0 hol ∼ = R ⎝ −i  −i  0 ⎠ , 0 0 0n ⎛

which is a one-dimensional subalgebra of su(1, 1) ⊂ su( p, q), p + q = n + 2, for  = −1, and sl(2, R) ⊂ sl(n + 2, R) for  = 1. Proof Since (M, g, J ) is Ricci-flat, (5.2.18) becomes ∇ R = 4θ ⊗ R. Taking a symmetric sum in the previous formula and applying the second Bianchi identity, we have that θ ∧ RW U = 0 for every W, U . But from the -Kähler symmetries of R we also have (θ ◦ J ) ∧ RW U = 0. These force the curvature to be of the form R = k(θ ∧ (θ ◦ J )) ⊗ (θ ∧ (θ ◦ J )), for some function k.

5.2 The Pseudo- and Para-Kähler Cases

145

On the other hand, since (M, g, J ) is real analytic, the infinitesimal holonomy algebra coincides with the holonomy algebra (see [79, Ch.  II]). Recall that the ∞ ml , where infinitesimal holonomy algebra at p ∈ M is defined as hol = l=0 m0 = Span{R X Y / X, Y ∈ T p M} and   ml = Span ml−1 ∪ {(∇ Z l . . . ∇ Z 1 R) X Y / Z 1 , . . . , Z l , X, Y ∈ T p M} . As a simple computation shows, one has ∇θ = θ ⊗ θ + (2λ + )(θ ◦ J ) ⊗ (θ ◦ J ). It is easy to see that this, together with the recurrence formula ∇ R = 4θ ⊗ R, implies that m0 = m 1 = . . . = ml for every l ∈ N, so that hol = m0 . Now, since R = k(θ ∧ (θ ◦ J )) ⊗ (θ ∧ (θ ◦ J )), the space m0 is the one-dimensional space generated by the endomorphism A: Tp M → Tp M ξ, J ξ → 0 q1  → J ξ q2 → ξ X i , J X i → 0. This endomorphism is expressed as ⎞ ⎛ i i 0 1⎝   −i  −i  0 ⎠ b 0 0 0n with respect to the -complex orthonormal basis 



√ √ 1 1 1 √ (q1 + i  q2 ), √ q1 − s |b|ξ + i  √ q2 − s |b|J ξ , |b| |b| |b| 

X i + i  J X i , where g(q1 , q1 ) = b and s is the sign of b.



As a consequence of Proposition 5.2.8 we have that for  = ±1 and λ = 0, − 2 , (M, g, J ) is an Osserman manifold with a 2-step nilpotent Jacobi operator. It is also easy to see that (M, g, J ) is a manifold such that all curvature polynomials invariant under the group of diffeomorphisms vanish (VSI manifold for short, cf. [105]). Finally, it is worth noting that making use of Theorem 1.2.15, if (M, g, J ) is connected and simply-connected, then it is the product of a 2n-dimensional -complex

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flat and totally geodesic manifold, and a 4-dimensional Walker -Kähler manifold with a parallel null -complex vector field. The similarities between this kind of manifold and the structure of plane waves will be explored in detail in Sect. 5.4.

5.2.3 Local Form of the Metrics We have seen (Propositions 5.2.5 and 5.2.6) that an -Kähler manifold (M, g, J ) admitting a degenerate homogeneous -Kähler structure of linear type is Ricci-flat and satisfies ζ = λξ for some constant λ ∈ R. As stated in Corollary 5.2.7, this implies that the only possible values for λ are λ = 0 and λ = − 2 . Hereafter M is assumed to be non-flat and of dimension 2n + 4. The case λ = − 2 : Substituting the value λ = − 2 into (5.2.2) we have S X Y = g(X, Y )ξ − g(ξ, Y )X + g(X, J Y )J ξ − g(ξ, J Y )J X + g(ξ, J X )J Y.  = 0 then implies Condition ∇ξ

∇ξ = θ ⊗ ξ,

which gives ∇θ = θ ⊗ θ,

∇(θ ◦ J ) = θ ⊗ (θ ◦ J ).

In particular dθ = 0, so that fixing a point p ∈ M there is a neighbourhood U and a function v : U → R such that θ = dv. We consider w1 = e−v , whence dw1 = −e−v dv = −w1 θ. We now take dw1 ◦ J = −w1 (θ ◦ J ). Differentiating we obtain d(dw1 ◦ J ) = −dw1 ∧ (θ ◦ J ) − w1 d(θ ◦ J ) = w1 θ ∧ (θ ◦ J ) − w1 θ ∧ (θ ◦ J ) = 0.

Therefore, reducing U if necessary, there is a function w2 : U → R such that dw2 = dw1 ◦ J . We consider the function w = w1 + i  w2 . Then dw = dw1 + i  (dw1 ◦ J ), so that dw is of type (1, 0) with respect to J and w : U → C is -holomorphic. In addition,

5.2 The Pseudo- and Para-Kähler Cases

147

∇dw = −dw1 ⊗ θ − w1 ∇θ − i  dw1 ⊗ (θ ◦ J ) − i  w1 ∇(θ ◦ J ) = 0, i.e., dw is a nowhere vanishing parallel 1-form. The function w : U → C defines a foliation of U by -complex hypersurfaces Hτ = w −1 (τ ), τ ∈ C (for those τ with w −1 (τ ) non-empty). Note that since the tangent space to Hτ is given by the kernel of dw, the hypersurfaces Hτ are tangent to the distribution Span{ξ, J ξ}⊥ . We consider the vector field 

Z = grad(w1 ) = dw1 . It is easy to see that by construction J Z = −grad(w2 ). These vector fields are written as Z = −w1 ξ,

J Z = −w1 J ξ,

so that ∇ Z = −dw1 ⊗ ξ − w1 ∇ξ = w1 θ ⊗ ξ − w1 θ ⊗ ξ = 0, and thus also ∇ J Z = 0. This implies in particular that Z and J Z are commuting -holomorphic Killing vector fields. We now look at the holonomy of g at p, which was computed in Proposition 5.2.8. Using the same notation as before, we consider the subspace E = Span{ξ, J ξ, q1 , q2 } ⊂ T p M. This subspace is invariant under the holonomy action and so is E ⊥ . In fact, the holonomy action on E ⊥ is trivial. This implies that, using the parallel transport with respect to ∇, we can extend an orthonormal basis {(X a )| p , (J X a )| p , a = 1, . . . , n} of E ⊥ to an orthonormal reference {X a , J X a , a = 1, . . . , n} on U, such that ∇ X a = 0 = ∇ J X a , a = 1, . . . , n. In particular, they are commuting -holomorphic Killing vector fields. In addition, for any smooth curve γ on U, we have       d   dw(X a )γ(t) = ∇γ(t) ˙ dw (X a ) + dw ∇γ(t) ˙ X a = 0,  dt t=0 whence the functions dw(X a ) are constant along γ and take the value 0 at p. This implies that X a and thus J X a are tangent to the foliation Hτ . Finally, note that since they are parallel, X a and J X a commute with Z and J Z . We have thus constructed a set of commuting para-holomorphic Killing vector fields {Z , J Z , X a , J X a } tangent to Hτ . Therefore, reducing U if necessary, we can take -complex coordinates {w, z, z a } on U such that ∂z = 21 (Z + i  J Z ), ∂z a = 21 (X a + i  J X a ). Note that since the distributions Span{∂w , ∂z } and Span{∂z a , a = 1, . . . , n} are invariant under holonomy, the vector fields X a and J X a are orthogonal to Span{∂w , ∂z }.

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5 Homogeneous Structures of Linear Type

We write z = z 1 + i  z 2 , z a = x a + i  y a and w = w 1 + i  w 2 , and rearrange the coordinates as {z 1 , z 2 , w 1 , w 2 , x a , y a }. The metric with respect to these coordinates is then given by ⎞ ⎛ 0 0 1 0 0 ... 0 ⎜0 0 0 − 0 . . . 0⎟ ⎟ ⎜ ⎜1 0 b 0 0 . . . 0 ⎟ ⎟ ⎜ ⎟ ⎜ (5.2.21) g = ⎜0 − 0 −b 0 . . . 0⎟ , ⎟ ⎜0 0 0 0 ⎟ ⎜ ⎟ ⎜ .. .. .. .. ⎝. . . .  ⎠ 0 0 0 0 for some function b, where we put a

ε 0  = diag( , a = 1, . . . , n), 0 −εa with εa = g(X a , X a ) ∈ {±1}. In addition, the -complex structure reads ⎛ 0 ⎜1 0 ⎜ ⎜ .. J =⎜ . ⎜ ⎝ 0 1

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠ 0

(5.2.22)

Imposing that ∂z 1 , ∂z 2 , ∂x a and ∂ ya are parallel, it is easy to see that b does not depend on z 1 , z 2 , x a , y a . Finally, computing the curvature tensor with respect to those coordinates we obtain R=

1   b(dw 1 ∧ dw 2 ) ⊗ (dw 1 ∧ dw 2 ), 2

where we put  = −

∂2 ∂2 + . 1 2 ∂(w ) ∂(w 2 )2

Defining F =  b and taking into account that dw 1 and dw 2 are parallel, we have that 1 ∇ R = d F ⊗ (dw 1 ∧ dw 2 ) ⊗ (dw 1 ∧ dw 2 ). 2 Recall that formula (5.2.18) together with the Ricci-flatness of g gave that ∇ R = 4θ ⊗ R.

5.2 The Pseudo- and Para-Kähler Cases

149

Comparing these two formulas for ∇ R we have that d F = 4Fθ, where θ can be written as θ=−

1 dw 1 . w1

Note that by construction w1 = 0. The system of partial differential equations is thus ⎧ 4 ∂F ⎪ ⎨ = − F, 1 ∂w w1 ⎪ ⎩ ∂ F = 0, ∂w 2 which has solution F=

R0 , (w 1 )4

for some constant R0 ∈ R. We have thus proved the following. Proposition 5.2.9 Let (M, g, J ) be an -Kähler manifold of dimension 2n + 4, n ≥ 0, admitting a degenerate homogeneous -Kähler structure of linear type with ζ = − 2 ξ. Then, each p ∈ M has a neighbourhood -holomorphically isometric to an open subset of (C )n+2 with the -Kähler metric g = dw 1 dz 1 − dw 2 dz 2 + b(dw 1 dw 1 − dw 2 dw 2 ) +

n 

εa (d x a d x a − dy a dy a ),

a=1

(5.2.23) where εa = ±1, and the function b = b(w 1 , w 2 ) only depends on the coordinates {w 1 , w 2 } and satisfies R0  b = (w 1 )4 for R0 ∈ R − {0}. The strongly degenerate case λ = 0: Substituting the value λ = 0 into (5.2.2) we have that the homogeneous structure S takes the form S X Y = g(X, Y )ξ − g(ξ, Y )X + g(X, J Y )J ξ − g(ξ, J Y )J X.  = 0 then implies Condition ∇ξ

∇ξ = θ ⊗ ξ,

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5 Homogeneous Structures of Linear Type

which gives ∇θ = θ ⊗ θ − (θ ◦ J ) ⊗ (θ ◦ J ), ∇(θ ◦ J ) = θ ⊗ (θ ◦ J ) − (θ ◦ J ) ⊗ θ. We consider the -complex form α = θ + i  (θ ◦ J ), which is of type (1, 0) with respect to the -complex structure J . As a straightforward computation shows, ∇α = α ⊗ α, so that dα = 0. This implies in particular that α is an -holomorphic 1-form. Fixing a point p ∈ M, by the closeness of α, there is a neighbourhood U of p and an -holomorphic function v : U → C such that α = dv. We consider the -holomorphic function w = e−v , where the exponential must read e x+i y = e x (cos y + i  sin y) for  = −1 and e x+i y = e x (cosh y + i  sinh y) for  = 1. Differentiating, we obtain that dw = −wα, so that ∇dw = 0. This means that dw is a nowhere vanishing parallel -holomorphic 1-form on U. The function w : U → C defines a foliation of U by -complex hypersurfaces Hτ = w −1 (τ ), τ ∈ C (for those τ with w −1 (τ ) non empty). Note that since the tangent space to Hτ is given by the kernel of dw, the hypersurfaces Hτ are tangent to the distribution Span{ξ, J ξ}⊥ . Writing w = w 1 + i  w 2 , we take the vector fields J Z = −(dw 2 ) . Z = (dw 1 ) , These vector fields are obviously tangent to the foliation given by Hτ , and since ∇dw = 0 we have ∇ Z = 0 and ∇ J Z = 0. This implies in particular that Z , J Z are commuting -holomorphic Killing vector fields. Making use of Proposition 5.2.8, and by the same arguments used for the case λ = −/2, we take coordinates {w, z, z a }, a = 1, . . . , n, such that ∂z a = 21 (Z + i  J Z ) and ∇∂z a = 0. Writing z = z 1 + i  z 2 , z a = x a + i  y a , and w = w 1 + i  w 2 , with respect to real coordinates {z 1 , z 2 , w 1 , w 2 , x a , y a } the metric g and the complex structure J take the form (5.2.21) and (5.2.22) respectively, where b does not depend on z 1 , z 2 , x a , y a . As a straightforward computation shows, R=

1   b(dw 1 ∧ dw 2 ) ⊗ (dw 1 ∧ dw 2 ), 2

and θ=

(w 1 )2

−1 (w 1 dw 1 − w2 dw 2 ). − (w 2 )2

5.2 The Pseudo- and Para-Kähler Cases

151

Finally, imposing ∇ R = 4θ ⊗ R and defining F =  b, we obtain the system of partial differential equations ⎧ ∂F −4w 1 ⎪ ⎪ ⎨ = F, ∂w 1 (w 1 )2 − (w 2 )2 2 ∂F 4w ⎪ ⎪ ⎩ = F, 2 1 2 ∂w (w ) − (w 2 )2 which has solution F=

R0 (w 1 )2

− (w 2 )2

2 ,

for some constant R0 ∈ R. Note that since w = e−v we always have (w 1 )2 − (w 2 )2 = 0. We have thus proved the following. Proposition 5.2.10 Let (M, g, J ) be an -Kähler manifold of dimension 2n + 4, n ≥ 0, admitting a strongly degenerate homogeneous -Kähler structure of linear type S. Then each p ∈ M has a neighbourhood -holomorphically isometric to an open subset of (C )n+2 with the -Kähler metric g = dw 1 dz 1 − dw 2 dz 2 + b(dw 1 dw 1 − dw 2 dw 2 ) +

n 

εa (d x a d x a − dy a dy a ),

a=1

(5.2.24) where εa = ±1, and the function b only depends on the coordinates {w1 , w 2 } and satisfies R0  b = ((w 1 )2 − (w 2 )2 )2 for R0 ∈ R − {0}. The manifold ((C )n+2 , g) Propositions 5.2.10 and 5.2.9 give the local forms (5.2.24) and (5.2.23) of the metric of a manifold with a degenerate homogeneous -Kähler structure of linear type. This motivates the study of the space (C )2+n endowed with this particular -Kähler metric, which can thus be understood as the simplest instance of this type of manifold. In particular, the goal is to study the singular nature of this space. We shall restrict ourselves to the Lorentz -Kähler case, i.e., the case of metrics of index 2. Throughout this section, wλ must be understood as  w2λ =

w12 − w22 for λ = 0, for λ = −/2. w12

(5.2.25)

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5 Homogeneous Structures of Linear Type

In addition,  shall stand for the differential operator  = −

∂2 ∂2 + . 2 ∂w1 ∂w22

We thus consider the manifold (C )2+n = (R2n+4 , J0 ), where J0 is the standard -complex structure, with real coordinates {z 1 , z 2 , w 1 , w 2 , x a , y a }, endowed with the metric g = dw1 dz 1 − dw 2 dz 2 + b(dw 1 dw 1 − dw 2 dw 2 ) +

n 

(d x a d x a − dy a dy a ),

a=1

where b is a function of the variables (w1 , w 2 ) satisfying  b =

R0 , w4λ

R0 ∈ R − {0}.

(5.2.26)

(5.2.27)

As computed before, the curvature (1, 3)-tensor field of g is R=

 1 R0  (dw 1 ∧ dw 2 ) ⊗ (dw 1 ⊗ ∂z 2 ) + (dw 1 ∧ dw 2 ) ⊗ (dw 2 ⊗ ∂z 1 ) . 4 2 wλ

Since R0 = 0, the curvature exhibits a singular behavior at S = {wλ = 0}. This set can be understood as a singularity of g in the cosmological sense: the j geodesic deviation equation is governed by the components Rwz 1 w2 wi , i, j = 1, 2, of the curvature tensor field, making the tidal forces infinite at S. Indeed, we can compute a component of the curvature tensor with respect to an orthonormal parallel frame along a geodesic reaching the singular set in finite time, and see that it is singular (see [110]). Let γ be the geodesic with initial value γ(0) = (0, 0, 1, 0, . . . , 0) and γ˙ = (0, 0, −1, 0, . . . , 0). It is easy to see that this geodesic is of the form γ(t) = (z 1 (t), z 2 (t), 1 − t, 0, x a (t), y a (t)), for some functions z 1 (t), z 2 (t), x a (t), y a (t), a = 1, . . . , n, and reaches the singular set S at t = 1. Let E(t) = W 1 (t)∂w1 + W 2 (t)∂w2 + Z 1 (t)∂z 1 + Z 2 (t)∂z 2 + X a (t)∂x a + Y a (t)∂ y a be a vector field along γ. E is parallel if the following equations hold: W˙ 1 = 0, 1 1 Z˙ 1 − W 1 wz 1 w1 − W 2 wz 1 w2 = 0, X˙ a = 0,

W˙ 2 = 0, 2 2 Z˙ 2 − W 1 wz 1 w1 − W 2 wz 1 w2 = 0, Y˙ a = 0.

5.2 The Pseudo- and Para-Kähler Cases

153

We can thus obtain an orthonormal parallel frame {E 1 (t), . . . , E 4+2n (t)} with E 1 (t) and E 2 (t) of the form 1 E 1 (t) = √ ∂w1 + Z 11 (t)∂z 1 + Z 12 (t)∂z 2 + X 1a ∂x a + Y1a ∂ y a , |b(0)| 1 E 2 (t) = √ ∂w2 + Z 21 (t)∂z 1 + Z 22 (t)∂z 2 + X 2a ∂x a + Y2a ∂ y a , |b(0)| 1 1 ∂w1 , E 2 (0) = √|b(0)| ∂w2 , and b(0) is the value of b at w = 0. where E 1 (0) = √|b(0)| The curvature tensor applied to E 1 (t), E 2 (t) is

R E1 (t)E2 (t)E1 (t)E2 (t) =

1 1 R0 R0 = , 2b(0)2 ||w(t)||4λ 2b(0)2 (1 − t)4

which is singular at t = 1. Note that (C )2+n − S is connected and not simply-connected for  = −1 and λ = 0, while it is neither connected nor simply-connected for the other values. Moreover, (C )2+n − S has two connected components for λ = −/2 and  = ±1 and four connected components for λ = 0 and  = 1 (Table 5.1). Table 5.1 Singular set Singular set S  = −1

λ = − 2

λ=0

w2

w2 w1

S : (w1 )2 + (w2 )2 = 0 =1

w2

w1

S : w1 = 0

w2 w1

S : (w1 )2 − (w2 )2 = 0

w1 S : w1 = 0

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5 Homogeneous Structures of Linear Type

We finally show that degenerate homogeneous -Kähler structures of linear type are realized on the -Kähler manifold ((C )2+n − S, g). Proposition 5.2.11 For every data (b, R0 ) satisfying Eq. (5.2.27), ((C )2+n − S, g) admits a degenerate -Kähler homogeneous structure of linear type. Proof Let

 ξ=

−1 (w 1 ∂z 1 (w 1 )2 −(w 2 )2 − w11 ∂z 1

+ w 2 ∂z 2 ) for λ = 0, for λ = − 2 .

We take the tensor field S X Y = g(X, Y )ξ − g(ξ, Y )X + g(X, J Y )J ξ − g(ξ, J Y )J X − 2λg(ξ, J X )J Y.  = 0 and ∇  R = 0, where ∇  = ∇ − S, A straightforward computation shows that ∇ξ so that S is a degenerate homogeneous -Kähler structure of linear type (see (4.2.4) and (4.2.6)).

We close this section with the characterization of homogeneous structures of linear type for quaternion-Kähler manifolds, referring to [46, 89] for the proof. On one hand, the non-degenerate case is similar to the purely pseudo-Riemannian or -Kähler case. However, the degenerate case does not provide anything of interest. Theorem 5.2.12 Let (M, g, Q) be a connected -quaternion Kähler manifold of dimension 4n  8 admitting a homogeneous -quaternion Kähler structure of linear type S. If S is non-degenerate, then (M, g, Q) has constant -quaternion sectional curvature −4g(ξ, ξ), and ζ a = 0 for a = 1, 2, 3. If S is degenerate then (M, g, Q) is flat.

5.3 Infinitesimal Models, Homogeneous Models and Completeness In this section we shall make use of the definitions of infinitesimal model, transvection algebra and homogeneous model associated to a homogeneous structure (Sect. 2.4). Referring to the description of complex manifolds given in (1.3.3), the aim of this section is to prove the following results. Theorem 5.3.1 With the exception of CP0n and CHnn , the indefinite -complex space CP n locally admit a non-degenerate homogeneous -Kähler forms CPpn , CH pn and  structure of linear type. Theorem 5.3.2 Let (M, g, J ) be a connected and simply-connected -Kähler manifold with dim M  4 admitting a non-degenerate homogeneous -Kähler structure of linear type. If g is not definite then (M, g, J ) is not complete. On the other hand,

5.3 Infinitesimal Models, Homogeneous Models and Completeness

155

the homogeneous model associated to a degenerate homogeneous -Kähler structure of linear type is not complete. Remark 5.3.3 Note that Theorem 5.3.2 implies that the indefinite -complex space forms CPpn , CH pn and  CP n do not admit a globally defined homogeneous -Kähler structure of linear type, except for the definite cases CPnn and CH0n . We now explain the general procedure to prove Theorems 5.3.1 and 5.3.2. This procedure will be then specified for each case: degenerate and non-degenerate, and pseudo-Kähler and para-Kähler. Proof (Procedure for the proof of Theorems 5.3.1 and 5.3.2) The first step is to explicitly compute the infinitesimal model and the transvection algebra g = T p M ⊕  hol∇ associated to a homogeneous -Kähler structure of linear type. This is done by  = R − RS. obtaining the expression for R  Defining h = hol∇ , we next show that the transvection algebra (g, h) is regular, that is, H is a closed subgroup of G, where G is the simply-connected Lie group with Lie algebra g and H is its connected Lie subgroup with Lie algebra h . In order to prove this, we obtain a matrix realization of h and g in gl(N , R) for some N ∈ N, and we exponentiate it to see that the connected Lie subgroup of GL(N , R) with Lie algebra h is closed in GL(N , R), whence H must be closed in G. For the degenerate case this is done using the adjoint representation. For the non-degenerate CP n , since case, by Remark 1.3.2, we will only need to consider the spaces CH pn and  by Theorem 5.2.3 our spaces of linear type are locally ±-isometric to one of these models. CP n as symmetric Recalling the expressions (1.3.3) and (1.3.4) of CHsn and  spaces Isom/Isot, we identify g with a subalgebra of isom in such a way that h is the intersection of g and isot. This gives a matrix realization of g and h, and subgroups G ⊂ Isom and H ⊂ Isot, and we find that H is closed in Isom (which is closed in Gl(m, R)). We can thus take the homogeneous model G/H associated to S. Continuing with the non-degenerate case, the orbit of p = e Isot in the model space Isom/Isot is just G/H . Counting dimensions one sees that G/H is an open subset of Isom/Isot. Since by construction G/H admits a non-degenerate homogeneous -Kähler structure of linear type, this would prove Theorem 5.3.1. We now return to the general case. (M, g) is locally isometric to the homogeneous space G/H (see Proposition 3.1.21) and when (M, g) is simply-connected and complete, it will be globally isometric to G/H . To prove Theorem 5.3.2 we show that G/H is not complete. We consider a Lie algebra involution σ : g → g with σ(h) ⊂ h and restricting to an isometry for the Ad(H )-invariant metric on T p M. The map σ determines a Lie group involution σ : G → G with σ(H ) ⊂ H , and an involution σ on the homogeneous space G/H . Denote the fixed-point set of σ on X by X σ . Then the homogeneous spaces G σ /H σ and (G/H )σ are isometric. However, σ is an isometry, so (G/H )σ is a totally geodesic submanifold of G/H . By considering a sequence of such Lie algebra involutions, we can construct a chain of totally geodesic submanifolds

156

5 Homogeneous Structures of Linear Type

· · · ⊂ ((G/H )σ1 )σ2 ⊂ (G/H )σ1 ⊂ G/H. In our cases, we use this technique to construct a totally geodesic submanifold that we can show is not complete (Lemmas 5.3.4 and 5.3.5). It follows that G/H is not geodesically complete.

