Conformal Vector Fields, Ricci Solitons and Related Topics [1 ed.] 9789819992577, 9789819992584

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Infosys Science Foundation Series in Mathematical Sciences

Ramesh Sharma Sharief Deshmukh

Conformal Vector Fields, Ricci Solitons and Related Topics

Infosys Science Foundation Series

Infosys Science Foundation Series in Mathematical Sciences Series Editors Gopal Prasad, University of Michigan, Ann Arbor, USA Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA, USA Editorial Board Chandrashekhar Khare, University of California, Los Angeles, USA Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur, India Ritabrata Munshi, Tata Institute of Fundamental Research, Mumbai, India S. R. S. Varadhan, New York University, New York, USA Weinan E, Princeton University, Princeton, USA

The Infosys Science Foundation Series in Mathematical Sciences, a Scopusindexed book series, focuses on high-quality content in the domain of pure and applied mathematics, biomathematics, financial mathematics, operations research and theoretical computer science. With this series, Springer hopes to provide readers with monographs, textbooks, edited volume, handbooks and professional books of the highest academic quality on current topics in relevant disciplines. Literature in this series will appeal to a wide audience of researchers, students, educators and professionals across the world.

Ramesh Sharma · Sharief Deshmukh

Conformal Vector Fields, Ricci Solitons and Related Topics

Ramesh Sharma Department of Mathematics University of New Haven West Haven, CT, USA

Sharief Deshmukh Department of Mathematics King Saud University Riyadh, Saudi Arabia

ISSN 2363-6149 ISSN 2363-6157 (electronic) Infosys Science Foundation Series ISSN 2364-4036 ISSN 2364-4044 (electronic) Infosys Science Foundation Series in Mathematical Sciences ISBN 978-981-99-9257-7 ISBN 978-981-99-9258-4 (eBook) https://doi.org/10.1007/978-981-99-9258-4 Mathematics Subject Classification: 53-02, 53B30, 53B35, 53C21, 53C25, 53C44, 57N16, 83C20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

This book provides up-to-date information on conformal vector fields, Ricci solitons and related topics applicable to geometry and physics. More specifically, it presents the characterizations and classifications of Riemannian and Lorentzian manifolds (in particular, the space-times of general relativity) that admit conformal vector fields and Ricci solitons (including their generalizations). It has the following salient features: 1. It is the first ever attempt to bring out basic concepts of semi-Riemannian geometry and a comprehensive collection of a fairly large number of research works in the areas of conformal vector fields and Ricci solitons and related topics. 2. We have aimed at bringing together the researchers interested in differential geometry and the mathematical physics of general relativity by providing an invariant (index-free) as well as the index form of the main formulas and results. 3. Overall, the presentation is self-contained, fairly accessible and, in some cases, supported by cited references. 4. The material presented should stimulate and foster future research among researchers. Chapter 1 contains the prerequisite notions of manifolds and submanifolds. In particular, it sweeps through the ideas of smooth manifolds, semi-Riemannian metrics, connection, curvature quantities, submanifolds, semi-Riemannian and warped products. Chapter 2 deals with Lie groups and their Lie algebras. Subsequently, it provides the definition and computations of the Lie derivatives of geometric objects, commutation formulas between Lie derivative and covariant derivative as well as exterior derivative. These commutation formulas are very useful and have been exploited in later chapters. Chapter 3 covers conformal transformations and equations relating the geometric objects corresponding to a metric and conformally transformed metrics. It closes with brief introductions to Weyl tensor, and related tensors such as Cotton and Bach tensors.

v

vi

Preface

Chapter 4 offers a treatment of conformal vector fields, their integrability conditions and zeros. Its integrability conditions have been used very frequently in subsequent chapters. Chapter 5 describes standard operators, divergence theorems (including its restricted validity for semi-Riemannian manifolds) and integral as related to the existence of Killing and conformal vector fields on Riemannian manifolds, especially compact and those with constant scalar curvature. Chapter 6 provides an account of the kinematics of space-time manifolds, Einstein’s field equations, globally hyperbolic space-times, some important spacetimes, homothetic vector fields and Cauchy surfaces. Having a brief review of Petrov classification, this chapter provides characterization and classification of various space-times carrying a conformal vector field. It ends with a brief account of inheriting vector fields with the inverse temperature function as an interesting example. Chapter 7 begins with an introduction and properties of Hamilton’s Ricci flow that was used along with surgery, by Perelman to prove Poincare and Thurston’s conjectures. Then it talks about Ricci solitons which are self-similar solutions of Ricci flow and are important in understanding singularity models. Ricci solitons are also critical points of Perelman’ entropy functionals. Results on gradient (mainly shrinking) have been surveyed mostly from Riemannian point of view, and some in the context of Lorentzian geometry with application to general relativity. Chapter 8 gives a brief review of almost Hermitian and contact metric manifolds, and provides classifications when they admit conformal vector fields. After touch basing with Kaehler–Ricci solitons, it investigates Ricci solitons whose metrics are special contact metrics. Interestingly, in the Sasakian case, the fundamental collineation tensor is preserved by the soliton vector field. The Sasakian Ricci soliton supports the Heisenberg group and in dimension 3, it is the nilpotent group. Chapter 9 gives a coverage on Ricci almost soliton and its classification when it is compact, or contact. Next, it describes generalized quasi-Einstein manifolds, their properties and classifications when they admit a holomorphically planar (in particular, closed) closed conformal vector fields. Chapter 10 starts with basic notion and definition of a Yamabe flow and a Yamabe soliton. Then it presents fundamental results on compact manifolds with or without a boundary. It also gives classification results on locally conformally flat Yamabe solitons. Finally, it presents results on Yamabe solitons in contact (in particular, Sasakian and K-contact) geometries. This book could be recommended as a nice resource for researchers as well as post-graduate students taking courses or pursuing research in differential geometry and/or general relativity. Both authors are grateful to all the authors of books and articles whose work has been used in preparing this book. We acknowledge both the University of New Haven (USA) and King Saud University (Saudi Arabia) for providing facilities and research support. Technical help from colleagues Omar Hijab and Sabir Umarov is