Lemma 5.3.4 The Lie group K with Lie algebra k = Span{A, V }, [A, V ] = V , and left-invariant metric given by g(A, A) = 1,

g(V, V ) = −1,

g(A, V ) = 0,

is not geodesically complete, time-like complete, null complete nor space-like complete. Proof The Levi-Civita connection of this metric is ∇ A A = 0,

∇ A V = 0,

∇V A = −V,

∇V V = −A.

Let γ be a curve in K and γ˙ its derivative. We write γ(t) ˙ = γ1 (t)A + γ2 (t)V . The geodesic equation thus implies 

γ˙ 1 − γ22 = 0, γ˙ 2 − γ1 γ2 = 0.

The solution to this system with space-like initial value γ1 (0) = 0, γ2 (0) = 1 is γ1 (t) = tan(t), γ2 = 1/ cos(t) which is defined for −π/2 < t < π/2. On the other hand, the null initial value γ1 (0) = 1 = γ2 (0) has solution γ1 (t) = γ2 (t) = 1/(1 − t) which is only defined for t < 1. Finally, the time-like initial value γ1 (0) = 1, γ √2 (0) = r , 0 < r < 1, has x(t) = s coth(st + k), y(t) = s/ sinh(st + k), where s = 1 − r 2 , tanh k = s. These solutions are only defined for t = −k/s.

Lemma 5.3.5 The Lie group K with Lie algebra k = Span{U, V }, [U, V ] = μ(V − U ), where μ ∈ R+ , and left-invariant metric given by g(U, U ) = s = ±1,

g(V, V ) = −s,

g(U, V ) = 0,

is not geodesically complete, time-like complete, null complete nor space-like complete. Proof The Levi-Civita connection of g is ∇U U = − √|b(1 p)| V, ∇V V = − √|b(1 p)| U,

∇U V = − √|b(1 p)| U, ∇V U = − √|b(1 p)| V.

Let γ be a curve in K and γ˙ its tangent vector. Setting γ(t) ˙ = u(t)U + v(t)V , the geodesic equation ∇γ˙ γ˙ = 0 implies

5.3 Infinitesimal Models, Homogeneous Models and Completeness

u˙ − √

1 (uv + v 2 ) = 0, |b( p)|

157

1 v˙ − √ (uv + u 2 ) = 0. |b( p)|

Changing variables to x = u + v and y = v − u, the equations transform into 1 x˙ − √ x 2 = 0, |b( p)|

1 y˙ + √ x y = 0. |b( p)|

Space-like and time-like initial values are obtained for example for x(0) = 1 and y(0) = ±1, and a null initial value is obtained for example for x(0) = 1 and y(0) = 0. For each of those cases, the solutions for x is x=

 |b( p)|

1 , 1 − μt

for some constant c ∈ R, which is only defined for t = 1/μ.



We now specify all the components involved in the proof of Theorems 5.3.1 and 5.3.2 for each case. Due to differences, we treat them separately. The convention R X Y Z = ∇[X,Y ] Z − ∇ X ∇Y Z + ∇Y ∇ X Z , R XS Y Z = SSX Y −SY X Z − S X SY Z + SY S X Z , will be used here.

5.3.1 The Non-degenerate Para-Kähler Case As above,  C denotes the set of para-complex numbers, and in this section e will stand for the imaginary para-complex unit e2 = +1, and z¯ the para-complex conjugation of z ∈  C. We first compute the infinitesimal model associated to S. Using formula (5.2.2) with ζ = 0 and  = 1, we obtain by direct calculation R XS Y Z =g(ξ, ξ) {g(Y, Z )X − g(X, Z )Y + g(Y, J Z )J X − g(X, J Z )J Y } − 2g(X, J Y ) {g(ξ, J Z )ξ + g(ξ, Z )J ξ} , and since (M, g, J ) has constant para-holomorphic sectional curvature, we have X Y Z = −2g(X, J Y ) {g(ξ, ξ)J Z − g(ξ, J Z )ξ − g(ξ, Z )J ξ} . R X Y acts trivially on R2 = Span{ξ, J ξ}. On the other hand, X Y ξ = 0 and thus R Now, R for Z ∈ Span{ξ, J ξ}⊥ , one has

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5 Homogeneous Structures of Linear Type

X Y Z = −2g(X, J Y )g(ξ, ξ)J Z , R X Y acts on U = Span{ξ, J ξ}⊥ as −2g(X, J Y )g(ξ, ξ)J . We conclude that so that R  ∇ 1  hol is one-dimensional and is generated by the element J = 2g(ξ,ξ) 2 Rξ J ξ . The remaining brackets are [Z 1 , Z 2 ] = 2g(Z 1 , J Z 2 )L 0 , [ξ, J ξ] = 2g(ξ, ξ)L 0 , [ξ, Z ] = g(ξ, ξ)J Z ,

[J ξ, Z ] = g(ξ, ξ)J Z ,

(5.3.28)

where Z 1 , Z 2 , Z ∈ U and L 0 = J ξ − g(ξ, ξ)J . The transvection algebra is thus given by g = RJ ⊕ Span{ξ, J ξ} ⊕ U. Next, the description (1.3.4) of  CP n as a symmetric space has Cartan decomposition sl(n + 1, R) = s(gl(n, R) ⊕ gl(1, R)) ⊕ m ⊂ so(n + 1, n + 1),

 0n v n  : v ∈ C . −v ∗ 0



with m=

We write  Cn = Rn + eRn . The algebra sl(n + 1, R) decomposes as sl(n + 1, R) = s (gl(n, R) ⊕ gl(1, R)) ⊕ a ⊕ n1 ⊕ n2 , ⎛

where a = RA0 ,

⎞ 0n−1 0 0 A0 = ⎝ 0 0 e ⎠ , 0 e0

is a maximal R-diagonalizable subalgebra of m, and ⎧⎛ ⎫ ⎞ ⎨ 0n−1 −ev v ⎬ n1 = ⎝−ev ∗ 0 0⎠ : v ∈  Cn−1 , ⎩ ⎭ −v ∗ 0 0 ⎧⎛ ⎫ ⎞ ⎨ 0n−1 0 0 ⎬ n2 = ⎝ 0 −eb b ⎠ : b ∈ R ⎩ ⎭ 0 −b eb are the eigenspaces of the positive restricted roots  + = {λ, 2λ} with λ(A0 ) = 1. We shall identify g with a subalgebra of sl(n + 1, R), following arguments analogous to those in [43].

5.3 Infinitesimal Models, Homogeneous Models and Completeness

159

First, it is obvious that J ∈ s (gl(n, R) ⊕ gl(1, R)) and since J acts trivially on Span{ξ, J ξ} and effectively on U , the space U can be identified with n1 and Span{ξ, J ξ} ⊂ RJ + a + n2 . Now, from (5.3.28) it easily follows that L 0 ∈ n2 , and since ξ has only real eigenvalues on g, we can take ξ = g(ξ, ξ)A0 up to a Lie algebra automorphism. Let ⎛ ⎞ 0n−1 0 0 X = ⎝ 0 −e 1⎠ . 0 −1 e Using a Lie algebra automorphism we can take L 0 = X , which gives J ξ = X + Cn−1 in the obvious way, from g(ξ, ξ)J . Finally, identifying U with n1 and n1 with  (5.3.28) we have [v, w] = 2g(v, J w)X . we obtain [v, w] = −2v, ewX , where v, w = From  the matrix expression of n1n−1 C . Comparing these two expressions we conRe j v¯ j w j , v, w ∈ U ≡ n1 ≡  clude that J is acting on U as multiplication by −e. Therefore, J must be given by J =

e diag((−2)n−1 , (n − 1)2 ), n+1

with powers denoting multiplicities. Exponentiating the Lie algebra spanned by J we obtain a closed subgroup of SL(n + 1, R). Regarding the Lie algebra involutions involved in the proof of Theorem 5.3.2, we take σ : g → g determined by J → −J , A0  → A0 , X + g(ξ, ξ)J → − (X + g(ξ, ξ)J ) , v ∈ n1 ≡  Cn−1 v → −v, and τ : gσ → gσ with A0  → A0 , (v1 , . . . , vn−2 , vn−1 )T → (−v1 , . . . , −vn−2 , vn−1 )T . We thus have

⎧⎛ 0n−2 ⎪ ⎪ ⎨⎜ 0 σ τ k = (g ) = ⎜ ⎝ 0 ⎪ ⎪ ⎩ 0

0 0 0 −es

0 0 0 et

⎫ ⎞ 0 ⎪ ⎪ ⎬ es ⎟ ⎟ /s, t ∈ R , et ⎠ ⎪ ⎪ ⎭ 0

and the chain of totally geodesic submanifolds K = (G σ )τ ⊂ G σ = (G/H )σ ⊂ G/H, where K is as in Lemma 5.3.4, and is incomplete.

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5.3.2 The Non-degenerate Pseudo-Kähler Case Throughout this subsection i denotes the imaginary complex unit. The computations for the infinitesimal model are completely analogous to those in the previous subsection, setting  = −1. We obtain that X Y Z = 2g(X, J Y )g(ξ, ξ)J Z , R 

so that hol∇ is the one-dimensional Lie algebra generated by J = remaining brackets are [Z 1 , Z 2 ] = −2g(Z 1 , J Z 2 )L 0 , [ξ, Z ] = g(ξ, ξ)J Z ,

1 2g(ξ,ξ)2

[ξ, J ξ] = 2g(ξ, ξ)L 0 , [J ξ, Z ] = g(ξ, ξ)J Z ,

ξ J ξ . The R

(5.3.29)

where Z 1 , Z 2 , Z ∈ U and L 0 = J ξ − g(ξ, ξ)J . The transvection algebra is g = RJ ⊕ Span{ξ, J ξ} ⊕ U, where U = Span{ξ, J ξ}⊥ . On the other hand, recall description (1.3.3) of CHsn as a symmetric space. The Riemannian case CH0n is studied in [43]. We then suppose s > 0, and for the sake of simplicity we also suppose 2s < n − 1, the opposite case being analogous. Let ε=

01 10

  and  = diag (1)n−2s−1 , (ε)s+1 .

We have su(n − s, s + 1) = {C ∈ gl(n + 1, C)/ C ∗  + C = 0, Tr(C) = 0}, so that su(n − s, s + 1) decomposes as su(n − s, s + 1) = s (u(n − s, s) ⊕ u(1)) ⊕ a ⊕ n1 ⊕ n2 , where a = R A0 , A0 = diag(0, . . . , 0, 1, −1), ⎧⎛ ⎧⎛ ⎫ ⎫ ⎞ ⎞ 0 0 v 0 0 0 ⎨ ⎨ ⎬ ⎬ n−1 n−1  ∗ n1 = ⎝−   v 0 0⎠ /v ∈ Cn−1 , n2 = ⎝ 0 0 ib⎠ /b ∈ R , ⎩ ⎩ ⎭ ⎭ 0 0 0 0 00   for   = diag (1)n−2s−1 , (ε)s . As in the para-Kähler case, we identify g with a subalgebra of su(n − s, s + 1). More precisely, we have that U is identified with n1 , ξ = g(ξ, ξ)A0 , and J ξ = L 0 + g(ξ, ξ)A0 with

5.3 Infinitesimal Models, Homogeneous Models and Completeness

161

⎛

⎞ 0n−1 0 0 L0 = ⎝ 0 0 i ⎠ . 0 00 In addition, from the matrix representation of n1 we obtain J =

  i diag (−2)n−1 , (n − 1)2 , n+1

so that the Lie algebra spanned by J gives a closed subgroup of SU(n − s, s + 1). Regarding the Lie algebra involutions involved in the proof of Theorem 5.3.2, we take σ : g → g defined by J → −J , A0  → A0 ,   X + g(ξ, ξ)J → − X + g(ξ, ξ)J , v → v, v ∈ n1 ≡ Cn−1 and τ : gσ → gσ with A0  → A0 , (v1 , . . . , vn−2 , vn−1 )T → (−v1 , . . . , −vn−1 , −vn−2 )T . Then,

⎧⎛ 0n−3 ⎪ ⎪ ⎪ ⎪ 0 ⎨⎜ ⎜ 0 k = (gσ )τ = ⎜ ⎜ ⎪ ⎪ ⎪⎝ 0 ⎪ ⎩ 0

0 0 0 t 0

0 0 0 −t 0

0 0 0 s 0

⎫ ⎞ 0 ⎪ ⎪ ⎪ ⎪ t ⎟ ⎬ ⎟ ⎟ −t ⎟ /s, t ∈ R , ⎪ ⎪ 0⎠ ⎪ ⎪ ⎭ −s

and we have the following chain of totally geodesic submanifolds: K = (G σ )τ ⊂ G σ = (G/H )σ ⊂ G/H, with K as in Lemma 5.3.4.

5.3.3 The Degenerate Case with λ = − 2 Defining p1 = ξ and p2 = J ξ, for the sake of simplicity we choose x ∈ M such that with respect to the basis { p1 , p2 , q1 , q2 , X a , J X a } and its dual { p 1 , p 2 , q 1 , q 2 , X a , J X a }, the curvature is written as Rx = R0 q 1 ∧ q 2 ⊗ (q 1 ⊗ p2 + q 2 ⊗ p1 ).

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5 Homogeneous Structures of Linear Type 

Let h = hol∇ . Substituting λ = −/2 into (5.2.2) we obtain by direct calculation that  are: the non-vanishing terms of R p2 q1 : R

q1 q2 Xa J Xa p1 , p2

q2 X a : R

→ → → → →

q1 q2 Xa J Xa p1 , p2

2 p2 2 p1 0 0 0

→ → → → →

q1 q2 : R

q2 J X a : R

−J X a −X a p2  p1 0

X a J X a : R

q1 q2 Xa J Xa p1 , p2

q1 q2 Xa J Xa p1 , p2

→ → → → →

→ → → → →

(R0 − b( p)) p2 (R0 − b( p)) p1 −J X a −X a 0

q1 q2 Xa J Xa p1 , p2

→ → → → →

−X a −J X a  p1  p2 0

−2 p2 −2 p1 0 0 0,

so that dimh = 2n + 2. Choosing endomorphisms A = 2(q 1 ⊗ p2 + q 2 ⊗ p1 ), K =

q2 X a , Ca = R q2 J X a , Ba = R

 1 (R0 − b( p))A − (X a ⊗ J X a + J X a ⊗ X a ) 2 a

as a basis of h, the transvection algebra g has non-vanishing brackets [Ba , Ca ] = A, [Ba , K ] = −Ca , [Ca , K ] = −Ba , [A, q1 ] = 2 p2 , [A, q2 ] = 2 p1 , [Ba , q1 ] = −J X a , [Ba , q2 ] = −X a , [Ba , X a ] = − p2 , [Ba , J X a ] = − p1 , [Ca , q1 ] = −X a , [Ca , q2 ] = −ya , [Ca , X a ] =  p1 , [Ca , J X a ] =  p2 , [K , X a ] = J X a , [K , J X a ] = X a , [ p1 , q1 ] = − p1 , [ p2 , q1 ] = −3 p2 − A, [ p2 , q2 ] = −2 p1 , [q1 , q2 ] = 2b( p) p2 − q2 − 21 (R0 − b( p))A + K , [q1 , X a ] = X a , [q1 , J X a ] = J X a , [q2 , X a ] = 2J X a − Ba , [q2 , J X a ] 2X a − Ca , [X a , J X a ] = 2 p2 + A. One can check that g is a solvable Lie algebra with a 3-step nilradical. Since g has trivial center, the adjoint representation is faithful and provides a matrix realization of g.

5.3 Infinitesimal Models, Homogeneous Models and Completeness

163

With respect to this realization, a straightforward computation shows that by exponentiation of h we obtain a connected Lie group H which is closed inside GL(4n + 6, R), so that (g, h) is regular. For instance, the matrix realization of h for n = 2 is ⎞ ⎛ 00 0 2λ A  λC1  −λ B1  λC2  −λ B2  0 0 0 0 0 0 ⎜ 0 0 2λ A 0 −λ B1 λC1  −λ B2 λC2  0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎜ 0 0 −λC1  −λ B1  0 λK  0 0 0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜ 0 0 −λ B1 −λC1  λ K 0 0 0 0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜ 0 0 −λC  −λ B  0 0 0 λK  0 0 0 0 0 0 ⎟ 2 2 ⎟ ⎜ ⎜ 0 0 −λ B −λC  0 0 λK 0 0 0 0 0 0 0 ⎟ 2 2 ⎟ ⎜ ⎜0 0 0 0 0 0 0 0 0 −λC1  −λC2  λ B1  λ B2  0 ⎟ ⎟ ⎜ ⎜0 0 0 0 0 0 0 0 0 0 0 λ K  0 −λC1  ⎟ ⎟ ⎜ ⎜0 0 0 0 0 0 0 0 0 0 0 0 λ K  −λC2  ⎟ ⎟ ⎜ ⎜0 0 0 0 0 0 0 0 0 λK 0 0 0 −λ B1 ⎟ ⎟ ⎜ ⎝0 0 0 0 0 0 0 0 0 0 λK 0 0 −λ B2 ⎠ 00 0 0 0 0 0 0 0 0 0 0 0 0 for λ A , λ B1 , λ B2 , λC1 , λC2 , λ K ∈ R. Regarding the Lie algebra involutions needed for the proof of Theorem 5.3.2, we take σ:

and

g→g A → −A Ba → −Ba Ca → Ca K → −K p1  → p1 p2  → − p2 q1  → q1 q2 → −q2 X a → X a J X a → −J X a ,

θ : gσ Ca p1 q1 Xa

→ gσ → −Ca  → p1  → q1 → −X a .

The subalgebra of fixed points is k = (gσ )θ = span { p1 , q1 }, and we have the chain of totally geodesic submanifolds K ⊂ Gσ,

G σ = (G/H )σ ⊂ G/H.

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5 Homogeneous Structures of Linear Type

Let s be √ the sign of b( p).√We define the left-invariant vector fields U = 1/( |b( p)|)q1 , V = U − s |b( p)| p1 in k. We thus have < U, U >= s, < V, V >= −s, < U, V >= 0, and [U, V ] = √|b(1 p)| (V − U ), where < ·, · > stands for the bilinear form inherited by k from gσ . The Lie algebra k is as in Lemma 5.3.5 with μ = √|b(1 p)| , so that K is not geodesically complete.

5.3.4 The Degenerate Case with λ = 0 Let again p1 = ξ and p2 = J ξ. We choose x ∈ M such that with respect to the basis { p1 , p2 , q1 , q2 , X a , J X a } and its dual { p 1 , p 2 , q 1 , q 2 , X a , J X a } the curvature is written as Rx = R0 q 1 ∧ q 2 ⊗ (q 1 ⊗ p2 + q 2 ⊗ p1 ). 

Let h = hol∇ . Substituting λ = 0 into (5.2.2) we obtain by direct calculation that the  are: non-vanishing terms of R

q1 q2 : R

p1 q2 : R

q1 q2 Xa J Xa p1 , p2

q1 q2 Xa J Xa p1 , p2

→ → → → →

→ → → → →

−2 p2 −2 p1 0 0 0

p2 q1 : R

(R0 − 2b( p)) p2 (R0 − 2b( p)) p1 0 0 0

q1 q2 Xa J Xa p1 , p2

X a J X a : R

→ → → → →

2 p2 2 p1 0 0 0

q1 q2 Xa J Xa p1 , p2

→ → → → →

−2 p2 −2 p1 0 0 0,

so that dimh = 1. Choosing the endomorphism A = 2(q 1 ⊗ p2 + q 2 ⊗ p1 ) as a basis of h, the transvection algebra g has non-vanishing brackets [A, q1 ] = 2 p2 , [A, q2 ] = 2 p1 , [ p1 , q1 ] = − p1 , [ p1 , q2 ] = p2 + A, [ p2 , q1 ] = −3 p2 − A, [ p2 , q2 ] = − p1 , [q1 , q2 ] = 2b( p) p2 − 21 (R0 − 2b( p))A, [q1 , X a ] = X a , [q1 , J X a ] = J X a , [q2 , X a ] = J X a , [q2 , J X a ] = X a , [X a , J X a ] = 2 p2 + A.

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165

One can check that g is a solvable Lie algebra with a 2-step nilradical. Since g has trivial center, the adjoint representation is faithful and provides a matrix realization of g. The matrix realization of h is ⎞ ⎛ 0 0 0 2t 0 . . . 0 ⎜0 0 2t 0 0 . . . 0⎟ ⎟ ⎜ ⎜0 0 0 0 0 . . . 0⎟ ⎟ ⎜ ⎜ .. .. .. .. .. .. ⎟ ⎝. . . . . .⎠ 0 0 0 0 0 ... 0 for t ∈ R. Exponentiating we obtain a connected Lie group H which is closed in GL(2n + 5, R), so that (g, h) is regular. Regarding the Lie algebra involutions needed for the proof of Theorem 5.3.2, we take σ:

g→g A → −A p1  → p1 p2  → − p2 q1  → q1 q2 → −q2 X a → X a J X a → −J X a ,

and θ : gσ p1 q1 Xa

→ gσ  → p1  → q1 → −X a .

The subalgebra of fixed points is k = (gσ )θ = span { p1 , q1 }, and we have the chain of totally geodesic submanifolds K ⊂ Gσ,

G σ = (G/H )σ ⊂ G/H.

Let s be √ the sign of b( p).√We define the left-invariant vector fields U = 1/( |b( p)|)q1 , V = U − s |b( p)| p1 in k. We thus have < U, U >= s, < V, V >= −s, < U, V >= 0, and [U, V ] = √|b(1 p)| (V − U ), where < ·, · > stands for the bilinear form inherited by k from gσ . The Lie algebra k is as in Lemma 5.3.5 with μ = √|b(1 p)| , so that K is not geodesically complete.

5.4 Relation with Homogeneous Plane Waves We exhibit the parallelism between certain kinds of (Lorentzian) homogeneous plane waves and Lorentz–Kähler spaces admitting degenerate homogeneous structures of linear type (by Lorentz–Kähler we mean pseudo-Kähler of index 2).

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5 Homogeneous Structures of Linear Type

Although as far as the authors know there is no formal definition of a plane wave in complex geometry, this relation could allow us to understand the latter spaces as a complex generalization of the former, at least in the important Lorentz–Kähler case, suggesting a starting point for a possible definition of complex plane wave. A plane wave is a Lorentz manifold M = Rn+2 with metric g = dudv + Aab (u)x a x b du 2 +

n  (d x a )2 , a=1

where (Aab )(u) is a symmetric matrix depending on the coordinate u called the profile of g. A plane wave is called homogeneous if the Lie algebra of Killing vector fields acts transitively on the tangent space at every point. Among homogeneous plane waves we will be interested in two types. A Cahen– Wallach space is defined as a plane wave with profile a constant symmetric matrix (Bab ), which makes it symmetric and geodesically complete. On the other hand, a singular scale-invariant homogeneous plane wave is a plane wave with profile (Bab )/u 2 , where (Bab ) is a constant symmetric matrix. Singular scale-invariant homogeneous plane waves are homogeneous but they are not symmetric. In addition, they present a singularity in the cosmological sense at {u = 0}, since the geodesic deviation equation (or Jacobi equation) governed by the curvature is singular at this set (see [110]). Note that these two kinds of spaces are composed by the twisted product of a totally geodesic flat wave front and a 2-dimensional manifold containing time and the direction of propagation. This 2-dimensional space gives the real geometric information of the total manifold and in particular it contains a null parallel vector field. They are all VSI, and the curvature information is contained in the profile Aab (u), since the only non-vanishing component of the curvature is given by Ruaub = −Aab (u),

a, b = 1, . . . , n.

Cahen–Wallach spaces are one of the possible indecomposable simply-connected Lorentzian symmetric spaces together with (R, −dt 2 ), the de Sitter, and the anti de Sitter spaces (see [24]). On the other hand, in [94] the following characterization is given. Theorem 5.4.1 Let (M, g) be a connected pseudo-Riemannian manifold of dimension n + 2 admitting a degenerate homogeneous pseudo-Riemannian structure of linear type with g(ξ, ξ) = 0. Then (M, g) is locally isometric to Rn+2 with metric ds 2 = dudv +

n Bab a b 2  x x du + εa (d x a )2 u2 a=1

for some symmetric matrix (Bab ) and εa = ±1, a = 1, . . . , n. Note that for Lorentzian signature this means that a manifold admitting a degenerate homogeneous structure of linear type is locally a singular scale-invariant

5.4 Relation with Homogeneous Plane Waves

167

homogeneous plane wave. Conversely, it is easy to see that every singular scaleinvariant homogeneous plane wave admits such a homogeneous structure with ξ = − u1 ∂v . In the Lorentz–Kähler case, according to [67] there is only one possibility for a simply-connected, indecomposable (and not irreducible), symmetric space of complex dimension 2 and signature (2, 2) with a null parallel complex vector field, that is, a manifold with holonomy γ1 =0,γ2 =0 holn=0



i i =R −i −i

in the notation of [67]. In order to get a plane wave structure, we add a plane wave γ1 =0,γ2 =0 ⊕ {0n }. Note front by considering a manifold (M, g, J ) with holonomy holn=0 that this is the holonomy algebra in Proposition 5.2.8 for  = −1. Proposition 5.4.2 Let (M, g, J ) be a locally symmetric Lorentz–Kähler manifold γ1 =0,γ2 =0 of dimension 2n + 4, n ≥ 0, with holonomy holn=0 ⊕ {0n }. Then the metric g is locally of the form g = dw1 dz 1 + dw 2 dz 2 + b(dw 1 dw 1 + dw 2 dw 2 ) +

n  (d x a d x a + dy a dy a ), a=1

(5.4.1) where the function b only depends on w1 and w 2 and satisfies b = b0 , b0 ∈ R − {0}. Proof Looking at the holonomy representation, there are two parallel isotropic (real) vector fields Z and J Z on M. Let α1 = g(·, Z ) and α2 = −α1 ◦ J , and consider the complex form α = α1 + 2 iα . Since ∇ Z = 0 = ∇ J Z , we have ∇α = 0, hence in particular α is holomorphic and closed. This means that locally there is a holomorphic function w : U → C such that dw = α. Since dw is non-zero at some point and it is parallel, we have that dw is nowhere vanishing. Hence if the set w−1 (λ), λ ∈ C, is non-empty, then it defines a complex hypersurface in U . Let p ∈ M and let q1 ∈ T p M be such that g(Z p , q1 ) = 1 with g(q1 , q1 ) = 0. The subspace E = Span{Z p , J Z p , q1 J q1 } is invariant under holonomy, hence so is E ⊥ . In fact, the holonomy action is trivial on E ⊥ . This implies that there are parallel vector fields E i , J E i , i = 1, . . . , n, which are an orthonormal basis of E ⊥ at every point. In addition, it is easy to see that Z , J Z , E i , J E i are always tangent to the hypersurfaces w−1 (λ). We can thus take coordinates {w 1 , w 2 , z 1 , z 2 , x a , y a } with w = w1 + iw 2 , ∂z 1 = Z , ∂z 2 = J Z , ∂x i = E i and ∂ y i = J E i , and such that

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5 Homogeneous Structures of Linear Type

g = dw 1 dz 1 + dw 2 dz 2 + b(dw 1 dw 1 + dw 2 dw 2 ) +

n  (d x a d x a + dy a dy a ). a=1

In addition, the only non-zero element of the curvature tensor field is R∂w1 ∂w2 ∂w1 ∂w2 =

1 b, 2

where  stands for the Laplace operator with respect to the variables (w 1 , w 2 ). The condition of being locally symmetric is then ∇ R = 0 ⇔ b = b0 , for b0 ∈ R − {0}.



The previous proposition suggests to consider the pseudo-Kähler manifold (C2+n , g) with g as in (5.4.1), as a natural Lorentz–Kähler analogue to Cahen– Wallach spaces. Note that equation b = b0 admits singular solutions. Nevertheless, as Cahen–Wallach spaces are simply-connected, in order to have an actual analogue we only consider non-singular functions b, so that (C2+n , g) is complete. On the other hand, since Lorentzian singular scale-invariant homogeneous plane waves are characterized by degenerate pseudo-Riemannian homogeneous structures of linear type, from Propositions 5.2.9 and 5.2.10, the natural analogues to these spaces are Lorentz–Kähler manifolds with degenerate homogeneous pseudo-Kähler structures of linear type. More precisely, the spaces (Cn+2 − {wλ = 0}, g) with wλ as in (5.2.25), g ¯ with g given in as in (5.2.26) with  = −1 and signature (2, 2 + 2n), and (Cn+2 , g) (5.4.1), are composed by the twisted product of a flat and totally geodesic complex manifold and a 2-dimensional complex manifold containing a null parallel complex vector field. Moreover, expressions (5.2.26) and (5.4.1) are the same except for the function b, which has a different Laplacian in each case. As a straightforward computation shows, the curvature tensor of both metrics is R=

1 b(dw 1 ∧ dw 2 ) ⊗ (dw 1 ∧ dw 2 ), 2

whence all the curvature information is contained in the Laplacian of the function b. For this reason, analogously to Lorentz plane waves, we call b the profile of the metric. It is worth noting that in the Lorentz case one goes from Cahen–Wallach spaces to singular scale-invariant homogeneous plane waves by making the profile be singular with a term 1/u 2 . Doing so, the space is no longer geodesically complete and a cosmological singularity at {u = 0} is created. In the same way, in the Lorentz–Kähler case one goes from metric (5.4.1) to (5.2.26) by making the profile singular with a term 1/w4λ , and again one transforms

5.4 Relation with Homogeneous Plane Waves Table 5.2 Relation between homogeneous plane waves Symmetric space Lorentz

Cahen-Wallach spaces

Lorentz–Kähler

Profile: A(u) = B(const.) Geodesically complete C2+n with metric (5.4.1) Profile: b = R0 (const.) Geodesically complete

169

Deg. homog. of linear type Singular s.-i. homog. plane wave Profile: A(u) = B/u 2 Geodesically incomplete C2+n − {wλ = 0} with metric (5.2.26) Profile: b = R0 /w4λ Geodesically incomplete

a geodesically complete space into a geodesically incomplete space, creating a singularity at {wλ = 0}. Finally, we also note that all these metrics are VSI. This reinforces the parallelism and exhibits a close relation between these two pairs of spaces (Table 5.2).

Chapter 6

Reduction of Homogeneous Structures

Symmetries represent a classical tool in reduction procedures, found in many different contexts, such as systems of differential equations, variational principles, symplectic or other geometric structures, etc. The philosophy is quite simple: given any object on a manifold M¯ and a group of symmetries H preserving that object, ¯ the existence of a similar object on the quotient M/H may simplify the study of the properties of the initial one. Very frequently, the existence of symmetries entails the homogeneity of the mani¯ although the group H with which the reduction is performed need not be the fold M, ¯ entire isometry group, so that the quotient M/H still possesses relevant geometric information. The choice of H depends on the problem or object under study. Some interesting situations arise regarding geometric structures (such as those defined by tensors) in the reduction framework: a geometric structure on the unreduced space M¯ becomes a different one on the orbit space. An instance of this situation is the reduction of Sasakian spaces to Kähler spaces by the action of one dimensional groups H . The goal of this chapter, which makes use of the arguments introduced in [45], is the study of homogeneous structures under reduction by subgroups of the group of isometries. In particular, this gives rise to new homogeneous structure tensors on the orbit space of the action which, in addition, may be associated to a different geometric ¯ structure on M/H . This procedure reveals and sheds light on some known properties of homogeneous structures and it is also a source of new homogeneous structure tensors as well as interesting examples. Finally, reduction can provide information about homogeneous structures in the unreduced space starting from homogeneous structures in the quotient space and conversely.

© Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_6

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6.1 Reduction by a Normal Subgroup of Isometries 6.1.1 Reduction of a Homogeneous Structure Let π : M¯ → M be an H -principal bundle, where M¯ is a pseudo-Riemannian manifold with metric g, ¯ and the fibers are non-degenerate with respect to g. ¯ Suppose that H acts on M¯ by isometries. Although it is not essential, throughout this chapter the action of isometries is understood as action on the left, and hence π is a left principal bundle. ¯ ⊥ its ¯ let Vx¯ M¯ be the vertical subspace at x¯ and Hx¯ M¯ = (Vx¯ M) For x¯ ∈ M, orthogonal complement with respect to g. ¯ As H acts by isometries, the decomposition (6.1.1) Tx¯ M¯ = Vx¯ M¯ ⊕ Hx¯ M¯ is a principal H -connection in the bundle M¯ → M. This connection is sometimes called the mechanical connection (see for instance [90, 95]) due to its presence in many situations in geometric mechanics. Furthermore, there is a unique pseudoRiemannian metric g in M such that the restriction π∗ : Hx¯ M¯ → Tπ(x) ¯ M is an ¯ Obviously, the metric g satisfies isometry for every x¯ ∈ M. g(X, Y ) ◦ π = g(X ¯ H, Y H)

for all X, Y ∈ X(M),

(6.1.2)

where X H and Y H denote the horizontal lift of X and Y with respect to the mechanical connection. To complete the notation, in the following we will denote by Z h ∈ Hx¯ M¯ the horizontal part of Z ∈ Tx¯ M¯ with respect to the mechanical connection. Proposition 6.1.1 In the situation above, if ∇¯ is the Levi-Civita connection of g¯ on ¯ then the Levi-Civita connection ∇ of the reduced metric g on M is given by M, ∇ X Y = π∗ (∇¯ X H Y H )

for all X, Y ∈ X(M).

(6.1.3)

Proof Since the structure group H acts by isometries, it also acts by affine trans¯ The vector field ∇¯ X H Y H is thus projectable and the operator formations of ∇. ¯ H D X Y = π∗ (∇ X Y H ) is well defined. As a direct computation shows, D fulfills the properties of a linear connection on M. From (6.1.2), for X, Y, Z ∈ X(M) we have g(D X Y, Z ) ◦ π + g(Y, D X , Z ) ◦ π = g(( ¯ ∇¯ X H Y H )h , Z H ) + g(Y ¯ H , (∇¯ X H Z H )h ) = g( ¯ ∇¯ X H Y H , Z H ) + g(Y ¯ H , ∇¯ X H Z H ) = X H (g(Y ¯ H , Z H )). Hence, g(D X Y, Z ) + g(Y, D X Z ) = X (g(Y, Z )), so that the connection D is metric. Finally, as [X, Y ] H = [X H , Y H ]h , the torsion tensor of D is

6.1 Reduction by a Normal Subgroup of Isometries

173

T (X, Y ) = D X Y − DY X − [X, Y ] = π∗ (∇¯ X H Y H − ∇¯ Y H X H − [X H , Y H ]) = 0, whence D is the Levi-Civita connection of g.



We now suppose that H is a normal subgroup of the isometry group G¯ that ¯ The normality of H induces a well-defined action of the group acts freely on M. ¯ ¯ G = G/H on M = M/H as L: G×M → M ([a], ¯ [x]) ¯ → L [a] ¯ = [L a¯ (x)], ¯ ¯ ([ x])

(6.1.4)

where [a] ¯ and [x] ¯ denote the classes modulo H of a¯ ∈ G¯ and x¯ ∈ M¯ respectively, ¯ The action of G need not be effective. If and L a¯ denotes the action of G¯ on M. it is not, we replace G by G/N , where N is the kernel of the map G → Diff(M), a → L a , a ∈ G. One can easily prove the following result. Proposition 6.1.2 The group G acts on (M, g) by isometries. In particular, if G¯ acts transitively, that is, M¯ is homogeneous, so does G and M is homogeneous as well. Remark 6.1.3 Note that Proposition 6.1.2 shows that the horizontal distribution is ¯ This means that the mechanical connection is G-invariant, ¯ invariant under G. an important fact that will be used in Sect. 6.2. Let now x¯ ∈ M¯ and x = π(x) ¯ ∈ M. We denote by K¯ the isotropy group of x¯ ¯ and by K the corresponding isotropy group of x under the under the action of G, action of G. We also denote their Lie algebras by k¯ and k respectively. ¯ Lemma 6.1.4 Let τ : G¯ → G = G/H be the quotient homomorphism. Then K = τ ( K¯ ) and the restriction τ | K¯ : K¯ → K is an isomorphism of Lie groups. Proof It is obvious from (6.1.4) that τ ( K¯ ) ⊂ K . Let now k ∈ K and take a¯ ∈ G¯ such that k = τ (a). ¯ For any x ∈ M, we have x = L k (x) = π(L a¯ (x)), ¯ and then L a¯ (x) ¯ is ¯ = x, ¯ so in the same fiber as x. ¯ Hence, there exists an h ∈ H such that L h ◦ L a¯ (x) that h a¯ ∈ K¯ . Since τ (h a) ¯ = τ (a) ¯ = k we have k ∈ τ ( K¯ ). For the injectivity of τ | K¯ , let k¯1 , k¯2 ∈ K¯ be such that τ (k¯1 ) = τ (k¯2 ). There exists an h ∈ H such that h k¯1 = k¯2 . Then k¯1−1 h k¯1 = k¯1−1 k¯2 , so k¯1−1 k¯2 ∈ K¯ ∩ H . But since  H acts freely, k¯1−1 k¯2 = e¯ and so, k¯1 = k¯2 . ¯ ¯ K¯ is reductive with reductive decomposition g¯ = m ¯ ⊕ k. Suppose now that M¯ = G/ Let μ¯ be the infinitesimal action of g¯ at the point x, ¯ that is, μ¯ : g¯ → Tx¯ M¯ ξ¯ → dtd t=0 L ex p(t ξ) ¯ ¯ ( x).

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6 Reduction of Homogeneous Structures

Then for all k¯ ∈ K¯ , the following diagram is commutative μ

g¯ ¯ Ad(k)

 μ¯

g¯

Tx¯ M¯

(6.1.5)

(L k¯ )∗

Tx¯ M¯

¯ and the canonical ¯ gives an isomorphism μ¯ : m ¯ → Tx¯ M, The restriction of μ¯ to m  ¯ ⊕ k¯ is determined connection ∇¯ with respect to the reductive decomposition g¯ = m by its value at x, ¯ given by      ¯ X¯ Y¯ = μ¯ [μ¯ −1 ( X¯ ), μ¯ −1 (Y¯ )]m¯ , ∇ x¯

¯ X¯ , Y¯ ∈ Tx¯ M.

(6.1.6)

¯ g) Theorem 6.1.5 Let ( M, ¯ be a connected reductive homogeneous pseudoRiemannian manifold, and let G¯ be a group of isometries acting transitively and ¯ Let H  G¯ be a normal subgroup acting freely on M. ¯ Then, every effectively on M. homogeneous structure tensor S¯ associated to G¯ induces a homogeneous struc¯ ture tensor S associated to G = G/H in the reduced pseudo-Riemannian manifold ¯ M = M/H . ¯ For any Proof Let x¯ ∈ M¯ and x = π(x) ¯ ∈ M, and let g¯ be the Lie algebra of G. ¯ the restricted isomorphism ¯ ⊕ k¯ associated to S, reductive decomposition g¯ = m ¯ → Tx¯ M¯ induces a decomposition μ¯ : m ¯h ¯ =m ¯v ⊕m m from g¯ which is Ad( K¯ )-invariant by the commutativity of (6.1.5). Let g = g¯ /h be the Lie algebra of G and μ : g → Tx M the corresponding infinitesimal action at x. For any ξ¯ ∈ g¯ , by (6.1.4) we have ¯ = π∗ ¯ ξ) π∗ ◦ μ( = = = =



 d  L ¯ ¯ ( x) dt t=0 exp(t ξ)   π ◦ L exp(t ξ) ¯ ¯ ( x)

 d  dt t=0  d  L ¯ ¯ (π( x)) dt t=0 τ (exp(t ξ))  d  L ¯ (x) dt t=0 exp(tτ∗ (ξ)) ¯ μ ◦ τ∗ (ξ),

6.1 Reduction by a Normal Subgroup of Isometries

175

which means that the following diagram is commutative g¯

μ¯

Tx¯ M¯ π∗



τ∗

g

μ

(6.1.7)

Tx M.

¯ h and m ¯ v give commutative diagrams Restrictions to m μ¯

¯v m τ∗

Vx¯ M¯



¯ v) τ ∗ (m

μ

π∗

{0}

¯h m τ∗

μ¯

Hx¯ M¯ π∗



¯ h) τ ∗ (m

μ

(6.1.8)

Tx M

¯ h → τ ∗ (m ¯ h ) and μ : τ∗ (m ¯ h ) → Tx M are isomorphisms, and showing that τ∗ : m v ¯ ) ⊂ k. In addition, by Lemma 6.1.4 the restriction of τ∗ : g¯ → g to k¯ is an isoτ ∗ (m ¯ h ), morphism of Lie algebras from k¯ to k. Therefore, denoting by m the image τ∗ (m we have g = m ⊕ k. (6.1.9) ¯ = ξ. ¯ = k and τ∗ (ξ) ¯ h be such that τ (k) Let k ∈ K and ξ ∈ m, and let k¯ ∈ K¯ and ξ¯ ∈ m We have: ¯ Adk (ξ) = Adτ (k) ¯ (τ∗ (ξ))

¯ = μ−1 ◦ L τ (k) ¯ ◦ μ(τ∗ (ξ)) ¯ = μ−1 ◦ L τ (k) ¯ ξ)) ¯ ◦ π∗ (μ( ¯ = μ−1 ◦ π∗ ◦ L k¯ (μ( ¯ ξ)) ¯ ¯ = μ−1 ◦ π∗ ◦ μ(Ad k¯ (ξ)) −1 ¯ = μ ◦ μ ◦ τ∗ (Adk¯ (ξ))   ¯ . = τ∗ Adk¯ (ξ)

¯ h is Ad( K¯ )-invariant, we deduce that Adk (m) ⊂ τ∗ (m ¯ h ) = m, whence (6.1.9) Since m is a reductive decomposition. The homogeneous structure tensor associated to (6.1.9) at x is given by (Sx ) X Y = (∇Y ξ ∗ )x ,

X, Y ∈ Tx M,

where ξ ∗ is the fundamental vector field associated to ξ ∈ m with ξx∗ = μ(ξ) = X . ¯ = ξ. Then, ¯ h be such that τ∗ (ξ) Let ξ¯ ∈ m   (Sx ) X Y = (∇Y ξ ∗ )x = π∗ (∇¯ Y H (ξ ∗ ) H )x¯     = π∗ (∇¯ Y H ξ¯∗ ) − π∗ (∇¯ Y H (ξ¯∗ )v )x¯ .

176

6 Reduction of Homogeneous Structures

Let Z¯ ∈ Tx¯ M¯ be a horizontal vector. Since ξ¯x∗¯ is horizontal       g¯ (∇¯ Y H (ξ¯∗ )v )x¯ , Z¯ = Y H g¯ (ξ¯∗ )v , Z¯ − g¯ (ξ¯∗ )vx¯ , ∇¯ Y H Z¯ = 0. Hence, by (6.1.8),   (Sx ) X Y = π∗ ( S¯ x¯ ) X H Y H ,

X, Y ∈ Tx M.

(6.1.10)

Finally, we extend Sx to the whole M with the action of G to obtain a homogeneous structure tensor S.  We shall call the tensor field S the reduced homogeneous structure tensor. Corollary 6.1.6 The reduced homogeneous structure can be expressed as   S X Y = π∗ S¯ X H Y H ,

X, Y ∈ X(M).

(6.1.11)

Proof Let a¯ ∈ G¯ and a = τ (a) ¯ ∈ G. We proved that the horizontal lift of (L a )∗ (X ) is (L a¯ )∗ (X H ) for all X ∈ X(M). This, together with the invariance of S¯ under G¯ and the invariance of S under G, gives (6.1.11). 

6.1.2 The Space of Tensors Reducing to a Given Tensor Suppose that we are in the situation described in Theorem 6.1.5 and we have a homogeneous structure tensor S associated to G in the reduced manifold M. Let g = m ⊕ k be a reductive decomposition associated to S. Making use of (6.1.7), we define subspaces of g¯ ¯ ¯ h = τ∗−1 (m) ∩ μ¯ −1 (Hx¯ M) m

and

¯ v = h. m

Therefore, the decomposition ¯ ¯ ⊕ k, g¯ = m

with

¯ h, ¯ =m ¯v ⊕m m

(6.1.12)

¯ it is obvious that Ad( K¯ )(h) ⊂ h. is reductive. Indeed, since H is normal in G, h ¯ ¯ we have ¯ ¯ ¯ ¯ , as μ(Ad ¯ ¯ ξ)), On the other hand, for k ∈ K andξ ∈ m k¯ (ξ)) = (L k¯ )∗ (μ( h ¯ ¯ ¯ ¯ ¯ . μ(Ad ¯ k¯ (ξ)) ∈ Hx¯ M and τ∗ Ad k¯ (ξ) ∈ m, and then Ad k¯ (ξ) ∈ m The homogeneous structure tensor associated to this decomposition at x¯ is given by ¯ η] ¯ − B([η, ¯ m¯ , ξ) ¯ + B([ζ, ¯ ξ] ¯ m¯ , η), ¯ m¯ , ζ) ¯ ζ] ¯ 2( S¯ x¯ ) X¯ Y¯ Z¯ = B([ξ,

¯ X¯ , Y¯ , Z¯ ∈ Tx¯ M, (6.1.13) ¯ η, ¯ are such that their images under μ¯ are X, Y, Z , and B is the bilinear where ξ, ¯ ζ¯ ∈ m ¯ inherited from g¯ x¯ . Note that we are exactly in the situation of the proof form on m

6.1 Reduction by a Normal Subgroup of Isometries

177

of Theorem 6.1.5, so that the homogeneous structure tensor S¯ associated to (6.1.12) reduces to S. We can construct all other homogeneous structures in M¯ associated to G¯ by ¯ in (6.1.12) to the graph changing m ¯ ¯ ϕ = {X + ϕ(X )/ X ∈ m} m ¯ The condition that the new homo¯ h → k. of an Ad( K¯ )-equivariant map ϕ : h ⊕ m geneous structure tensors reduce to S is equivalent to ϕ |m¯ h = 0. The family of homogeneous structure tensors that reduce to S is thus parameterized by the set of ¯ Ad( K¯ )-equivariant maps ϕ : h → k. For the sake of simplicity we will denote by ϕ both the map ϕ : h → k¯ and its ¯ =h⊕m ¯ h . The expression of the homogeneous structure extension by zero to m ϕ ¯ to m ¯ ϕ, B tensor S¯ associated to this map is the same as in (6.1.13) by changing m ϕ ϕ ¯ ¯ ¯ ¯ ¯ ¯ , and ξ, η, ¯ ζ to ξ = ξ + ϕ(ξ), η¯ = η¯ + ϕ(η), ¯ to the induced bilinear form B in m ¯ ∈m ¯ ϕ , respectively. As ζ¯ = ζ¯ + ϕ(ζ) ¯ η] ¯ ϕ(η)] ¯ η] ¯ ϕ(η)] ¯ m¯ ϕ + [ξ, ¯ m¯ ϕ + [ϕ(ξ), ¯ m¯ ϕ + [ϕ(ξ), ¯ m¯ ϕ [ξ¯ , η¯ ]m¯ ϕ = [ξ, ¯ ϕ(η)] and [ϕ(ξ), ¯ m¯ ϕ = 0, we have that       ¯ η] ¯ η] ¯ ϕ(η)] ¯ m¯ ϕ , ζ¯ ¯ m¯ ϕ , ζ¯ + B ϕ [ξ, ¯ m¯ ϕ + [ϕ(ξ), B ϕ [ξ¯ , η¯ ]m¯ ϕ , ζ¯ = B ϕ [ξ,     ¯ ϕ(η)] ¯ η], ¯ η] ¯ + [ϕ(ξ), ¯ ζ¯ , = B [ξ, ¯ m¯ , ζ¯ + B [ξ, ¯ →m ¯ ϕ mapping ξ¯ to where one has to take into account that the isomorphism m ¯ is an isometry with respect to B and B ϕ . Therefore, ξ¯ + ϕ(ξ)  1  ¯ ϕ ¯ η], B [ξ, ϕ(η)] ¯ + [ϕ(ξ), ¯ ζ¯ (6.1.14) ( S¯ x¯ ) X¯ Y¯ Z¯ = ( S¯ x¯ ) X¯ Y¯ Z¯ + 2     ¯ + [ϕ(η), ¯ ξ¯ + B [ζ, ¯ ϕ(ξ)] ¯ + [ϕ(ζ), ¯ ξ], ¯ η¯ . − B [η, ¯ ϕ(ζ)] ¯ ζ], The summands involving B define a tensor field P ϕ globally defined in M¯ by the ¯ this tensor ¯ More precisely, for any y¯ ∈ M, ¯ with y¯ = L a¯ (x), ¯ a¯ ∈ G, left action of G. is  1  ¯ ¯ η], B y¯ [ξ, ϕ y¯ (η)] ¯ + [ϕ y¯ (ξ), ¯ ζ¯ 2   ¯ + [ϕ y¯ (η), ¯ ξ¯ − B y¯ [η, ¯ ϕ y¯ (ζ)] ¯ ζ],   ¯ ϕ y¯ (ξ)] ¯ + [ϕ y¯ (ζ), ¯ ξ], ¯ η¯ , + B y¯ [ζ,

ϕ

(Py¯ ) X¯ Y¯ Z¯ =

¯ where for X¯ , Y¯ , Z¯ ∈ Ty¯ M, ¯ ¯ y¯ := Ad(a)( ¯ m), m

¯ k¯ y¯ := Ad(a)( ¯ k),

(6.1.15)

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6 Reduction of Homogeneous Structures

ϕ y¯ := Ad(a) ¯ ◦ ϕ ◦ Ad(a¯ −1 ) : h → k¯ y¯ , ¯ y¯ induced from g¯ y¯ by B y¯ is the bilinear form on m ¯ ¯ y¯ → Ty¯ M, μ¯ y¯ := (L a¯ )∗ ◦ μ¯ ◦ Ad(a¯ −1 ) : m ¯ η, ¯ y¯ are such that their images by μ¯ y¯ are X¯ , Y¯ , Z¯ respectively. We have and ξ, ¯ ζ¯ ∈ m thus proved the following. Proposition 6.1.7 Under the hypotheses of Theorem 6.1.5, let S be a homogeneous structure tensor in M associated to G. Then the space of homogeneous structure tensors in M¯ associated to G¯ and reducing to S is a vector space isomorphic to the ¯ The isomorphism is explicitly given space of Ad( K¯ )-equivariant maps ϕ : h → k. by ϕ → S¯ ϕ = S¯ + P ϕ , where S¯ is the homogeneous structure associated to the decomposition (6.1.12) and P ϕ is given in (6.1.15).

6.2 Reduction in a Principal Bundle In this section we describe a reduction procedure in a more general framework. We ¯ g) have a group H acting by isometries on a pseudo-Riemannian manifold ( M, ¯ that ¯ The manifold M¯ need not be is endowed with a homogeneous structure tensor S. globally homogeneous or, even if there is a group G¯ of isometries acting transitively, we do not assume H to be a normal subgroup. We loose the conditions required ¯ in the previous section and, as we will see soon, the projection of S¯ to M = M/H (equipped with the projected metric) cannot be guaranteed. We now explore the conditions needed for the existence of that projection. We first look again at the situation studied in the previous section. In that case we ¯ had that the mechanical connection was G-equivariant, so that its connection form ¯ ω was Ad(G)-equivariant, i.e., ¯ · ω, L a∗¯ ω = Ad(a)

¯ a¯ ∈ G,

(6.2.16)

where Ada¯ · ω denotes the 1-form in M¯ with values in h given by (Ada¯ · ω)( X¯ ) = Ada¯ (ω( X¯ )).  ¯ = ∇¯ − S¯ associated Recall that by Proposition 1.4.12, the canonical connection ∇ ¯ ⊕ k¯ at x¯ is characterized by the following to the reductive decomposition g¯ = m  ¯ along the ¯ the parallel displacement with respect to ∇ property: for every ξ¯ ∈ m, ¯ from x¯ to γ(t), is equal to (L exp(t ξ) curve γ(t) = L exp(t ξ) ¯ ( x), ¯ )∗ . This implies

6.2 Reduction in a Principal Bundle

  ∇¯ X¯ ω x¯ = adμ¯ −1 ( X¯ ) · ωx¯ ,

179

X¯ ∈ Tx¯ M¯

 ¯ ¯ under G, and, by the invariance of ∇   ∇¯ X¯ ω y¯ = adμ¯ −1 ¯ · ω y¯ , y¯ ( X )

¯ X¯ ∈ Ty¯ M, ¯ y¯ ∈ M,

(6.2.17)

 ¯ is proportional to itself that is, the covariant derivative of ω by the connection ∇ by a suitable linear operator. We note that, in particular, if H is contained in the ¯ the linear operator is null, hence ω is invariant under G. ¯ If H is just a center of G, normal subgroup not contained in the center, the condition (6.2.17) comes from the equivariance of ω. We now turn to the general case. Namely, we begin with any homogeneous pseudo¯ g) Riemannian structure S¯ on a pseudo-Riemannian manifold ( M, ¯ where a group H ¯ ¯ acts by isometries, and such that M → M/H = M is a principal bundle with nondegenerate fibers. The preceding discussion suggests that for the reduction procedure to be possible, we may need to add an algebraic condition for the mechanical connection analogous to (6.2.17). This suggestion is correct, as the following result shows. ¯ g) Theorem 6.2.1 Let ( M, ¯ be a pseudo-Riemannian manifold. Let π : M¯ → M be a principal bundle with non-degenerate fibers, and with structure group H acting on M¯ by isometries. Let ω be the connection 1-form of the mechanical connection. Let S¯ be an H -invariant homogeneous pseudo-Riemannian structure with associated  ¯ such that AS-connection ∇  ¯ =α·ω (6.2.18) ∇ω for some 1-form α on M¯ taking values in End(h). Then, the tensor field S defined by   S X Y = π∗ S¯ X H Y H ,

X, Y ∈ X(M),

(6.2.19)

is a homogeneous pseudo-Riemannian structure on (M, g), where g is the reduced Riemannian metric. Proof We first observe that the H -invariance of S¯ implies that S¯ X H Y H is projectable, so that S is well defined. Since the structure group H acts by isometries, the Levi ¯ = ∇¯ − S¯ is also Civita connection ∇¯ of g¯ is H -invariant, which implies that ∇ H -invariant. From (6.2.18) we have that for all X, Y ∈ X(M),      ¯ X H ω (Y H ) = −α(X H ) · ω(Y H ) = 0, ¯ X H Y H ) = X H ω(Y H ) − ∇ ω(∇  ¯ X H Y H is horizontal. If we define ∇  = ∇ − S, ∇ being the Levi-Civita so that ∇  H X H Y H . Hence by H -invariance, connection of g, then ∇¯ X H Y projects to ∇    ¯ XH Y H. X Y H = ∇ ∇

(6.2.20)

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6 Reduction of Homogeneous Structures

We now prove that S satisfies  = 0, ∇g

 = 0, R ∇

 S = 0, ∇

(6.2.21)

 (the curvature tensor of ∇)  and S are seen as tensor fields of type (0, 4) where R and (0, 3) respectively. Recall that by Proposition 2.2.2, the above equations are equivalent to the Ambrose–Singer equations. For the first equation, taking into account (6.2.20), we have 

 U X, Y ) ◦ π − g(X, ∇ U Y ) ◦ π U g (X, Y ) ◦ π = U (g(X, Y )) ◦ π − g(∇ ∇      H  H H H H H U X ) , Y − g¯ X , (∇ U Y ) H = U g(X , Y ) − g¯ (∇ ¯         ¯ H X H , Y H − g¯ X H , ∇ ¯ HYH = U H g(X ¯ H , Y H ) − g¯ ∇    ¯ U H g¯ (X H , Y H ) = ∇

U

U

 ¯ g¯ = 0. For the third equation, let  = 0 since ∇ for U, X, Y ∈ X(M), whence ∇g U, X, Y, Z ∈ X(M). Again by (6.2.20), we have 

U S ∇

 XY Z

  ◦ π = U (S X Y Z ) ◦ π − S∇U X Y Z ◦ π     − S X ∇U Y Z ◦ π − S X Y ∇U Z ◦ π   = U H S¯ X H Y H Z H − S¯(∇ X ) H Y H Z H U

− S¯ X H (∇U Y ) H Z H − S¯ X H Y H (∇U Z ) H   = U H S¯ X H Y H Z H − S¯ ¯ H H H ∇U H X Y Z

− S¯ X H ∇¯ H Y H Z H − S¯ X H Y H ∇U H Z H U    = ∇¯ U H S¯ H H H , X Y Z

 ¯ S¯ = 0. We now prove the second equation in (6.2.21). Let  which vanishes as ∇ R¯ be  ¯ From (6.2.20), for X, Y, Z ∈ X(M) we first have the curvature tensor of ∇.    X Y Z ) H = ∇ ¯ X H (∇ ¯ Y H (∇ ¯ [X,Y ] H Z H Y Z ) H − ∇ X Z ) H − ∇ (R      ¯ X H (∇ ¯ YH ZH) − ∇ ¯ Y H (∇ ¯ XH Z H) − ∇ ¯ [X H ,Y H ]h Z H =∇       ¯ X H (∇ ¯ YH ZH) − ∇ ¯ Y H (∇ ¯ XH Z H) − ∇ ¯ [X H ,Y H ] Z H + ∇ ¯ [X H ,Y H ]v Z H =∇  ¯ [X H ,Y H ]v Z H . = R¯ X H Y H Z H + ∇

6.2 Reduction in a Principal Bundle

181

For X, Y, Z , W ∈ X(M) one has    X Y Z W ◦ π =  ¯ [X H ,Y H ]v Z H , W H R R¯ X H Y H Z H W H + g¯ ∇    ¯ (X H ,Y H )∗ Z H , W H , = R¯ X H Y H Z H W H − g¯ ∇

(6.2.22)

where by (X H , Y H )∗ we denote the fundamental vector field associated to ¯ let I(x) ¯ be the non-degenerate bilinear form in (X H , Y H ) ∈ h. For any x¯ ∈ M, h defined as ξ, η ∈ h. I(x)(ξ, ¯ η) = g(ξ ¯ x∗¯ , ηx∗¯ ), Applying Koszul’s formula for ∇¯ and taking into account that [X H , ξ ∗ ] = 0 for any X ∈ X(M) and ξ ∈ h, we have        ¯ (X H ,Y H )∗ Z H , W H = g¯ ∇¯ (X H ,Y H )∗ Z H , W H − g¯ S¯(X H ,Y H )∗ Z H W H g¯ ∇ =

 1  I (X H , Y H ), (Z H , W H ) − S¯(X H ,Y H )∗ Z H W H , 2

 ¯ Applying the previous equation and (6.2.22), a direct ¯ = ∇¯ − S. where, as usual, ∇ computation shows that      ¯  ¯ UH  U R ◦ π = R ∇ ∇ XY ZW

(6.2.23)

XHYH ZHWH

  1 − U H I((X H , Y H ), (Z H , W H )) 2  1   ¯ U H X H , Y H ), (Z H , W H ) + I (∇ 2  1   ¯ U H Y H ), (Z H , W H ) + I (X H , ∇ 2  1   ¯ UH Z H, W H) + I (X H , Y H ), (∇ 2  1   ¯ UH W H) + I (X H , Y H ), (Z H , ∇ 2   + U H S¯(X H ,Y H )∗ Z H W H − S¯(∇¯ H X H ,Y H )∗ Z H W H U

− S¯(X H ,∇¯

UH

Y H )∗ Z H W H

− S¯(X H ,Y H )∗ Z H (∇¯

UH

− S¯(X H ,Y H )∗ (∇¯

WH)

UH

Z H )W H

.

On the other hand, by (6.2.18),         ¯ Y H ω (X H ) = dω(X H , Y H ) − ω  ¯ X H ω (Y H ) − ∇ 0= ∇ T¯ X H Y H ,

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6 Reduction of Homogeneous Structures

 ¯ Since ( X¯ , Y¯ ) = dω( X¯ h , Y¯ h ), by definition, where  T¯ is the torsion tensor field of ∇. we have   (X H , Y H ) = ω  T¯ X H Y H . Making use of

 T¯ X H Y H = S¯Y H X H − S¯ X H Y H ,

 ¯ S¯ = 0, one has that together with (6.2.18) and ∇    ¯ U H  (X H , Y H ) = α(U H ) · (X H , Y H ). ∇

(6.2.24)

Now, from ω([X H , Y H ]v ) = −(X H , Y H ) and (6.2.18) we get      ¯ U H [X H , Y H ]v = −U H (X H , Y H ) + α(U H ) · (X H , Y H ), ω ∇

(6.2.25)

so that       ¯ U H [X H , Y H ]v , [Z H , W H ]v U H I (X H , Y H ), (Z H , W H ) = g¯ ∇    ¯ U H [Z H , W H ]v + g¯ [X H , Y H ]v , ∇   =I U H (X H , Y H ), (Z H , W H )   − I α(U H ) · (X H , Y H ), (Z H , W H )   + I (X H , Y H ), U H (Z H , W H )   − I (X H , Y H ), α(U H ) · (Z H , W H ) . In addition, by (6.2.24) and (6.2.25),      ¯ U H Y H ) = −ω ∇ ¯ U H [X H , Y H ]v , ¯ U H X H , Y H ) + (X H , ∇ (∇ whence    ¯ U H Y H )∗ = ∇ ¯ U H (X H , Y H )∗ , ¯ U H X H , Y H )∗ + (X H , ∇ (∇

(6.2.26)

 ¯ U H [X H , Y H ]v is vertical. Substituting the preceding formulas and grouping since ∇ terms, (6.2.23) becomes      ¯  ¯ UH  U R ∇ ◦ π = R ∇ XY ZW XHYH ZHWH  1  + I (∇¯ U H )(X H , Y H ), (Z H , W H ) 2

6.2 Reduction in a Principal Bundle

183

 1  I α(U H ) · (X H , Y H ), (Z H , W H ) 2  1   ¯ U H )(Z H , W H ) + I (X H , Y H ), (∇ 2  1  − I (X H , Y H ), α(U H ) · (Z H , W H )  2  ¯ U H S¯ . − ∇

−

(X H ,Y H )∗ Z H W H

 = 0. This completes U R Taking into account (6.2.24) and (6.2.26), we deduce that ∇ the proof of Theorem 6.2.1.  Remark 6.2.2 In the situation of Theorem 6.2.1, if the homogeneous structure tensor ¯ one could ask whether S¯ is associated to a Lie group G¯ acting by isometries in M, ¯ H can be seen as a normal subgroup of G and if the projected tensor S is associated ¯ to the group G = G/H . The answer is not necessarily positive. More precisely, for a connected, simply ¯ if we construct the group G¯ from S¯ followconnected and complete manifold M, ing the proof of the Ambrose–Singer Theorem (see Sect. 2.2), one can see that the normality of H is not guaranteed and the group G¯ needs not project to the group G constructed in M from S by the same method. An example of this situation will be shown in Sect. 6.2.1 (Hopf fibration, case λ = 0). In the next result we explore the preservation of the homogeneous structure classes of Proposition 4.1.2 under the projection given in Theorem 6.2.1. Proposition 6.2.3 The classes {0}, S1 , S3 , S1 ⊕ S2 and S1 ⊕ S3 are invariant under the reduction procedure. Proof By the expression of the reduced structure tensor (6.2.19), it is obvious that if ¯ Recall S¯ = 0 then S = 0. Let S¯ be a tensor field in the class S1 given a vector field ξ.  ¯ Since S¯ is H -invariant the vector field ξ¯ is also that ξ¯ is parallel with respect to ∇. ¯ We have ξ H = ξ¯h and H -invariant, and then projectable. Let ξ be the projection of ξ. then ¯ Z H ) − g(Y ¯ g(X ¯ H , Y H )g( ¯ ξ, ¯ H , ξ) ¯ H, ZH) S X Y Z ◦ π = g(X = g(X ¯ H , Y H )g(ξ ¯ H , Z H ) − g(Y ¯ H , ξ H )g(X ¯ H, ZH) = g(X, Y )g(ξ, Z ) ◦ π − g(Y, ξ)g(X, Z ) ◦ π, whence S ∈ S1 . By a similar argument one proves that the class S1 ⊕ S2 is also invariant. Regarding the classes S3 and S1 ⊕ S3 , they are characterized by algebraic conditions which are clearly preserved by the reduction formula (6.2.19).  The remaining classes S2 and S2 ⊕ S3 are characterized by conditions involving the vanishing of the trace c12 . Let x ∈ M and {ei }i=1,...,n be an orthonormal basis of Tx M. For X ∈ Tx M,

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6 Reduction of Homogeneous Structures

c12 (S)(X ) =



εi Sei ei X =



i

H ¯ εi S¯eiH eiH X H = c12 ( S)(X )−



i

ε j S¯ V j V j X H ,

j

(6.2.27) ¯ x¯ ∈ π −1 (x). where {V j } j=1,...,r is an orthonormal basis of the vertical subspace Vx¯ M,  ¯ = ∇¯ − S¯ one has From ∇   ¯ V j V j , X H ) = −g( ¯ V j X H , V j ), ¯ ∇¯ V j V j , X H ) − g( ¯ ∇ ¯ ∇¯ V j X H , V j ) + g( ¯ ∇ S¯ V j V j X H = g( where the vectors V j , j = 1, ..., r , are extended to unitary and respectively orthogonal vertical vector fields. As from (6.2.18) we have  ¯ V j X H ) = V j (ω(X H )) − α(V j ) · ω(X H ) = 0, ω(∇ the second summand in the formula for S¯ V j V j X H is zero, and then H ¯ ∇¯ V j X H , V j ) = g(B(V ¯ S¯ V j V j X H = −g( j , V j ), X ),

where B denotes the second fundamental form of the fiber π −1 (x) at x. ¯ Inserting this into (6.2.27) we obtain that H ¯ c12 (S)(X ) = c12 ( S)(X )−



H H ¯ ε j g(B(V ¯ ¯ X H ), j , V j ), X ) = c12 ( S)(X ) − g(H,

j

where H denotes the mean curvature operator (trace of B) of the fiber at x. ¯ We have then proved the following. Proposition 6.2.4 The classes S2 and S2 ⊕ S3 are invariant under reduction if and ¯ g) only if the fibers of the principal bundle π : ( M, ¯ → (M, g) are minimal Rieman¯ nian submanifolds of ( M, g). ¯ Remark 6.2.5 Propositions 6.2.3 and 6.2.4 (when the fibers are minimal) do not exclude that a homogeneous structure S¯ in a class Si ⊕ S j reduces to a tensor S belonging to a smaller class Si or S j , or even to the null tensor. We shall show some examples of these situations in the next section.

6.2.1 Examples (a) The real hyperbolic space RH (n) → RH (n − 1) The real n-dimensional hyperbolic space (RH (n), g): ¯ RH (n) = {( y¯ 0 , y¯ 1 , . . . , y¯ n−1 ) ∈ Rn / y¯ 0 > 0},

6.2 Reduction in a Principal Bundle

g¯ =

185 n−1 1 j d y¯ ⊗ d y¯ j , ( y¯ 0 )2 j=0

is a symmetric space, as RH (n) = SO(n − 1, 1)/O(n − 1). For the sake of simplicity we confine ourselves to the case n = 4, the generalization for arbitrary n being straightforward. Taking into account the Iwasawa decomposition S O(3)AN of the connected component of the identity of the isometry group, all the subgroups of isometries acting transitively on RH (4) are of the type G¯ = F N , where F is a connected closed subgroup of SO(3)A, with nontrivial projection to A (see [43]). In particular, we now consider G¯ = SO(2)AN . Geometrically, if we see SO(2) as the isotropy group of the point x¯ = (1, 0, 0, 0), its Lie algebra k¯ consists of infinitesimal rotations generated by ∂ ∂ r = y¯ 2 3 − y¯ 3 2 . ∂ y¯ ∂ y¯ ¯ = a ⊕ n, which is the Lie algebra of the factor AN , gives a reductive The subspace m decomposition ¯ ¯ ⊕ k. g¯ = m Let a ∈ a, n 1 , n 2 , n 3 ∈ n be the generators of a and n respectively, where n i is the infinitesimal translation in RH (4) in the direction of ∂/∂ y¯ i . All other reductive ¯ ϕ + k¯ associated to g¯ and k¯ are given by the graph of any decompositions g¯ = m equivariant map ϕ : m → k. As a computation shows, all these equivariant maps are of the form m→k ϕ(λ0 ,λ1 ) : a → λ0 r n 1 → λ1 r n 2 , n 3 → 0, with λ0 , λ1 ∈ R. The homogeneous structure tensors associated to this 2-parameter family of reductive decompositions are S¯ (λ0 ,λ1 )  

3 k k 0 0 2 3 1 2 3 , = ( y¯10 )3 d y ¯ ⊗ d y ¯ ∧ d y ¯ − λ d y ¯ ⊗ d y ¯ ∧ d y ¯ − λ d y ¯ ⊗ d y ¯ ∧ d y ¯ 0 1 k=1  ¯ = ∇¯ − S¯ (λ0 ,λ1 ) (where ∇¯ is the Levi-Civita connecand the canonical connection ∇ tion of g) ¯ is thus given by  ¯ ∂0 ∂0 = − 10 ∂0 , ∇ y¯  ¯ ∂0 ∂3 = − 10 ∂3 − ∇ y¯

λ0 ∂ , y¯ 0 2

 ¯ ∂0 ∂1 = − 10 ∂1 , ∇ y¯  ¯ ∂1 ∂2 = λ01 ∂3 , ∇ y¯

 ¯ ∂0 ∂2 = − 10 ∂2 + ∇ y¯  ¯ ∂1 ∂3 = − λ01 ∂2 , ∇ y¯

λ0 ∂ , y¯ 0 3

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6 Reduction of Homogeneous Structures

where ∂k stands for

∂ . ∂ y¯ k

Let H  R be the subgroup of RH (4) given by H = {(1, λ, 0, 0)/λ ∈ R}.

We take the H -principal bundle RH (4) → RH (3) ( y¯ 0 , y¯ 1 , y¯ 2 , y¯ 3 ) → ( y¯ 0 , y¯ 2 , y¯ 3 ) with mechanical connection form ω = d y¯ 1 . We have that  ¯ = ∇ω



1 0 d y¯ · ω, y¯ 0

where we have identified h  R and End(h)  R. From Theorem 6.2.1, the family of homogeneous structure tensors S¯ (λ0 ,λ1 ) can be reduced to RH (3). If (y 0 , y 1 , y 2 ) are the standard coordinates of RH (3), these reduced homogeneous structure tensors form a one-parameter family S

λ0

1 = 0 3 (y )

 2

 dy ⊗ dy ∧ dy − λ0 dy ⊗ dy ∧ dy k

k

0

0

1

2

.

k=1

Note that in the expression of both S¯ (λ0 ,λ1 ) and S λ0 , the first summand is the standard S1 structure of RH (4) and RH (3), respectively. The other summands are of type S2 ⊕ S3 since they have null trace, which makes S¯ (λ0 ,λ1 ) and S λ0 of type S1 ⊕ S2 ⊕ S3 in the generic case. In the special case λ0 = 0 we will have a reduction of the generic class S1 ⊕ S2 ⊕ S3 to the class S1 . (b) The Hopf fibration S 3 → S 2 Let S 3 ⊂ R4  C2 be the 3-sphere with its standard Riemannian metric with full isometry group O(4). The natural action of U(2) in C2 defines a transitive and effective action of U(2) on S 3 given by

U(2) → SO ⎛ (4) ⎞ Re(a) −Im(a) Re(b) −Im(b)  ⎜Im(a) Re(a) Im(b) Re(b) ⎟ . ab ⎟ → ⎜ ⎝ Re(c) −Im(c) Re(d) −Im(d)⎠ cd Im(c) Re(c) Im(d) Re(d) The isotropy group at x¯ = (1, 0, 0, 0) ∈ S 3 is K¯ =

 10 ∈ U(2)/z ∈ U(1) 0z



6.2 Reduction in a Principal Bundle

187

with Lie algebra k¯ = Span



00 0i

 .

It is easy to see that the complement  ¯ = Span m

   0 1 0i i 0 , , −1 0 i 0 0 −i

¯ ⊕ k¯ a reductive decomposition. The rest of the complements m ¯ makes u(2) = m ¯ ¯ ⊕ k are obtained as the graph of Ad( K¯ )giving reductive decompositions u(2) = m ¯ One can check that these decompositions are exhausted ¯ → k. equivariant maps ϕ : m by the one-parameter family of complements  ¯ λ = Span m

    0 1 0i i 0 00 , , +λ , λ ∈ R. −1 0 i 0 0 −i 0i

From (6.1.13), the expression of the homogeneous structure tensor S¯ λ associated to each reductive decomposition computed at Tx¯ S 3 is given by ( S¯ λ )x¯ = (λ − 1)d x¯ 2 ⊗ d x¯ 3 ∧ d x¯ 4 + d x¯ 3 ⊗ d x¯ 2 ∧ d x¯ 4 − d x¯ 4 ⊗ d x¯ 2 ∧ d x¯ 3 , (6.2.28) where (x¯ 1 , x¯ 2 , x¯ 3 , x¯ 4 ) is the natural system of coordinates in R4 . Let H be the subgroup of U(2) isomorphic to U(1) given by  z0 /z ∈ U(1) . 0z

 H=

It is easy to check that H is a normal subgroup of U(2) acting freely on S 3 . Reduction by the action of H gives the Hopf fibration S 3 → S 2 with vertical and horizontal subspaces at x¯  ∂ , Vx¯ S = Span ∂ x¯2 

3

 ∂ ∂ Hx¯ S = Span . , ∂ x¯3 ∂ x¯4 

3

Since all the terms of S¯ λ have the vertical factor d x¯ 2 , it is obvious that they all reduce to the structure tensor S = 0 on S 2 , describing S 2 as a symmetric space. Note that this is what one can expect, since S 2 only admits the zero homogeneous structure tensor [116]. For the case λ = 0 one can compute the transvection algebra associated to S¯ λ , obtaining the Lie algebra of a Lie group acting transitively by isometries on S 3 . As  ¯ = ∇¯ − S¯ 0 is trivial a simple computation shows, the holonomy of the connection ∇ 3 and one obtains the reductive decomposition Te S ⊕ {0}  su(2), which describes the action of SU(2)  S 3 on itself. We then have an example of a homogeneous

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6 Reduction of Homogeneous Structures

 ¯ = α · ω as in Theorem 6.2.1 (ω being the Riemannian structure S¯ 0 satisfying ∇ω mechanical connection form of the Hopf fibration S 3 → S 2 ), but for which the structure group of the fibration (H = U(1)) cannot be seen as a normal subgroup of the group (G¯ = SU(2)) obtained from the transvection algebra. Remark 6.2.6 There are no more reducible tensors than those described above as the other groups acting transitively on S 3 are SO(4), which has no normal subgroups, and SU(2)  S 3 . In addition, this procedure can be adapted to Berger 3-spheres, for which a family of homogeneous structures is computed in [66]. All reducible structures of this family reduce to S = 0 on S 2 as expected. (c) The Hopf fibrations S 7 → S 4 and S 7 → CP 3 The groups acting isometrically and transitively on S 7 are SO(7), SU(4), Sp(2)Sp(1), U(4) and Sp(2)U(1) (see [107]). The first two groups do not have any normal subgroups and hence do not fit into the reduction scheme. The group G¯ = Sp(2)Sp(1) has the normal subgroup H = Sp(1) = SU(2), which gives the Hopf fibration S 7 → S 4 . In this case, a similar computation to the fibration S 3 → S 2 shows that the corresponding homogeneous Riemannian structures in the 7-sphere reduce to the null tensor on S 4 , the only homogeneous structure in the four dimensional sphere. We analyze the remaining two groups. Let ij denote the 4 × 4 complex matrix with 1 in the i-th row and the j-th column and the rest zeros. Let S 7 be the standard 7-sphere as a Riemannian submanifold of C4 with the usual Hermitian inner product. The standard action of the unitary group U(4) on C4 gives a transitive and effective action by isometries on S 7 . The isotropy group K¯ at x¯ = (1, 0, 0, 0) ∈ S7 is isomorphic to U(3) and we can ¯ ⊕ k¯ with decompose u(4) = m k¯ = and

  0 0 : A ∈ u(3) 0 A

¯ = Span{i11 , 1j − 1j , i(1j + 1j ), j = 1, 2, 3}. m

¯ ⊕ k¯ is the unique reductive decomposition of u(4) with One can check that u(4) = m ¯ From (6.1.13), identifying R8  C4 and taking its natural coordinates respect to k. (x¯ 1 , . . . , x¯ 8 ), the homogeneous structure tensor S¯ associated to this decomposition at Tx¯ S 7 reads S¯ x¯ = d x¯ 3 ⊗ d x¯ 2 ∧ d x¯ 4 − d x¯ 4 ⊗ d x¯ 2 ∧ d x¯ 3 + d x¯ 5 ⊗ d x¯ 2 ∧ d x¯ 6 −d x¯ 6 ⊗ d x¯ 2 ∧ d x¯ 5 + d x¯ 7 ⊗ d x¯ 2 ∧ d x¯ 8 − d x¯ 8 ⊗ d x¯ 2 ∧ d x¯ 7 . (6.2.29) As a simple computation shows, this tensor belongs to the class S2 ⊕ S3 . Let H be the subgroup of U(4) isomorphic to U(1) given by

6.2 Reduction in a Principal Bundle

189

H = {z · Id/z ∈ U(1)} , where Id is the 4 × 4 identity matrix. It is obvious that H is a normal subgroup of U(4) the action of which on S 7 is free. The reduction of S 7 by the action of H gives the Hopf fibration S 7 → CP 3 , from which the complex projective space inherits the Fubini–Study metric. The vertical and horizontal subspaces at x¯ are  ∂ , Vx¯ S = Span ∂ x¯2 

7

 ∂ ∂ Hx¯ S = Span . ,..., ∂ x¯3 ∂ x¯8 

7

As in the Hopf fibration S 3 → S 2 , the homogeneous structure tensor S¯ reduces to S = 0, describing U(4) CP 3 = U(3) × U(1) as a symmetric space. Let H denote the quaternion algebra. We now view the 7-sphere S7 =

  q1 ∈ H2 : |q1 |2 + |q2 |2 = 1 q2

as a Riemannian submanifold of H2 with the standard quaternion inner product. The group Sp(2)U(1) acts on H2 by    q z q1 q1 =A 1 , ∈ H2 , A ∈ Sp(2), z ∈ U (1), (A, z) · q2 q2 q2 z where z stands for complex conjugation. This action restricts to a transitive and effective action by isometries on S 7 . The isotropy group at x¯ = (1, 0) ∈ S 7 is K¯ =

 z 0 , z /q ∈ Sp(1), z ∈ U(1)/Z2 , 0q



which is isomorphic to Sp(1)U(1). The Lie algebra of Sp(2)U(1) is given by sp(2) ⊕ u(1), where  sp(2) = Span

    0 1 i 0 0i j0 0 j , , , , , −1 0 00 i 0 00 j 0       k0 0k 00 00 00 , , , , 00 k0 0i 0 j 0k

and u(1) = Span{i}. The isotropy algebra is thus k¯ = Span



    i 0 00 00 00 + i, , , . 00 0i 0 j 0k

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6 Reduction of Homogeneous Structures

¯ to be Taking m  Span

       0 1 i 0 0i j0 0 j k0 0k , , , , , , −1 0 00 i 0 00 j 0 00 k0

¯ ⊕ k¯ is a reductive decomposition. All other reductive we have that sp(2) ⊕ u(1) = m decompositions associated to sp(2) ⊕ u(1) and k¯ are given by a one-parameter family ¯ λ , λ ∈ R, which the graph of the Ad( K¯ )-equivariant maps of complements m are  i 0 i 0 ¯ → k,¯ where ϕλ maps ϕλ : m to λ + λi and the rest of elements of the 00 00 basis to zero. Identifying H2  R8 , the homogeneous structure tensor S¯ λ associated to each ¯ λ ⊕ k¯ is computed at Tx¯ S 7 as reductive decomposition sp(2) ⊕ u(1) = m ( S¯ λ )x¯ = d x¯ 5 ⊗ d x¯ 2 ∧ d x¯ 6 + d x¯ 5 ⊗ d x¯ 3 ∧ d x¯ 7 + d x¯ 5 ⊗ d x¯ 4 ∧ d x¯ 8 −λd x¯ 2 ⊗ d x¯ 5 ∧ d x¯ 6 + (1 + 2λ)d x¯ 2 ⊗ d x¯ 3 ∧ d x¯ 4 +λd x¯ 2 ⊗ d x¯ 7 ∧ d x¯ 8 + d x¯ 6 ⊗ d x¯ 5 ∧ d x¯ 2 + d x¯ 6 ⊗ d x¯ 3 ∧ d x¯ 8 −d x¯ 6 ⊗ d x¯ 4 ∧ d x¯ 7 + d x¯ 3 ⊗ d x¯ 2 ∧ d x¯ 4 + d x¯ 4 ⊗ d x¯ 2 ∧ d x¯ 3 −d x¯ 7 ⊗ d x¯ 3 ∧ d x¯ 5 − d x¯ 7 ⊗ d x¯ 2 ∧ d x¯ 8 + d x¯ 7 ⊗ d x¯ 4 ∧ d x¯ 6 −d x¯ 8 ⊗ d x¯ 4 ∧ d x¯ 5 + d x¯ 8 ⊗ d x¯ 2 ∧ d x¯ 7 − d x¯ 8 ⊗ d x¯ 3 ∧ d x¯ 6 . Let H = {(Id, w)/ w ∈ U(1)} ⊂ Sp(2)U(1), where Id is the identity of Sp(2), it is easy to see that H is a normal subgroup of Sp(2)U(1) isomorphic to U(1). Reduction by the action of H gives again the Hopf fibration π : S 7 → CP 3 with π(x) ¯ = [1 : 0 : 0 : 0] ∈ CP 3 . The vertical and horizontal subspaces of π at x¯ are  Vx¯ S 7 = Span

 ∂ , ∂ x¯2

 Hx¯ S 7 = Span

 ∂ ∂ . ,..., ∂ x¯3 ∂ x¯8

Let (t 1 , . . . , t 6 ) : CP 3 − {z 0 = 0} → R6 be the coordinate system around x = [1 : 0 : 0 : 0] given by              [z 0 : z 1 : z 2 : z 3 ] → Re zz01 , Im zz01 , Re zz20 , Im zz20 , Re zz03 , Im zz03 . The reduced homogeneous structure tensor S at Tx CP 3 is Sx = dt 3 ⊗ dt 1 ∧ dt 5 + dt 3 ⊗ dt 2 ∧ dt 6 + dt 4 ⊗ dt 1 ∧ dt 6 − dt 4 ⊗ dt 2 ∧ dt 5 + dt 5 ⊗ dt 2 ∧ dt 4 − dt 5 ⊗ dt 1 ∧ dt 3 − dt 6 ⊗ dt 2 ∧ dt 3 − dt 6 ⊗ dt 1 ∧ dt 4 . It is easy to check that for all λ ∈ R, S¯ λ is of type S2 ⊕ S3 but neither of S2 nor of S3 , and S is also a strict S2 ⊕ S3 structure. Note that in the latter and the former examples the class S2 ⊕ S3 is preserved by the reduction procedure. This fact is expected from

6.2 Reduction in a Principal Bundle

191

Proposition 6.2.4, since the fibers of the Hopf fibration are totally geodesic and in particular minimal Riemannian submanifolds of S 7 .

6.3 Application to Cosymplectic and Sasakian Homogeneous Structures of Linear Type We recall that a vector field X¯ on M¯ is said to be regular if it is nowhere vanishing. In that case, every point x¯ ∈ M¯ has a neighbourhood U with coordinates (x 1 , . . . , x 2n+1 ) whose intersection with any integral curve of X¯ is given by x 1 = const., . . . , x 2n+1 = const. The vector field X¯ is called strictly regular if, in addition, all the integral curves are homeomorphic. Definition 6.3.1 An almost contact structure (φ, ξ, η) is 1. (strictly) regular if the vector field ξ is (strictly) regular, and 2. invariant if φ and η are invariant under the 1-parameter group generated by ξ. Remark 6.3.2 It is easy to prove that in the cosymplectic and Sasakian cases, the invariance property is automatically satisfied, since Lξ φ = 0 = Lξ η. ¯ We denote by Let (φ, ξ, η) be a regular almost contact structure on a manifold M. ¯ M the orbit space (which is a smooth manifold) and by π : M → M the projection. The following results can be found in [99]. Theorem 6.3.3 Let (φ, ξ, η) be a strictly regular invariant almost contact structure ¯ and let H be the 1-parameter group generated by ξ. Then on M, 1. π : M¯ → M is a principal H -bundle over M, and 2. η is a connection form on π : M¯ → M. Hereafter we will only consider strictly regular invariant almost contact structures. Theorem 6.3.4 The (1, 1)-tensor field defined by   Jx X = π∗ φX H ,

X ∈ Tx M,

where X H denotes the horizontal lift of X with respect to the connection η, is an almost complex structure on M. Moreover, [φ, φ] = 0 if and only if J is a complex structure and η is a flat connection. On the other hand, [φ, φ] + 2η ⊗ ξ = 0 if and only if J is a complex structure and (J X, J Y ) = (X, Y ), where π ∗  = dη. ¯ Note that We now consider an almost contact metric structure (φ, ξ, η, g) ¯ on M.  since η = εξ , the connection η coincides with the mechanical connection on π : M¯ → M with respect to g. ¯

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6 Reduction of Homogeneous Structures

¯ Theorem 6.3.5 ([99]) Let (φ, ξ, η, g) ¯ be an almost contact metric structure on M, and let g and J be the reduced metric and almost complex structure on M. If (φ, ξ, η, g) ¯ is either cosymplectic or Sasakian, then (M, g, J ) is pseudo-Kähler. Let S¯ be a homogeneous structure, either cosymplectic or Sasakian, invariant  ¯ the linear connection associated under the 1-parameter group generated by ξ and ∇  ¯ ¯ = 0, we are in the to S. Since η defines the mechanical connection on π and ∇η H ¯ situation of Theorem 6.2.1, so that S X Y = π∗ ( S X H Y ) is a homogeneous pseudoRiemannian structure on M. ¯ Proposition 6.3.6 If S¯ is a homogeneous cosymplectic or Sasakian structure on M, then S is a homogeneous pseudo-Kähler structure on M.  = ∇ − S, where ∇ is the Levi-Civita connection of g. Then ∇ X Y = Proof Let ∇  H ¯ ¯ ¯ ¯ ¯ π∗ (∇ X H Y ). Since η(φ( X )) = 0, we have that φ( X ) is horizontal for all X ∈ X( M). We thus have 

   X (J Y ) − J ∇ X Y X J Y = ∇ ∇        ¯ X H (J Y ) H − π∗ φ ∇ ¯ XH Y H = π∗ ∇        ¯ X H φY H − φ ∇ ¯ XH Y H = π∗ ∇     ¯ XH φ Y H = π∗ ∇ =0

 J = 0. for every X, Y ∈ X(M), whence ∇



Proposition 6.3.7 If S¯ is of linear type then S is of linear type. Moreover, if S¯ is non-degenerate (resp. degenerate) then S is non-degenerate (resp. degenerate). ¯ Since S¯ is invariant Proof Let χ¯ and ζ¯ be the basic vector fields determining S. under the one-parameter group generated by ξ, it is easy to prove that so are χ¯ ¯ In particular, there are vector fields χ and ζ on M such that π∗ (χ) and ζ. ¯ = χ and ¯ π∗ (ζ) = ζ. A simple inspection shows that S X Y = π∗ ( S¯ X H Y H ) takes the form (4.2.3) for the vector fields χ and ζ, so that S is of linear type. It is obvious that g( ¯ χ, ¯ χ) ¯ = g(χ, χ).  In view of the previous propositions, we can study invariant homogeneous cosymplectic and Sasakian structures of linear type via the reduction procedure and making use of the results on homogeneous pseudo-Kähler structures of linear type obtained in Chap. 5. ¯ g, Proposition 6.3.8 Let ( M, ¯ φ, ξ, η) be a cosymplectic manifold of dimension 2n + ¯ 5 admitting an invariant homogeneous cosymplectic structure of linear type S. ¯ g, 1. If S¯ is non-degenerate, then ζ¯ = 0 and ( M, ¯ φ, ξ, η) is of constant φ-sectional curvature k = −4g(χ, ¯ χ). ¯

6.3 Application to Cosymplectic and Sasakian …

193

¯ g, 2. If S¯ is degenerate, then ζ¯ = λχ¯ with λ ∈ {0, 1/2}, and ( M, ¯ φ, ξ, η) is locally isometric to Cn+2 × R with metric g¯ = g + εdt 2 and its standard cosymplectic structure, where g is the metric on Cn+2 given by (5.2.26) for  = −1. Proof 1. Since the reduced homogeneous structure S is a non-degenerate homogeneous pseudo-Kähler structure of linear type, we have by Proposition 5.2.3 that (M, g, J ) has constant holomorphic sectional curvature c = −4g(π∗ χ, ¯ π∗ χ). ¯ Let X¯ be a vector in the contact distribution. By the curvature formulas for pseudoRiemannian submersions [101], we have      3  K¯ X¯ , φ X¯ = K π∗ X¯ , π∗ φ X¯ + g¯ [ X¯ , φ X¯ ]V , [ X¯ , φ X¯ ]V , 4 where X¯ is unitary and K¯ and K denote sectional curvatures on M¯ and M respectively. ¯ g, As ( M, ¯ φ, ξ, η) is cosymplectic, the curvature of the connection η vanishes, so that [ X¯ , φ X¯ ]V = 0. We thus conclude that       K¯ X¯ , φ X¯ = K π∗ X¯ , π∗ φ X¯ = K π∗ X¯ , J π∗ X¯ ¯ π∗ χ) ¯ = −4g( ¯ χ, ¯ χ). ¯ = −4g(π∗ χ,     ¯ g¯ = 0, and ∇ ¯ S¯ = 0 implies ∇ ¯ χ¯ = 0, where ∇ ¯ Note that g( ¯ χ, ¯ χ) ¯ is constant since ∇ ¯ is the ASK-connection associated to S. 2. Since S is a degenerate homogeneous pseudo-Kähler structure of linear type ¯ we have by Corollary 5.2.7 that π∗ ζ¯ = determined by the vector fields π∗ χ¯ and π∗ ζ, ¯ ¯ as ζ¯ and χ¯ are basic by definition. λπ∗ χ¯ with λ ∈ {0, 1/2}. This implies ζ = λχ, Let now H be the one-parameter group generated by ξ. Since the manifold ¯ g, ( M, ¯ φ, ξ, η) is cosymplectic, the curvature of the connection η on π : M¯ → M vanishes. This implies that for every p¯ ∈ M¯ we can choose a trivialization  : ¯ = Tπ( p) ¯ = (π( p), ¯ e), such that ∗ (Hq¯ M) π −1 (U ) → U × H with ( p) ¯ U + {0} for −1 ¯ all q¯ ∈ π (U ), where e is the neutral element of H and Hq¯ M is the horizontal subspace at q¯ with respect to η, which coincides with the contact distribution Span{ξ}⊥ . Taking a coordinate t ∈ (−δ, δ) on a neighbourhood of H around e, we obtain a diffeomorphism  : V → U × (−δ, δ), where V is a certain neighbourhood around ¯ = gU + εdt 2 , where gU is the reduced p. ¯ By construction we have that ( −1 )∗ (g) metric g on M restricted to U . Reducing U if necessary we have that gU is holomorphically isometric to an open set W of Cn+2 with metric (5.2.26) for  = −1. Finally, it is easy to see that the cosymplectic structure on M¯ transforms by  into the standard cosymplectic structure of Cn+2 × R restricted to the open set W × (−δ, δ).  ¯ g, Corollary 6.3.9 Let ( M, ¯ φ, ξ, η) be a cosymplectic manifold of dimension 2n + 5 admitting an invariant degenerate homogeneous cosymplectic structure of linear type. Then g¯ is Ricci-flat and the holonomy algebra is given by the one-dimensional Lie algebra

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6 Reduction of Homogeneous Structures

⎞ i i 0 hol(g) ¯ ∼ = R ⎝ −i −i 0 ⎠ + {0} ⊂ su(1, 1) + {0} ⊂ Lie(U ( p, q) × 1), 0 0 0n ⎛

with p + q = n + 2. Proof This result follows immediately from the fact that g is Ricci-flat and has holonomy algebra ⎛ ⎞ i i 0 hol(g) ∼ = R ⎝ −i −i 0 ⎠ . 0 0 0n  ¯ g, Proposition 6.3.10 Let ( M, ¯ φ, ξ, η) be a Sasakian manifold admitting an invari¯ ant homogeneous Sasakian structure of linear type S. ¯ g, 1. If S¯ is non-degenerate, then ζ¯ = 0 and ( M, ¯ φ, ξ, η) is of constant φ-sectional curvature k = −4g(χ, ¯ χ) ¯ + 3. 2. If S¯ is degenerate, then ζ¯ = λχ¯ with λ ∈ {0, 1/2}, and there is a set of coordinates {z 1 , z 2 , w 1 , w 2 , x a , y a , t} such that the metric takes the form g¯ = g + εηdt, where g is given by (5.2.26) for  = −1. Proof 1. Since the reduced homogeneous structure S is a non-degenerate homogeneous pseudo-Kähler structure of linear type, we have by Proposition 5.2.3 that ¯ π∗ χ). ¯ Let (M, g, J ) has constant holomorphic sectional curvature c = −4g(π∗ χ, X¯ be a vector in the contact distribution. By the curvature formulas for pseudoRiemannian submersions [101]      3  K¯ X¯ , φ X¯ = K π∗ X¯ , π∗ φ X¯ + g¯ [ X¯ , φ X¯ ]V , [ X¯ , φ X¯ ]V , 4 where K¯ and K denote sectional curvatures on M¯ and M respectively, and X¯ is ¯ g, unitary. As ( M, ¯ φ, ξ, η) is Sasakian, the curvature form of the connection η is given by dη = , so that [ X¯ , φ X¯ ]V = 2dη( X¯ , φ X¯ ) = 2( X¯ , φ X¯ ). We thus conclude that     K¯ X¯ , φ X¯ = K π∗ X¯ , J π∗ X¯ + 3( X¯ , φ X¯ )2 = −4g(π∗ χ, ¯ π∗ χ) ¯ + 3 = −4g( ¯ χ, ¯ χ) ¯ + 3. Note that by the same reason as before, g( ¯ χ, ¯ χ) ¯ is constant.

6.3 Application to Cosymplectic and Sasakian …

195

2. Since S is a degenerate homogeneous pseudo-Kähler structure of linear type ¯ we have by Corollary 5.2.7 that π∗ ζ¯ = determined by the vector fields π∗ χ¯ and π∗ ζ, ¯ λπ∗ χ¯ with λ ∈ {0, 1/2}, so that ζ¯ = λχ. ¯ we consider a Let now H be the 1-parameter group generated by ξ. Given p¯ ∈ M, trivialization of the principal bundle π : M¯ → M, i.e., we consider a diffeomorphism ¯ = (π( p), ¯ e) and e is the neutral element of H .  : π −1 (U ) → U × H , where ( p) Let  ξ = ∗, p¯ (ξ p¯ ) ∈ h. There is an open interval (−δ, δ) such that f : (−δ, δ) → H given by f (t) = exp(t  ξ) is a diffeomorphism onto its image. We thus consider the map  −1

id× f

F : U × (−δ, δ) → U × H → π −1 (U ), which is a diffeomorphism onto a certain neighbourhood of p. ¯ Then, the pullback of the metric g¯ by F is F ∗ g¯ = gU + εF ∗ ηdt, where gU is the reduced metric g restricted to U . Reducing U if necessary, we can take coordinates {z 1 , z 2 , w 1 , w 2 , x a , y a } on U such that gU is expressed as (5.2.26) for  = −1. We have thus constructed coordinates {z 1 , z 2 , w 1 , w 2 , x a , y a , t} around p¯ with respect to which g¯ = g + εηdt, with g given by (5.2.26) for  = −1.  Note that due to the non-integrability of the contact distribution, in the previous −1 ∗ ¯ = Tπ( p) proof ∗ (Hq¯ M) ¯ U + {0} for general q¯ ∈ π (U ). This means that F η  = dt, so that the components of η with respect to {z 1 , z 2 , w 1 , w 2 , x a , y a , t} do not identically vanish. Actually, if we write η = ηz 1 dz 1 + ηz 2 dz 2 + ηw1 dw 1 + ηw2 dw 2 + ηx a d x a + η y a dy a + dt, we have to impose that dη = . Since π ∗ ω = , where ω is the symplectic form associated to g and J , we have that ∂β ηα − ∂α ηα = ωαβ ,

α, β = z 1 , z 2 , w 1 , w 2 , x a , y a ,

⎛

with

0 ⎜0 ⎜ ⎜0 ⎜ ⎜ ω = ⎜1 ⎜0 ⎜ ⎜ .. ⎝.

0 0 −1 0 0 .. .

0 1 0 b 0 .. .

−1 0 −b 0 0 .. .

0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟, ⎟ ⎟ ⎟ ⎠

... ... ... ...

0 0 0 0 

and = diag

0 −εa , a = 1, . . . , n . εa 0

Chapter 7

Where All This Fails: Non-reductive Homogeneous Pseudo-Riemannian Manifolds

As shown in the previous chapters, reductive homogeneous pseudo-Riemannian manifolds are completely characterized by the existence of a homogeneous pseudoRiemannian structure, and the geometric properties of these spaces are reflected by the properties of such a structure. A homogeneous Riemannian manifold is always reductive (see Proposition 1.4.8). Therefore, the study of homogeneous Riemannian manifolds can be completely carried out with the techniques illustrated so far. On the other hand, in proper pseudoRiemannian settings, a peculiar phenomenon occurs: there exist some homogeneous pseudo-Riemannian manifolds which are not reductive, in the sense that Definition 1.4.7 does not apply to a group G of isometries acting transitively on them: Definition 7.0.1 A pseudo-Riemannian manifold (M, g) is a non-reductive homogeneous space if there exists a group of isometries G of (M, g), acting transitively on M, but the Lie algebra g of G cannot be decomposed as g = h ⊕ m, where h is the isotropy subalgebra and m an ad(H )-invariant subspace of g. Thus, if (M, g) is non-reductive, there exists no complement m of h in g such that Ad H (m) ⊂ m. The general approach to the classification of non-reductive homogeneous pseudoRiemannian manifolds was outlined in detail in [59]. The starting point is Cartan’s method for the classification of homogeneous spaces, already used by Cartan himself to classify three-dimensional simply-connected Riemannian homogeneous spaces with a group of isometries of dimension at least four. The same method was also applied to classify four-dimensional homogeneous Riemannian [72] and pseudoRiemannian manifolds (see [80] and references therein). In principle, the local classification of homogeneous pseudo-Riemannian manifolds can be completely carried out for any given dimension n and signature ( p, q), p + q = n. In fact, it corresponds to considering all subalgebras of so( p, q), up to conjugation with respect to GL(n, R). © Springer Nature Switzerland AG 2019 G. Calvaruso and M. Castrillón López, Pseudo-Riemannian Homogeneous Structures, Developments in Mathematics 59, https://doi.org/10.1007/978-3-030-18152-9_7

197

198

7 Where All This Fails: Non-reductive Homogeneous …

Given a subalgebra h of so( p, q), assuming that h acts naturally on Rn , there is a one to one correspondence between faithful generalized modules (h, Rn ) and corresponding subalgebras of so( p, q), which leads to the determination of all the models of n-dimensional homogeneous pseudo-Riemannian manifolds of the prescribed signature. Consider a non-degenerate symmetric bilinear form η 0 on Rn , of signature ( p, q), and the Lie group O( p, q) ⊂ GL(n, R) preserving η 0 . Let (M n , g) denote a pseudo-Riemannian manifold of signature ( p, q),   O(M) = u p : Rn → T p M : g(u p (X ), u p (Y )) = η 0 (X, Y ) and π : O(M) → M the orthonormal frame bundle. Given X ∈ o( p, q), let X u∗ = d exp(t X e )) |t=0 denote the corresponding infinitesimal vector field. dt (u For all X ∈ o( p, q) and Z ∈ Tu (O(M)), the canonical Rn -valued one-form θ and the o( p, q)-valued connection one-form ω satisfy i Z θ = u −1 π∗ (Z ),

i X ∗ ω = X,

dθ = −ω ∧ θ,

where i denotes the left interior multiplication. The curvature two-form  = dω + ω ∧ ω then satisfies i X ∗  = 0,

 ∧ θ = 0,

d =  ∧ ω − ω ∧ 

and the equivariance of θ, ω,  is expressed, for any a ∈ O( p, q), by a ∗ θ = a −1 (θ), a ∗ ω = ad(a −1 )ω, a ∗  = ad(a −1 ). Given an isometry f of (M, g), its lift to the frame bundle is defined by φ f (u) = f ∗ u, u ∈ F(M). It obviously leaves invariant O(M) ⊂ L(M), and θ, ω,  are invariant under φ f . If (G/H, g) is a homogeneous pseudo-Riemannian manifold, put σ := [H ] ∈ G/H . Consider an orthonormal frame u σ ∈ O(G/H ) at σ and the linear isotropy representation ρ : H → O( p, q) defined by u σ ρ(h) = (L h )∗ u σ . Define  : G → O(G/H ) by (k) = k∗ u σ , for all k ∈ G. Then, π ◦  : G → G/H coincides with the canonical projection from G to G/H . Moreover, since  is equivariant with respect to both the left action of G on G and the linear isotropy representation, it satisfies (kh) = (k)ρ(h),

(k1 k2 ) = φk1 ◦ (k2 ).

One then introduces on the Lie algebra g of G the G-invariant forms θˆ :=  ∗ θ,

ωˆ :=  ∗ ω,

ˆ :=  ∗  

7 Where All This Fails: Non-reductive Homogeneous …

199

and obtains the following. Lemma 7.0.2 Let (G/H, g) be an n-dimensional homogeneous pseudo-Riemannian manifold. Denote by g and h the Lie algebras of G and H respectively. There exists a monomorphism ρ∗ : h → o( p, q), an Rn -valued form θˆ on g, and an o( p, q)-valued one-form ωˆ on g, such that (i) ker θˆ = h,

(ii) ω| ˆ h = ρ∗ ,

ˆ (iii) d θˆ = −ωˆ ∧ θ.

ˆ = d ωˆ + ωˆ ∧ ωˆ satisfies Moreover,  ˆ = 0 (X ∈ h), iX 

ˆ ∧ θˆ = 0, 

ˆ = ˆ ∧ ωˆ − ωˆ ∧ . ˆ d

The above result can be partially inverted in the following way. Lemma 7.0.3 Let h be a Lie algebra and ρ∗ : h → o( p, q) a monomorphism. Consider g := h ⊕ Rn , and suppose there exist an Rn -valued form θ on g, and an o( p, q)valued one-form ω on g, satisfying (i) ker θ = h,

(ii) ω|h = ρ∗ ,

(iii) i X (dω + ω ∧ ω) = 0,

for all X ∈ h, where dθ = −ω ∧ θ. If  = dω + ω ∧ ω satisfies  ∧ θ = 0,

d =  ∧ ω − ω ∧ ,

then g is a Lie algebra, with brackets described by α([X, Y ]) = dα(X, Y ), for all α ∈ g∗ and X, Y ∈ g. Thus, the classification of homogeneous pseudo-Riemannian manifolds, of dimension n and signature ( p, q), corresponds to the classification (up to conjugation) of all subalgebras h ⊂ o( p, q) satisfying the conditions of the above lemma. In particular, in order to identify the non-reductive examples, one needs to discard all the reductive homogeneous pseudo-Riemannian manifolds, which are characterized by the existence of a G-invariant connection (Theorem 2.2.1 of Ambrose and Singer), that is, to detect all the Lie algebras h, up to conjugacy, which satisfy Lemma 7.0.3 but do not admit any G-invariant connection. Clearly, the computational difficulty of the classification problem following Cartan’s method is strongly influenced by the dimension. In this chapter we confine ourselves to low dimensional spaces, more precisely, to dimension four, the first dimension with non-reductive examples. There is a vast field of work to do in higher dimensions, where, to our knowledge, no corresponding classifications have been obtained so far.

200

7 Where All This Fails: Non-reductive Homogeneous …

7.1 Classification and Invariant Metrics in Dimension Four In dimension two and three, all homogeneous pseudo-Riemannian manifolds are reductive. In fact, the two-dimensional case follows from the fact that if O( p, q)/ρ(H ) is reductive homogeneous, so is G/H (see Lemma 4.2 and Corollary 4.5 in [59]). The three-dimensional case was proved in Theorem 2.1 of [59]. Alternatively, it also follows from Theorem 4.3.3, since Lorentzian symmetric spaces and Lorentzian Lie groups are the models of all three-dimensional homogeneous Lorentzian manifolds. Non-reductive examples show up starting from dimension four. The following classification of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds, obtained following the procedure we outlined above, was given by Fels and Renner [59]. Theorem 7.1.1 Let (G/H, g) be a four-dimensional homogeneous pseudoRiemannian manifold, where H is connected. If G/H is not reductive, then the Lie algebra pair (g, h) is isomorphic to one of the following: (I) Cases with Both Lorentzian and Neutral Metrics (A1) g = a1 is the decomposable 5-dimensional Lie algebra sl(2, R) ⊕ s(2), where s(2) is the 2-dimensional solvable algebra. There exists a basis {e1 , ..., e5 } of a1 such that the non-zero products are [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 , [e4 , e5 ] = e4 , and the isotropy subalgebra is h = Span{h 1 = e3 + e4 }. (A2) g = a2 is the one-parameter family of 5-dimensional Lie algebras A5,30 of [102]. There exists a basis {e1 , ..., e5 } of a2 such that the non-zero products are [e2 , e5 ] = αe2 , [e1 , e5 ] = (α + 1)e1 , [e2 , e4 ] = e1 , [e3 , e5 ] = (α − 1)e3 , [e4 , e5 ] = e4 , [e3 , e4 ] = e2 , for any value of α ∈ R, and the isotropy is h = Span{h 1 = e4 }. (A3) g = a3 is one of the 5-dimensional Lie algebras A5,37 or A5,36 in [102]. There exists a basis {e1 , ..., e5 } of a3 such that the non-zero products are [e1 , e4 ] = 2e1 , [e2 , e3 ] = e1 , [e2 , e4 ] = e2 , [e2 , e5 ] = −εe3 , [e3 , e4 ] = e3 , [e3 , e5 ] = e2 , with ε = 1 for A5,37 and ε = −1 for A5,36 , and the isotropy is h = Span{h 1 = e3 }. (II) Cases with Lorentzian Metrics Only (A4) g = a4 is the 6-dimensional Schrödinger Lie algebra sl(2, R)  n(3), where n(3) is the 3-dimensional Heisenberg algebra. There exists a basis {e1 , ..., e6 } of a4 , where the non-zero products are

7.1 Classification and Invariant Metrics in Dimension Four

201

[e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 , [e1 , e4 ] = e4 , [e3 , e4 ] = e5 , [e4 , e5 ] = e6 , [e1 , e5 ] = −e5 , [e2 , e5 ] = e4 , and the isotropy is h = Span{h 1 = e3 + e6 , h 2 = e5 }. (A5) g = a5 is the 7-dimensional Lie algebra sl(2, R)  A14,9 , with A14,9 as in [102]. It admits a basis {e1 , ..., e7 } such that the non-zero products are [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e1 , e5 ] = −e5 , [e1 , e6 ] = e6 , [e3 , e6 ] = e5 , [e4 , e7 ] = 2e4 , [e2 , e3 ] = e1 , [e2 , e5 ] = e6 , [e6 , e7 ] = e6 . [e5 , e6 ] = e4 , [e5 , e7 ] = e5 , The isotropy is h = Span{h 1 = e1 + e7 , h 2 = e3 − e4 , h 3 = e5 }. (III) Cases with Neutral Metrics Only (B1) g = b1 is the 5-dimensional Lie algebra sl(2, R)  R2 , admitting a basis {e1 , ..., e5 }, where the non-zero products are [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 , [e1 , e4 ] = e4 , [e3 , e4 ] = e5 , [e1 , e5 ] = −e5 , [e2 , e5 ] = e4 , and the isotropy is h = Span{h 1 = e3 }. (B2) g = b2 is the 6-dimensional Schrödinger Lie algebra a4 = sl(2, R)  n(3), but with isotropy h = Span{h 1 = e3 − e6 , h 2 = e5 }. (B3) g = b3 is the 6-dimensional Lie algebra (sl(2, R)  R2 ) × R. It admits a basis {u 1 , ..., u 4 , h 1 = u 5 , h 2 = u 6 } such that h = Span{h 1 , h 2 } and the non-zero products are [h 1 , u 2 ] = u 1 , [h 1 , u 3 ] = −u 4 , [h 2 , u 2 ] = −2h 2 , [h 2 , u 3 ] = −u 2 , [u 2 , u 3 ] = −2u 3 , [h 2 , u 4 ] = u 1 , [u 1 , u 2 ] = −u 1 , [u 1 , u 3 ] = u 4 , [u 2 , u 4 ] = −u 4 . Remark 7.1.2 The Lie algebra B3 corresponds to the case 2.51 .2 in the work of Komrakov [80], as explained in [59, p. 302]. In Theorem 7.1.1 we report this Lie algebra as it appears in [80], since the case listed in [59, Theorem 2.4] does not correspond to it. Some corrections were also needed and have been made for the Lie brackets of the Lie algebra B1. The following Table 7.1 shows the correspondence between four-dimensional pseudo-Riemannian non-reductive spaces, as classified in [59] into types A1–A5 and B1–B3, and the corresponding cases in [80]. The following result, also proved in [59], clarifies the structure of the cases classified in Theorem 7.1.1. Theorem 7.1.3 Let (G/H, g) be a simply-connected non-reductive pseudoRiemannian homogeneous space of dimension four. Then

202

7 Where All This Fails: Non-reductive Homogeneous …

Table 7.1 Non-reductive pseudo-Riemannian four-manifolds

Non-reductive space [59] A1 A2 A3 A4 A5 B1 B2 B3

Komrakov’s classification [80] 1.41 .1 1.41 .2 1.41 .3 and 1.41 .4 2.52 .1 3.22 .1 1.31 .1 2.51 .1 2.51 .2

(i) G/H is diffeomorphic to R4 , and (ii) if G is the full isometry group then G/H is equivalent to one of the cases in Theorem 7.1.1, excluding A5. Conversely, for any Lie algebra pair in Theorem 7.1.1 except A5, there exists a pseudo-Riemannian metric on R4 such that: its isometry group acts transitively, the Lie algebra of that isometry group is g and the isotropy subalgebra is (conjugate to) h. We now explain how to deduce an explicit description of the homogeneous metrics of the above examples, following the method outlined by Komrakov in [80]. Let M = G/H again denote a homogeneous manifold, with H connected, and g and h the Lie algebra of G and the isotropy subalgebra respectively. The quotient m = g/h can be identified with a subspace of g complementary to h. It is crucial to emphasize here that m need not be ad H -invariant. The pair (g, h) uniquely defines the isotropy representation ρ : h → gl(m),

ρ(x)(y) = [x, y]m for all x ∈ h, y ∈ m.

Consider a basis {h 1 , ..., h r , u 1 , ..., u n } of g, where {h j } and {u i } are bases of h and m, respectively. Any bilinear form g on m is completely determined by the matrix, which we denote again by g, of its components with respect to the basis {u i }. Correspondingly, for any x ∈ h, we denote by ρ(x) both the isotropy representation of x and the matrix which describes it with respect to {u i }. Then, g is invariant if and only if t ρ(x) · g + g · ρ(x) = 0 for all x ∈ h. In particular, invariant pseudo-Riemannian metrics on the homogeneous space M = G/H are in one to one correspondence with non-degenerate invariant symmetric bilinear forms g on m. For the cases classified in Theorem 7.1.1, denoting by the symmetric tensor product, we have the following (see [27, 33]).

7.1 Classification and Invariant Metrics in Dimension Four

203

Theorem 7.1.4 Let (G/H, g) denote a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold. For each of the cases listed in Theorem 7.1.1, let m = Span{u 1 , u 2 , u 3 , u 4 } denote a complement to the Lie subalgebra h in g, {θ1 , θ2 , θ3 , θ4 } the basis of m∗ dual to {u 1 , u 2 , u 3 , u 4 }. Then, invariant pseudoRiemannian metrics g are classified as follows: (A1) With respect to m = Span{u 1 = e1 , u 2 = e2 , u 3 = e5 , u 4 = e3 − e4 }, the invariant metrics are of the form   g = a θ1 θ1 − 21 θ1 θ3 + θ2 θ4 + b θ2 θ2 + c θ2 θ3 + q θ3 θ3 , a(a − 4q) = 0. (7.1.1) In particular, these metrics are Lorentzian (of signature either (3, 1) or (1, 3)) if a(a − 4q) < 0, and neutral when a(a − 4q) > 0. (A2) For m = Span{u 1 = e1 , u 2 = e2 , u 3 = e3 , u 4 = e5 }, we have   g = a −θ1 θ3 + θ2 θ2 + b θ3 θ3 + c θ3 θ4 + q θ4 θ4 , aq = 0. (7.1.2) In particular, these invariant metrics are Lorentzian if aq > 0 and neutral if aq < 0. (A3) m = Span{u 1 = e1 , u 2 = e2 , u 3 = e4 , u 4 = e5 } and   g = a θ1 θ4 + θ2 θ2 + b θ3 θ3 + c θ3 θ4 + q θ4 θ4 , ab = 0. (7.1.3) They are Lorentzian when ab > 0 and neutral for ab < 0. (A4) m = Span{u 1 = e1 , u 2 = e2 , u 3 = e3 − e6 , u 4 = e4 } and the invariant metrics (all Lorentzian) are given by   g = a θ1 θ1 + θ2 θ3 + 21 θ4 θ4 + b θ2 θ2 , a = 0.

(7.1.4)

(A5) m = Span{u 1 = e1 − e7 , u 2 = e2 , u 3 = e3 + e4 , u 4 = e6 } and the invariant metrics (all Lorentzian) are of the form   g = a θ1 θ1 + 41 θ2 θ3 + 18 θ4 θ4 , a = 0.

(7.1.5)

(B1) Taking m = Span{u 1 = e1 , u 2 = e2 , u 3 = e4 , u 4 = e5 }, the invariant metrics (necessarily neutral) are of the form   g = a θ1 θ3 + θ2 θ4 + b θ2 θ2 + c θ2 θ3 + q θ3 θ3 , a = 0. (7.1.6) (B2) m = Span{u 1 = e1 , u 2 = e2 , u 3 = e3 + e6 , u 4 = e4 } and the invariant metrics (all neutral) are of the form   g = a θ1 θ1 + θ2 θ3 − 21 θ4 θ4 + b θ2 θ2 , a = 0.

(7.1.7)

(B3) m = Span{u 1 , ..., u 4 } and the invariant metrics (all neutral) are of the form

204

7 Where All This Fails: Non-reductive Homogeneous …

  g = a θ1 θ3 + θ2 θ4 + b θ3 θ3 , a = 0.

(7.1.8)

Proof We proceed case by case. For each of the cases listed in Theorem 7.1.1, we first determine a complement (as simple as possible) to h in g and describe with respect to the basis {u i } the isotropy representation for the basis of the isotropy subalgebra h. We then express the invariance condition t ρ(x) · g + g · ρ(x) = 0, x ∈ h, of a symmetric two-tensor g in terms of the basis {u i } and write down the (non-degenerate) solutions in terms of the dual basis {θi }. We report below the details for the case A1, the remaining cases being very similar. For g = a1 = Span{e1 , . . . , e5 }, the isotropy subalgebra is given by h = Span {h 1 = e3 + e4 }. So, we can take as a complement m = Span{u 1 = e1 , u 2 = e2 , u 3 = e5 , u 4 = e3 − e4 }. We then have the following isotropy representation H1 = ρ(h 1 ), described in terms of a corresponding matrix with respect to {u i }: ⎛

0 ⎜0 H1 = ⎜ ⎝0 1

−1 0 0 0

0 0 0 − 21

⎞ 0 0⎟ ⎟. 0⎠ 0

Expressing the invariancy condition t H1 · g + g · H1 = 0, we find that invariant metrics g are precisely the symmetric tensors described with respect to {u i } by symmetric matrices of the form ⎞ ⎛ a 0 − a2 0 ⎜ 0 b c a⎟ ⎟ g=⎜ ⎝−a c q 0 ⎠ , 2 0 a 0 0 that is, of the form (7.1.1) with respect to the basis {θi } dual to {u i }. It is straightforward to check that they are non-degenerate whenever a(a − 4q) = 0. Both Lorentzian and signature (2, 2) invariant metrics are allowed, for a(a − 4q) < 0 and a(a − 4q) > 0, respectively. 

7.2 Geometry of Four-Dimensional Examples In Theorem 7.1.4 we gave an explicit description of all invariant metrics on fourdimensional non-reductive homogeneous pseudo-Riemannian manifolds (G/H, g). In order to investigate their geometric properties, first of all we observe that, just like in the case of invariant metrics, all invariant tensors on G/H are in a one to one correspondence with corresponding tensors on m, invariant under the isotropy representation.

7.2 Geometry of Four-Dimensional Examples

205

Moreover, we need to calculate the Levi-Civita connection and the curvature of (M = G/H, g). Again following [80], an invariant nondegenerate symmetric bilinear form g on m uniquely defines its invariant linear Levi-Civita connection ∇, which may be described in terms of the corresponding homomorphism of h-modules  : g → gl(m), described by (x)(ym ) = [x, y]m for all x ∈ h and y ∈ g. Explicitly, it is given by (x)(ym ) =

1 [x, y]m + v(x, y), 2

for all x, y ∈ g,

(7.2.9)

where v : g × g → m is the h-invariant symmetric mapping uniquely determined by the condition 2g(v(x, y), z m ) = g(xm , [z, y]m ) + g(ym , [z, x]m ),

for all x, y, z ∈ g.

The curvature tensor is then described in terms of the corresponding mapping R : m × m → gl(m) (x, y) → [(x), (y)] − ([x, y]),

(7.2.10)

and all the other needed tensors can then be derived by a similar argument. For example, the Ricci tensor  of g corresponds to the bilinear symmetric tensor  : m × m → R on m explicitly described, in terms of its components with respect to {u i }, by 4 (u i , u j ) = Rri (u r , u j ), i, j = 1, .., 4, (7.2.11) r =1

where Rri = R(u r , u i ). As an example, we complete below the details for case (A1), the remaining cases can be treated in exactly the same way. Putting [i] := (u i ) for all indices i = 1, ..., 4, we find that in case (A1), the Levi-Civita connection of any invariant metric g, as described in (7.1.1) with respect to {u i }, is completely determined by ⎛

0 ⎜0 [1] = ⎜ ⎝0 0 ⎛

0 0 1 0 0 0 − ab − ac

c 0 a ⎜ 0 1 2 [3] = ⎜ ⎝ 0 0 b − ac − 2a

⎞ 0 0⎟ ⎟, 0⎠ −1

0 0

⎛

8bq 0 − a(a−4q) ⎜ −1 0 ⎜ [2] = ⎜ 0 − 4bc ⎝ a(a−4q)

− ab

⎞

0 0 ⎟ ⎟, 0 0 ⎠ 0 − 21

[4] = 0.

Consequently, the curvature tensor is determined by

4bc a(a−4q)

c a 1 2

0

⎞ 1 0⎟ ⎟ , 0⎟ ⎠

b − 2a 0

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7 Where All This Fails: Non-reductive Homogeneous …

⎛

R12

R14

R24

⎞ ⎛ − ac −1 0 − ac 0 1 ⎜ ⎟ ⎜ 0 0 − 0 0 0 ⎜ ⎟ 2 , R13 = ⎜ =⎜ 12b ⎝0 0 0 0 0 ⎟ ⎝ ⎠ a−4q c 4b 12bc b 0 − 2ac − a(a−4q) 0 a a a ⎛ b(a+4q) ⎛ ⎞ 0 − 2a(a−4q) − 2ac 0 −1 0 0 ⎜ 1 ⎜0 0 0 0⎟ − ac − 14 ⎜ 2 ⎜ ⎟ ⎜ R23 = ⎜ =⎜ ⎟, 2b 0 − a−4q ⎝0 0 0 0⎠ ⎝ 0 1 c2 1 0 −2 0 − ab − bc(3a−4q) a 2 (a−4q) a2 ⎛ ⎞ ⎞ ⎛ 0 21 0 0 0 0 00 ⎜ 0 0 0 0⎟ ⎜ 0 −1 0 0 ⎟ ⎜ ⎟ ⎟, R =⎜ = ⎜ ⎟. 34 ⎝0 0 0 0⎠ ⎝ 0 0 0 0⎠ b c 0 a a 1 −1 0 1 0 0 1 0

b(a+20q) a (a−4q)

2

⎞ 0 0⎟ ⎟, 0⎠ 0 ⎞ − 21 ⎟ 0 ⎟ ⎟, 0 ⎟ ⎠ c a

4

and so, with respect to {u i }, the Ricci tensor  is given by ⎛

−2

⎜ ⎜ 0 =⎜ ⎜ ⎝ 1 0

0 2b(a+12q) a(a−4q) − 2c a

−2

1

0

⎞

⎟ − 2c −2 ⎟ a ⎟. ⎟ − 21 0 ⎠ 0 0

Applying the procedure we have just illustrated, several geometric properties of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds can be investigated. We give below only some of them, but we would like to draw the readers’ attention to several other important topics present in the literature, such as for example, among others: the classification of invariant Yang–Mills connections [114]; Walker structures, that is, parallel degenerate line and plane fields [32]; and symmetries [35]. Curvature properties. Einstein manifolds are pseudo-Riemannian manifolds (M, g) whose Ricci tensor  satisfies  = λg for some real constant λ. By the argument we illustrated above, in the case of a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold, this condition translates, in purely algebraic terms, for the corresponding tensors on m. The following classification ([27, 33]) completes and partially corrects the one given in [59], where only the Einstein examples of nonconstant curvature were reported. Theorem 7.2.1 An invariant metric g of a non-reductive homogeneous four-manifold M = G/H is Einstein if and only if one of the following conditions holds: (i) M is of type A2 and either α = 2/3 or g satisfies b = 0; (ii) M is of type A3 and g satisfies q = −εb; (iii) M is of type A4 and g satisfies b = 0;

7.2 Geometry of Four-Dimensional Examples

(iv) (v) (vi) (vii)

M M M M

is of type is of type is of type is of type

207

A5; B1 and g satisfies bq = c2 ; B2 and g satisfies b = 0; B3.

In particular, g is Einstein (equivalently, Ricci-parallel) and not of constant sectional curvature if and only if either g = a2 with α = 2/3 (and b = 0), or g = b3 (Ricci-flat) and b = 0. Further classifications related to some curvature properties can be obtained by the same argument, that is, translating them into algebraic equations for the corresponding tensors defined on m. For example, we have (see [32]): Theorem 7.2.2 Let g be an invariant pseudo-Riemannian metric on a fourdimensional non-reductive homogeneous pseudo-Riemannian manifold M = G/H . Then: (I) g is locally conformally flat and not of constant curvature if and only if one of the following cases occurs: (i) g = a1 , with b = 0; (ii) g = a2 , with α = 2 and b = 0. (II) If g is of neutral signature and not locally conformally flat, then, g is either self-dual or anti-self-dual if and only if one of the following cases occurs: 1. if g = a1 , then g is self-dual if and only if either q = 0 or 3a + 4q = 0; 2. if g = b1 , then g is self-dual if and only if q = 0; 3. if g = b3 , then g is always anti-self-dual. The description of the curvature of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds also permits the explicit determination of the holonomy algebras of these spaces. In fact, yet another well-known theorem by Ambrose and Singer [6] states that the Lie algebra of the holonomy group of (M, g) is spanned by the curvature operators Rx y , x, y ∈ T p M, together with their parallel translates. The following classifications, showing several cases where the holonomy is special (that is, the holonomy algebra does not coincide with the full algebra so( p, q)), were obtained in [28]. Theorem 7.2.3 Let (M, g) be a non-reductive homogeneous Lorentzian fourmanifold. Then, its holonomy algebra is 1. 2. 3. 4.

sl(2, R) for invariant metrics with b = 0 on a space of type A1; R for invariant metrics with α = 0 = b on a space of type A2; trivial for invariant metrics with α = b = 0 on a space of type A2; so(1, 3) in the remaining cases.

Theorem 7.2.4 Let (M, g) be a non-reductive homogeneous pseudo-Riemannian four-manifold of neutral signature (2, 2). Then, its holonomy algebra is 1. sl(2, R) ⊕ s2 , where s2 is the only 2-dimensional solvable (non-abelian) Lie 2 algebra, for invariant metrics with either q = 0 and b = cq , or q = 0 = c on a space of type B1;

208

7 Where All This Fails: Non-reductive Homogeneous … 2

2. sl(2, R) ⊕ R for invariant metrics with b = cq = 0 on a space of type B1; 3. sl(2, R) for invariant metrics with b = 0 on a space of type A1; 4. R for invariant metrics with b = α = 0 on a space of type A2; for invariant metrics with b = c = q = 0 on a space of type B1; for invariant metrics with b = 0 on a space of type B3; 5. trivial for invariant metrics with α = b = 0 on a space of type A2; for invariant metrics with b = c = q = 0 on a space of type B1; for invariant metrics with b = 0 on a space of type B3; 6. so(2, 2) in the remaining cases. Invariant symplectic, complex and Kähler structures. Invariant symplectic structures on a homogeneous space M = G/H are in a one to one correspondence with non-degenerate invariant skew-symmetric bilinear forms  on m such that d(x, yz) = −([x, y]m , z) + ([x, z]m , y) − ([y, z]m , x) = 0, for all x, y, z ∈ m. Hence, we can determine the matrices (i j ) associated to invariant skew-symmetric bilinear forms  on m, with respect to the corresponding basis {u i }, for every non-reductive four-dimensional homogeneous space. By a direct calculation, we get ⎞ ⎞ ⎛ 0 0 0 0 0 12 0 0 ⎜0 0 ⎜ −12 0 23 0 ⎟ 23 0 ⎟ ⎟ ⎟ ⎜ A1 : ⎜ ⎝ 0 −23 0 0 ⎠ , A2, B3 : ⎝ 0 −23 0 34 ⎠ , 0 0 −34 0 0 0 0 0 ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 12 0 0 ⎜0 0 ⎜ ⎟ 0 24 ⎟ ⎟ , A4, B2 : ⎜ −12 0 0 24 ⎟ , A3 : ⎜ ⎝0 0 ⎝ 0 0 34 ⎠ 0 0 0 ⎠ 0 −24 −34 0 0 −24 0 0 ⎛ ⎞ ⎞ ⎛ 0000 0 12 13 0 ⎜0 0 0 0⎟ ⎜ −12 0 23 13 ⎟ ⎟ ⎟ A5 : ⎜ B1 : ⎜ ⎝0 0 0 0⎠, ⎝ −13 −23 0 3,4 ⎠ . 0 −1,3 −34 0 0000 ⎛

Therefore, there exist invariant non-degenerate 2-forms only on the non-reductive homogeneous space corresponding to case B1. Moreover, in this case we find that d = 0 if and only if 2,3 = 0. Thus, we have the following (see [27]). Theorem 7.2.5 A four-dimensional non-reductive homogeneous pseudo-Riemannian manifold M = G/H admits an invariant symplectic form if and only if g = b1 . In this case, with respect to the basis {u i } of m, invariant symplectic forms are given by ⎛

0 12 13 ⎜ −12 0 0 ⎜ =⎝ −13 0 0 0 −13 −34

⎞ 0 13 ⎟ ⎟, 34 ⎠ 0

12 34 − 213 = 0.

7.2 Geometry of Four-Dimensional Examples

209

Next, invariant complex structures on a homogeneous space M = G/H are in a one to one correspondence with complex structures J on m which commute with the isotropy representation. This leads to the following classification [27]. Theorem 7.2.6 A four-dimensional non-reductive homogeneous pseudo-Riemannian manifold M = G/H admits an invariant almost complex structure if and only if g = b1 . In particular, in this case there exists a two-parameter family of invariant complex structures Jt,s described, with respect to the basis {u i } of m, by ⎛ ⎜ Jt,s = ⎜ ⎝

−s 0 0 − (1+s t

2

)

0 0 −s −t (1+s 2 ) s t 0 0

⎞ t 0⎟ ⎟, 0⎠ s

for arbitrary real constants s and t = 0. The results above imply that invariant pseudo-Hermitian (in particular, pseudoKähler) structures may only exist on the non-reductive four-dimensional homogeneous space M = G/H corresponding to the Lie algebra b1 . Consider the complex structures Jt,s described in Theorem 7.2.6 and the pseudoRiemannian metrics g given by (7.1.6). Requiring the compatibility of Jt,s with g, that is, g(J u i , J u k ) = g(u i , u k ), for every i, k, and checking the compatibility of  with Jt,s , that is, (Jt,s u l , Jt,s u k ) = (u l , u k ), for every l, k = 1, . . . , 4, we have the following. Theorem 7.2.7 Let M = G/H denote the four-dimensional non-reductive homogeneous pseudo-Riemannian manifold corresponding to the Lie algebra g = b1 and g, Jt,s and  the invariant pseudo-Riemannian metrics, complex structures and symplectic forms over M, respectively. Then, (M, g, Jt,s ) is a homogeneous pseudoHermitian manifold if and only if 1 + s2 s c = q, b = q. t t2 Hence, any invariant complex structure Jt,s on M determines a two-parameter family of pseudo-Hermitian structures, depending on two arbitrary real constants a = 0 and q. In particular, each complex structure Jt,s corresponds to a one-parameter family of pseudo-Kähler structures (ga , Jt,s , a ) where ga is the (flat) pseudo-Riemannian metric determined by conditions b = c = q = 0, and  is completely determined by 2 a, 34 = ta, 13 = sa. coefficients 12 = 1+s t

210

7 Where All This Fails: Non-reductive Homogeneous …

A similar argument leads to the classification of invariant (almost) para-complex, para-Hermitian and para-Kähler structures on four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds [32]. Theorem 7.2.8 A four-dimensional non-reductive homogeneous pseudo-Riemannian manifold M = G/H admits an invariant almost para-complex structure if and only if g = b1 . In this case, there exists a two-parameter family of invariant paracomplex structures Jλ,μ described, with respect to the basis {u i } of m, by ⎞ ⎛ √ √ 1 1 (bλ − qμ) (±q λμ + 1 + cλ) λ ± λμ + 1 2a √ a ⎟ ⎜ 0 ± λμ + 1 0 ⎟, √−λ Jλ,μ = ⎜ ⎠ ⎝ 0 μ ∓ λμ + 1 0 √ √ 1 1 (bμ − qλ) ∓ λμ + 1 −μ − a (±b λμ + 1 + cμ) 2a (7.2.12) for arbitrary real constants λ, μ satisfying λμ + 1 ≥ 0. In particular: (i) Jλ,μ is para-Hermitian if and only if one of the following cases occurs: ± 4cμ + 4b = 0 (with c2 − bq ≥ 0), or (a) either λ = 0 and μ is a root of qμ2 √ (b) λ and μ satisfy bλ − qμ = cλ ± q λμ + 1 = 0. (ii) (Jλ,μ , g) is almost para-Kähler if and only if b = c = q = 0. In this case, (Jλ,μ , g) is para-Kähler.

7.3 Explicit Invariant Metrics We already recalled (Theorem 7.1.3) that simply-connected non-reductive pseudoRiemannian homogeneous spaces M = G/H of dimension four are diffeomorphic to R4 . Moreover, with the exception of case A5, for each of the other types there is a homogeneous metric on R4 with the corresponding isometry group G. In [59], the explicit form of some invariant Einstein metrics of these spaces were given. Some invariant explicit metrics also appeared in [114]. For all invariant metrics on non-reductive pseudo-Riemannian homogeneous manifolds, an explicit description in coordinates was obtained in [28], using the argument described below. Every element in a connected Lie group G can be written as a finite product of exponentials of the elements of g (see for example [100, Proposition 2.52]). For the connected Lie groups G of a non-reductive homogeneous pseudo-Riemannian manifold, not of type A5, we can show that every element of G can be written as a finite product of exponentials of the elements of the basis of the Lie algebra g, whose description we gave in Sect. 7.1. In fact, the following result was proved in [28]. Proposition 7.3.1 Let M = G/H be a non-reductive pseudo-Riemannian homogeneous four-manifold (not of type A5), where G is the connected Lie group whose Lie algebra g is the one described in the previous Sect. 7.1. Then, there exists a local chart σ : U → G, defined in a neighbourhood U of the identity element of G, such that

7.3 Explicit Invariant Metrics

211

σ(x1 , . . . , xr ) = exp(x1 u 1 ) . . . exp(x4 u 4 ) exp(x5 h 1 ) . . . exp(xr h s ), where r and s are the dimension of G and H , respectively. The above proposition permits us to prove the following. Theorem 7.3.2 Let M be a non-reductive pseudo-Riemannian homogeneous fourmanifold. (I) If M is not of type A5, then it is locally isometric to R4 , equipped with a pseudo-Riemannian metric g, which takes the following explicit form (in terms of some real constants a, b, c, q): (A1) g = (4bx22 + a)d x12 + 4bx2 d x1 d x2 − (4ax2 x4 − 4cx2 + a)d x1 d x3 +4ax2 d x1 d x4 + bd x22 − 2(ax4 − c)d x2 d x3 + 2ad x2 d x4 + qd x32 , on the whole of R4 , whenever a(a − 4q) = 0. (A2) g = −2ae2αx4 d x1 d x3 + ae2αx4 d x22 + be2(α−1)x4 d x32 +2ce(α−1)x4 d x3 d x4 + qd x42 , on the whole of R4 , whenever aq = 0. (A3) if ε = 1: g = 2ae2x3 d x1 d x4 + ae2x3 cos(x4 )2 d x22 + bd x32 + 2cd x3 d x4 + qd x42 , on the open subset where cos(x4 ) = 0, whenever ab = 0; if ε = −1: g = 2ae2x3 d x1 d x4 + ae2x3 cosh(x4 )2 d x22 + bd x32 + 2cd x3 d x4 + qd x42 , on the whole of R4 , whenever ab = 0. (A4) g = ( a2 x42 + 4bx22 + a)d x12 + 4bx2 d x1 d x2 + ax2 (4 + x42 )d x1 d x3 +a(1 + 2x2 x3 )x4 d x1 d x4 + bd x22 + a2 (4 + x42 )d x2 d x3 +ax3 x4 d x2 d x4 + a2 d x42 , on the whole of R4 , whenever a = 0. (B1) g = (q(x32 + 4x3 x2 x4 + 4x22 x42 ) + 4cx2 x3 + 8cx22 x4 + 2ax3 + 4bx22 )d x12 +2(q(x3 x4 + 2x2 x42 ) + 4cx2 x4 + cx3 + 2bx2 )d x1 d x2 +2(q(x3 + 2x2 x4 ) + 2cx2 + a)d x1 d x3 + 4ax2 d x1 d x4 +(q x42 + 2cx4 + b)d x22 + 2(q x4 + c)d x2 d x3 + 2ad x2 d x4 + qd x32 ,

212

7 Where All This Fails: Non-reductive Homogeneous …

on the whole of R4 , whenever a = 0. (B2) ax 2

g = (a − 24 + 4bx22 )d x12 + 4bx2 d x1 d x2 − ax2 (x42 − 4)d x1 d x3 −a(1 + 2x2 x3 )x4 d x1 d x4 + bd x22 − 21 a(x42 − 4)d x2 d x3 −ax3 x4 d x2 d x4 − 21 ad x42 , on the open subset where x4 = ±2, whenever a = 0. (B3) g = −2ae−x2 x3 d x1 d x2 + 2ae−x2 d x1 d x3 + 2(2bx32 − ax4 )d x22 −4bx3 d x2 d x3 + 2ad x2 d x4 + bd x32 , on the whole of R4 , whenever a = 0. (II) If M is of type A5, then there exists a real constant a = 0 such that M is locally isometric to (R2 \ {(0, 0)}) × R2 , and the invariant metric takes the following explicit form 4 d x1 d x2 + a4 d x1 d x4 + A5 : g = − ax 4x2 1 − ax d x2 d x4 + a8 d x32 , 4x2

a(2+2x4 x1 +x32 ) d x2 d x2 8x22

−

ax3 d x2 d x3 4x2

on the open subset where x2 = 0, whenever a = 0. Proof The result is proved proceeding case by case, considering as G and H the connected Lie groups as described in Tables 1 and 2 of [114] and using the matrix realization of G. We describe below the full details for cases A1 and A5, while the other cases may be obtained by the same argument used for A1. Consider the Lie algebra g = a1 and its basis {X 1 = u 1 , . . . , X 4 = u 4 , X 5 = h 1 }. By Proposition 7.3.1, there exists a local chart σ : U → G such that σ(x1 , . . . , x5 ) = exp(x1 X 1 )exp(x2 X 2 ) . . . exp(x5 X 5 ). Then, it is well known that in the local coordinates (x1 , . . . , x5 ), the Maurer–Cartan forms are the components of σ −1 dσ. On the other hand, one has (see for example [104]) σ −1 dσ = exp(−x5 X 5 ) . . . exp(−x1 X 1 ) · d x1 · X 1 exp(x1 X 1 ) . . . exp(x5 X 5 ) + exp(−x5 X 5 ) . . . exp(−x2 X 2 ) · d x2 · X 2 exp(x2 X 2 ) . . . exp(x5 X 5 ) + · · · + exp(−x5 X 5 ) · d x5 · X 5 exp(x5 X 5 ), where exp(−xi X i )X j exp(xi X i ) = X j − xi [X i , X j ] +

xi2 [X i , [X i , 2!

X j ]] − . . . .

We can now calculate the Maurer–Cartan forms in the local coordinates (x1 , . . . , x5 ). Using the description of the Lie brackets for the case A1 given in Sect. 2 (rewritten for the basis {xi }), we get

7.3 Explicit Invariant Metrics

213

σ −1 dσ = d x1 ((1 + 2x2 x4 + 2x2 x5 )X 1 + 2x2 X 2 +(−x5 − x2 x42 − x4 − x52 x2 − 2x5 x2 x4 )(X 4 + X 5 )) x 2 +x 2 +d x2 ((x4 + x5 )X 1 + X 2 − ( 4 2 5 + x4 x5 )(X 4 + X 5 )) x −x +d x3 (X 3 + 5 2 4 (X 4 − X 5 )) + d x4 X 4 + d x5 X 5 . Hence, the Maurer–Cartan forms for g are given by = (1 + 2x2 x4 + 2x2 x5 )d x1 + (x4 + x5 )d x2 , = 2x2 d x1 + d x2 , = d x3 , x 2 +x 2 = (−x5 − x2 x42 − x4 − x52 x2 − 2x5 x2 x4 )d x1 − ( 4 2 5 + x4 x5 )d x2 4 + x5 −x d x3 + d x4 , 2 x 2 +x 2 θ5 = (−x5 − x2 x42 − x4 − x52 x2 − 2x5 x2 x4 )d x1 − ( 4 2 5 + x4 x5 )d x2 4 − x5 −x d x3 + d x5 2

θ1 θ2 θ3 θ4

and their restrictions to the local coordinates (x1 , x2 , x3 , x4 ) on M = G/H are given by θ1 = (1 + 2x2 x4 )d x1 + x4 d x2 , θ2 = 2x2 d x1 + d x2 , θ3 = d x3 , x2 θ4 = −x4 (1 + x2 x4 )d x1 − ( 24 )d x2 − x24 d x3 + d x4 . The conclusion for the case A1 follows substituting these one-forms into Eq. (7.1.1). The case A5 needs a special treatment, being the only one where G is not the full isometry group (Theorem 7.1.3). We make use of the description obtained in [114], starting from the Lie group G = S L(2, R) φ K , where ⎧⎛ −2y ⎫ ⎞ 4 −y2 e−y4 y3 e−y4 −2y1 ⎪ ⎪ ⎪ e ⎪ ⎨ ⎬ −y4 ⎜ 0 ⎟ e 0 y 3 ⎟ , K = ⎜ /y ∈ R ⎝ 0 0 e−y4 y2 ⎠ i ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 0 1     a11 a12 S L(2, R) = A = /det (A) = 1 , a21 a22 K and the homomorphism φ : S L(2, R) → Aut(K ) is φ A = Addiag(1,A,1) . The group multiplication is the standard semi-direct product of Lie groups, that is

(k1 , A1 ) ·φ (k2 , A2 ) = (k1 φ A1 (k2 ), A1 A2 ). Then, a homogeneous pseudo-Riemannian manifold of type A5 is given by M = G/H , where the connected, closed Lie subgroup H of G is

214

7 Where All This Fails: Non-reductive Homogeneous …

⎧⎛⎛ −2r e 1 −r3 e−r1 ⎪ ⎪ ⎨ ⎜⎜ 0 e−r1 ⎜ H= ⎜ ⎝ ⎝ 0 0 ⎪ ⎪ ⎩ 0 0

⎫ ⎞ ⎞ 0 2r2 ⎪ ⎪   r ⎬ 1 ⎟ 0 0 ⎟ 0 e ⎟ ⎟ . /r , ∈ R e−r1 r3 ⎠ r2 er1 e−r1 ⎠ i ⎪ ⎪ ⎭ 0 1

Left-invariant Maurer–Cartan forms of G = S L(2, R) φ K are given in terms of the parameters (y1 , . . . , y4 , a11 , . . . , a22 ) by ω1 ω3 ω5 ω7

= dy4 , ω 2 = e y4 (a21 dy3 − a11 dy2 ), = e y4 (a22 dy3 − a12 dy2 ), ω 4 = e2y4 (−2dy1 + y2 dy3 − y3 dy2 ), = a12 da21 − a22 da11 , ω 6 = a12 da22 − a22 da12 , = a11 da21 − a21 da11 .

Moreover, by formulas (3.3)–(3.5) in [114], one has that a12 e y4 , a22 e y4 , e y4 (y2 a12 − y3 a22 ), e y4 (y2 a12 − y3 a22 ), a21 e−y4 + b1 a22 e y4 − 21 y2 e y4 (y2 a12 − y3 a22 ) are invariant under the H -right action and so, they induce on G/H the coordinates   (x1 , x2 , x3 , x4 ) = a12 e y4 , a22 e y4 , y2 x1 − y3 x2 , a21 e−y4 + b1 x2 − 21 b2 x3 , where (x1 , x2 ) = (0, 0). Consider the basis {u 1 , . . . , u 4 , h 1 , . . . , h 3 } of the Lie algebra, as given in Sect. 7.1, 7 denote its dual basis of 1-forms. Expressing θ1 , . . . , θ7 by means of and let {θi }i=1 1 7 ω , . . . , ω , we get θ1 = − 21 (ω 1 + ω 5 ), θ2 = −ω 6 , θ5 = 21 (ω 1 − ω 5 ),

θ3 = − 41 (ω 4 − 2ω 7 ), θ4 = ω 3 ,

θ6 = 41 (ω 4 + 2ω 7 ), θ7 = −ω 2 .

Setting a22 = y7 = 0, a11 = 1+aa1222 a21 , a12 = y5 and a21 = y6 and starting from (7.1.5), we can now write the invariant metric g in terms of the parameters (y1 , . . . , y7 ). Finally, we apply the transformation (x1, x2 , x3 , x4 )  = y5 e y4 , y7 e y4 , (y2 y5 − y3 y7 )e y4 , y6 e−y4 + y1 y7 e y4 − 21 y2 (y2 y5 − y3 y7 )e y4 to write the metric g in terms of the coordinates (x1 , . . . , x4 ), and this ends the proof.  The explicit form of the invariant metrics of four-dimensional non-reductive homogeneous pseudo-Riemannian spaces was crucial to the investigation of several properties of these manifolds. We list below some of the recent topics where such an explicit description was successfully used.

7.3 Explicit Invariant Metrics

215

Ricci and Yamabe Solitons We briefly recall that a Ricci soliton is a pseudo-Riemannian manifold (M, g), together with a smooth vector field X , such that L X g +  = λg,

(7.3.13)

where L X denotes the Lie derivative in the direction of X and λ is a real number. A Ricci soliton is said to be either shrinking, steady or expanding, depending on whether λ > 0, λ = 0 or λ < 0, respectively. Observe that Ricci solitons generalize Einstein metrics. In fact, an Einstein metric and a Killing vector field yield a trivial solution of the Ricci soliton equation. Ricci solitons are the self-similar solutions of the Ricci flow and are essential in understanding its singularities. We refer to the survey [39] for more information and further references on Ricci solitons. Recently, Ricci solitons have been studied by several authors in pseudo-Riemannian settings, and in particular on Lorentzian spaces. With respect to a system of local coordinates, the Ricci soliton equation (7.3.13) translates into an overdetermined system of PDEs, which is generally very difficult to solve. For this reason, the first approach to the study of homogeneous Ricci solitons is usually algebraic. A homogeneous Ricci soliton is a homogeneous space M = G/H , together with a G-invariant metric g, for which Eq. (7.3.13) holds. In particular, we may call it invariant if Eq. (7.3.13) holds for an invariant vector field. In this special case, the Ricci soliton equation is equivalent to a system of algebraic equations. An algebraic Ricci soliton is a reductive homogeneous manifold G/H , equipped with an invariant pseudo-Riemannian metric g, such that Q = c Id + D|m ,

(7.3.14)

where Q is the Ricci operator, c is a real number, and D|m is the projection on the Ad(h)-invariant complement m of h, of a derivation D of g. Any algebraic Ricci soliton is a Ricci soliton satisfying (7.3.13) with λ = c for a suitable smooth vector field X [86]. As proved in [75], homogeneous Riemannian Ricci solitons are necessarily algebraic. However, in Lorentzian settings, neither invariant nor algebraic Ricci solitons exhaust the whole class of solutions of (7.3.13), and four-dimensional non-reductive homogeneous pseudo-Riemannian spaces provided examples in this sense. The existence of Ricci solitons among four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds was first investigated algebraically in [27] and then completely solved in [33]. Using the explicit form of the invariant metrics reported in Theorem 7.3.2, the following classification result was obtained. Theorem 7.3.3 Let g denote an invariant pseudo-Riemannian metric of a nonreductive homogeneous four-manifold M = G/H . Then, g is a nontrivial Ricci soliton, satisfying Eq. (7.3.13) together with some suitable vector field X and real constant λ, if and only if one of the following conditions holds:

216

7 Where All This Fails: Non-reductive Homogeneous …

(A1) g satisfies b = 0, X =−

1 2 (ax3 ∂1 − 2ax2 x3 ∂2 + 2ax3 ∂3 + 2x3 (ax4 − c)∂4 ) , λ = − , 2a 2 a

so that the Ricci soliton is either expanding or shrinking (depending on the sign of a). (A2) g satisfies b = 0 and α satisfies one of the following non overlapping conditions: (i) α = 0, −1, 23 . In this case,  3α2 3α − 2  (a(α + 1)x1 + ce−(α+1)x4 )∂1 − a(α + 1)x3 ∂3 , λ = − , 2aq(α + 1) q (7.3.15) so that the Ricci soliton is either expanding or shrinking. (ii) α = −1. In this case, X=

X=

 5  3 (−ax1 + cx4 )∂1 + ax3 ∂3 , λ = − , 2aq q

(7.3.16)

so that the Ricci soliton is either expanding or shrinking. (iii) α = 0. In this case, X =−

 1  −x4 ce ∂1 + a∂4 , λ = 0. aq

(7.3.17)

(A4) g satisfies b = 0, X=

5 3 (x2 ∂2 − x3 ∂3 ) , λ = − , 2a a

so that the Ricci soliton is either expanding or shrinking. (B1) g satisfies one of the following partially overlapping conditions: (d1): bq = c2 . In this case, X=

15 3q (−q x2 ∂2 + (c + q x4 )∂4 ) , λ = 2 , 2 4a 2a

(7.3.18)

so that the Ricci soliton is either expanding, steady or shrinking. (d2): q = 0 = c. In this case,   bλ 15c X = λx3 ∂3 + λx4 + + 2 ∂4 , λ = 0, 2c 4a so that these Ricci solitons are either expanding or shrinking.

(7.3.19)

7.3 Explicit Invariant Metrics

217

(B2) g satisfies b = 0, X=

5 3 (x2 ∂2 − x3 ∂3 ) , λ = − , 2a a

(7.3.20)

so that the Ricci soliton is either expanding or shrinking. A Ricci soliton is said to be gradient if X = grad( f ) for some potential smooth function f . In this case, Eq. (7.3.13) becomes H ess( f ) +  = λg. The results on gradient Ricci solitons of non-reductive homogeneous four-manifolds are resumed in the following. Theorem 7.3.4 Let g denote an invariant pseudo-Riemannian metric of a nonreductive homogeneous four-manifold M = G/H . Then, g is a nontrivial gradient Ricci soliton if and only if one of the following conditions holds: (A2) one of the following not overlapping conditions holds: (1i) α = 0, −1, 23 and g satisfies c = 0 = b. In this case, X=

  3α − 2 grad x12 − x32 4q

is a vector field X of the form (7.3.15). (1ii) α = −1 and g satisfies c = 0 = b. In this case, X =−

  5 grad x12 − x32 4q

is a vector field X of the form (7.3.16). (1iii) α = 0 and g satisfies c = 0 = b. In this case, 1 X = − grad (x4 ) q is a vector field X of the form (7.3.17). (B1) g satisfies one of the following partially overlapping conditions: (2i) bq = c2 . In this case, X =−

  15 grad q x22 − 2cx4 − q x42 2 8a

is a vector field X of the form (7.3.18). (2ii) q = 0 = c. In this case,

218

7 Where All This Fails: Non-reductive Homogeneous …

  1 15c2 + 2a 2 bλ 2 2 x4 X = grad λ(x3 + x4 ) + 2 2a 2 c is a vector field X of the form (7.3.19). (B2) g satisfies b = 0. In this case, X=

  5 grad x22 − x32 4a

is a vector field X of the form (7.3.20). A pseudo-Riemannian manifold (M, g) is a Yamabe soliton if it admits a vector field Y such that LY g = (τ − μ)g, (7.3.21) where τ denotes the scalar curvature and μ is a real number. Compared with Ricci solitons, Yamabe solitons are a more rare phenomenon. The following classification was obtained in [33]. Theorem 7.3.5 Let g denote an invariant pseudo-Riemannian metric of a nonreductive homogeneous four-manifold M = G/H . Then, g is a nontrivial homogeneous Yamabe soliton if and only if M is of type B1 and g satisfies q = 0 = c. In this case, Eq. (7.3.21) holds with    b ∂4 , μ = 0, Y = −μ x3 ∂3 + x4 + 2c

(7.3.22)

so that the Yamabe soliton is either shrinking or expanding. This Yamabe soliton is lightlike. Moreover, it is a gradient Yamabe soliton, as Y =−

 μ b  grad x32 + x42 + x4 . 2 c

Homogeneous Geodesics and g.o. Spaces We briefly present some information on homogeneous geodesics and related topics. Definition 7.3.6 Let (M = G/H, g) be a pseudo-Riemannian homogeneous manifold. A geodesic γ through the base point Po ∈ M = G/H is called homogeneous if it is the orbit of a one-parameter subgroup. If γ is homogeneous with respect to some isometry group G  , then it is also homogeneous with respect to the maximal connected group of isometries G, but not conversely. Homogeneous geodesics of homogeneous Riemannian manifolds have been investigated by many authors. We refer to the surveys [8, 55] for more results and information about homogeneous geodesics. In [84], O. Kowalski and J. Szenthe

7.3 Explicit Invariant Metrics

219

proved that any homogeneous Riemannian manifold admits at least one homogeneous geodesic through each point. The study of reductive homogeneous pseudo-Riemannian manifolds resembles the Riemannian case from several points of view. With regard to the study of homogeneous geodesics, an algebraic formula gives the condition for a geodesic to be homogeneous. This formula is known under the name of the “Geodesic Lemma”, and its rigorous proof was given in [57]. It states that if M = G/H is a reductive homogeneous pseudo-Riemannian space, p a basic point of M and X ∈ g, then the curve γ(t) = exp(t X )( p) (the orbit of a one-parameter group of isometries) is a geodesic curve with respect to some parameter s if and only if [X, Z ]m , Vm  = kX m , Z  for all Z ∈ m,

(7.3.23)

where k ∈ R is some constant and g = m + h is a reductive decomposition of g. The Geodesic Lemma does not apply to the non-reductive case, where different techniques are required. Finally, Dusek [56] proved that if M = (G/H, ∇) is an arbitrary homogeneous affine manifold and p ∈ M, then there exists a homogeneous geodesic through p. In particular, applied to their Levi-Civita connection, this result solves the existence problem on pseudo-Riemannian homogeneous spaces as well, leading us to conclude that if (M = G/H, g) is a (not necessarily reductive) homogeneous pseudo-Riemannian manifold and p ∈ M, then M admits a homogeneous geodesic through p. Definition 7.3.7 A g.o. space is a coset representation (M = G/H, g) of a homogeneous pseudo-Riemannian manifold, all of whose geodesics are homogeneous. Naturally reductive spaces are necessarily g.o. In fact, for a naturally reductive space (M = G/H, g), the Levi-Civita connection of (M, g) and the canonical connection of the reductive split g = m ⊕ h have exactly the same geodesics. In particular, all of them are homogeneous. Riemannian g.o. spaces which are not naturally reductive appear starting from dimension 6. In pseudo-Riemannian settings, a more radical difference occurs, since there exist several non-reductive g.o. homogeneous pseudo-Riemannian manifolds. In particular, in dimension four, each of the cases A1–B3 admits some invariant pseudo-Riemannian g.o. metrics. The following classification was obtained in [28] using the explicit forms of the invariant metrics (Theorem 7.3.2) and the techniques introduced by Dusek for the affine case. Theorem 7.3.8 A four-dimensional non-reductive homogeneous pseudo-Riemannian manifold (M, g) is g.o. if and only if it corresponds to one of the following cases: A1 : g with b = 0; A2 : g with b = 0; A3 : g with q = −εb; A4 : g with b = 0; A5 : g arbitrary; B1 : g with bq = c2 ; B2 : g with b = 0; B3 : g arbitrary.

220

7 Where All This Fails: Non-reductive Homogeneous …

Conformal Geometry A pseudo-Riemannian manifold (M, g) is said to be conformally Einstein if the conformal class [g] of g contains an Einstein metric. Conformally Einstein manifolds (M n , g) are characterized by the existence of a solution ϕ : M → R to the conformally Einstein equation (n − 2)Hesϕ + ϕ = θg,

(7.3.24)

where  is the Ricci tensor of g and θ is some function on M. In such a case, g¯ = ϕ−2 g is Einstein. The above conformally Einstein equation (7.3.24) is in general very difficult to solve. Possible candidates are restricted by the property that the Bach tensor of a conformally Einstein manifold necessarily vanishes. Given a local orthonormal basis {e1 , . . . , en } of (M n , g), εi = g(ei , ei ), and denoting by W the Weyl tensor, the symmetric tensor field W [] (X, Y ) =



εi ε j W (ei , X, Y, e j )(ei , e j )

i, j

is called the Ricci-contraction of W . The Bach tensor is then defined by B = div1 div4 W +

n−3 W [] n−2

and necessarily vanishes for a conformally Einstein manifold. Using this property together with the explicit forms of the invariant metrics (Theorem 7.3.2), the following classification was obtained in [38]. Theorem 7.3.9 A four-dimensional non-reductive homogeneous pseudo-Riemannian manifold (M, g) is conformally Einstein if and only if (M, g) is either Einstein, locally conformally flat, or locally isometric to one of the following cases: A1:

g with b = 0 and either q = 0 or q = − 43 a. Solutions to the conformally Einstein equation (7.3.24) are given by  ϕ=

k1 e− 2 x3 1

k1 e x3 + k2 e

if q = 0, 1 2 x 3 −x 1

+ k3 x 2 e

1 2 x 3 +x 1

if q = − 43 a.

α = 1 and g with b = 0. Solutions to the conformally Einstein equation (7.3.24) are given by ϕ = e x4 φ(x3 ), with φ(x3 ) completely determined as the general solution (depending b φ. on the sign of bq = 0) to the ODE φ = 2q A3: g with b = −εq. Solutions to the conformally Einstein equation (7.3.24) are given by ϕ = A2:

7.3 Explicit Invariant Metrics

221

e x3 φ(x4 ), with φ(x4 ) completely determined as the general solution (depending φ. on the sign of b(b + εq)) to the ODE φ = − b−εq 2b Moreover, all the above cases are in the conformal class of a Ricci-flat metric, which is unique (up to a constant) in the case A1 with q = 0. Otherwise the space of Ricci-flat conformal metrics is either two or three-dimensional.

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Index

C Connection, 1, 6–12, 14, 15, 39, 66, 67, 71, 72, 76, 77, 80, 86, 172, 179, 187, 191– 194 Ambrose–Singer, 59, 60, 65, 67, 72, 91, 94, 179 Ambrose–Singer–Kiriˇcenko, 71, 77–81, 85–87, 90–92, 193 canonical, 1, 34, 38, 39, 43, 45, 47, 56– 58, 60, 72, 75, 85, 130, 174, 178, 185, 219 Cartan–Schouten, 57 curvature tensor (field), 9, 10, 13, 17, 20, 23, 27, 28, 30, 31, 33, 38, 67, 72, 78–81, 85–87, 90, 116, 117, 180 form, 6, 7, 38, 55, 178, 191 invariant, 8, 36–38, 199 Levi-Civita, 12–14, 16, 20, 23, 28, 29, 31, 36, 59, 62, 67, 92, 116, 130, 136, 156, 172, 173, 179, 185, 192, 205, 219 mechanical, 172, 173, 178, 179, 186, 188, 191, 192 metric, 12, 14, 44, 54, 76, 86 torsion (tensor field), 9, 15, 38, 67, 72, 78–80, 85–87, 91, 115 Cosymplectic manifold, 31, 112, 114, 191– 193 Curvature, 60, 67, 85, 133–135, 205–207 holomorphic, 20, 21, 134, 138, 139, 193, 194 paraholomorphic, 23, 24, 138, 139, 157 E Equations of Ambrose–Singer, 45, 52, 54, 56, 92, 95, 180

F Fundamental vector field, 6, 10, 47, 48, 175, 181

G G.o. spaces, 131, 218, 219 G-structure, 1, 5, 12, 14, 15, 18, 32, 37–39

H Holonomy, 1, 10, 15, 17, 25, 29, 92, 93, 144, 145, 147, 167, 194, 207 bundle, 12, 25, 47, 53, 68 group, 10, 11, 15–17, 20, 23, 25, 28, 31, 32, 43, 207 restricted group, 11 Homogeneous pseudo-Riemannian manifold, 35, 37, 91, 115, 116, 128–131 naturally reductive, 130, 131 non-reductive, 60, 65, 88, 89, 197, 200, 203, 211 reductive, 36, 91, 197 Homogeneous (pseudo-Riemannian) structure, 59, 60, 91–94, 96, 98, 101, 105, 106, 111, 116, 130, 171, 174, 176– 179, 183–185, 187–190, 192, 197 classification, 56, 58, 93, 96, 130, 197, 199 of linear type, 95, 98, 101, 105, 110, 111, 114, 115, 133–136, 138, 139, 142, 146, 149, 151, 154, 155, 165, 166, 168 on a Lie group, 54, 56, 58 reduced, 176, 186, 190, 193, 194 Hopf fibration, 183, 186–191

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230 Horizontal, 6, 7, 66, 176, 179, 187, 189, 190, 192, 193 curve, 7, 8, 10–12 distribution, 6, 7, 66, 68, 173 lift, 1, 7–9, 14, 26, 39, 67, 172, 176, 191 vector field, 7

I Isometry group, 14, 35, 42, 54, 115, 134, 171, 173, 185, 186, 202, 210, 213, 218

K Killing vector field, 14, 36, 61, 63, 70, 71, 88, 89, 147, 150, 166

L Locally homogeneous pseudo-Riemannian manifold, 35, 59–61, 65, 72, 74, 79, 80, 87, 89–91, 116, 118, 120, 121, 129 reductive, 65, 67, 71, 72 strongly reductive, 78–82 Locally symmetric pseudo-Riemannian manifold, 17, 117–122, 125, 126

N Nomizu construction, 52, 53, 72, 79, 85

P Para-Kähler manifold, 21, 23, 98, 146, 149, 151, 154, 210 Para-quaternion-Kähler manifold, 28–30, 106 Para-quaternion manifold, 28, 29 Plane waves Cahen-Wallach, 166 complex, 166

Index singular scale-invariant homogeneous, 135, 136, 166–168 Principal bundle, 1–6, 8–12, 35, 172, 178, 179, 184, 186, 191, 195 of orthonormal frames, 5, 12, 29, 47, 75, 86 Pseudo-group of diffeomorphisms, 60 Pseudo-group of isometries, 35, 61, 63–65, 67–71, 78, 79, 87 Pseudo-Kähler manifold, 18, 20, 89, 90, 96, 146, 149, 151, 154, 165, 168, 171, 192, 208, 209 Pseudo-quaternion-Kähler manifold, 24, 25, 27, 101, 109, 134, 154

R Ricci tensor, 13, 20, 23, 116, 117, 205, 206, 220

S Sasakian manifold, 31, 113, 114, 171, 191, 192, 194 Soliton Ricci, 215–218 Yamabe, 215, 218

T Transitive Lie groups (or Lie algebras), 2, 3, 6, 34, 35, 39, 60–64, 68–71, 89, 166, 173, 174, 178, 185–189, 197, 202 Transvection algebra, 52, 53, 72, 154, 155, 158, 160, 162, 164, 187, 188

V VSI space, 145, 166, 169

W Walker manifold, 126, 206