Preface

vii

also appreciated. Finally, it is a pleasure to thank Springer Nature for their continuous support and guidance in the preparation of the manuscript and excellent care in publishing it. West Haven, USA Riyadh, Saudi Arabia

Ramesh Sharma Sharief Deshmukh

Contents

1

Manifolds and Submanifolds Reviewed . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vector and Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Covariant and Exterior Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Semi-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Submanifolds and Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 6 12 15

2

Lie Group and Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lie Group and Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Properties of Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Computation of Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Commutation Formulas for Lie and Covariant Derivatives . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 22 22 24 25

3

Conformal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conformal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conformal Transformations of Geometric Objects . . . . . . . . . . . . 3.3 Other Tensors Related to Weyl and Ricci Tensors . . . . . . . . . . . . . 3.4 Isometries and Killing Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 30 33 37

4

Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Zeros of Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Zeros of Conformal Vector Fields on Semi-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Essential Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Lichnerowicz Conjecture in CR Geometry . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 41 42 43 44 45

ix

x

Contents

5

Integral Formulas and Conformal Vector Fields . . . . . . . . . . . . . . . . . . 5.1 Integration Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Some Classical Results on Conformal Vector Fields . . . . . . . . . . . 5.4 More Results on Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 52 54 58

6

Conformal Vector Fields on Lorentzian Manifolds . . . . . . . . . . . . . . . . 6.1 Space-Times and Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Some Important Space-Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Homothetic Vector Fields in General Relativity . . . . . . . . . . . . . . . 6.5 Homothetic Symmetry and Cauchy Surfaces . . . . . . . . . . . . . . . . . 6.6 Conformal Vector Fields on Space-Times . . . . . . . . . . . . . . . . . . . . 6.7 Inheriting Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 65 71 73 77 82 83

7

Ricci Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2 Ricci Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.3 Ricci Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.4 Some Results on Ricci Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8

Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Complex and Almost Complex Manifolds . . . . . . . . . . . . . . . . . . . 8.2 Contact Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Conformal Vector Fields on Almost Hermitian Manifolds . . . . . . 8.4 Conformal Vector Fields on Contact Metric Manifolds . . . . . . . . . 8.5 Kähler–Ricci Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Contact Metrics as Ricci Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Ricci Almost Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Generalized Quasi-Einstein Manifolds . . . . . . . . . . . . . . . . . . . . . . . 9.3 Generalized m-Quasi-Einstein Manifolds with Conformal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Generalized m-Quasi-Einstein Manifolds in Contact Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 107 112 116 118 120 126 129 129 133 139 140 141

Contents

10 Yamabe Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Yamabe Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Yamabe Flow and Yamabe Solitons . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Conformal Solutions of Yamabe Flow . . . . . . . . . . . . . . . . . . . . . . . 10.4 Yamabe Solitons in Contact Geometry . . . . . . . . . . . . . . . . . . . . . . 10.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

143 143 144 151 152 153 154

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 1

Manifolds and Submanifolds Reviewed

1.1 Differentiable Manifolds By a topology on a set . M, we mean a family .T of subsets of . M such that (i) the empty set .∅ and . M ∈ T , (ii) the intersection of any two members of .T is in .T and (iii) the union of an arbitrary collection of members of .T is in .T . The set . M is called a topological space with topology .T , and the members of .T are called open sets in . M. A neighborhood of a point . p ∈ M is an open set containing . p. A topological space is said to be Hausdorff provided any two points of it have non-intersecting neighborhoods. A map . f from a topological space . M to another topological space . N is continuous if the pre-image of each open set in . N is an open set in . M. If the map . f is a homeomorphism (i.e. it is one-to-one, onto, and both . f and . f −1 are continuous), then . M is said to be homeomorphic to . N . A topological space is said to be Hausdorff if any two distinct points have open disjoint neighborhoods. Definition: A topological Hausdorff space. M is said to be an.n-dimensional manifold if it has a countable basis of open sets and each point of . M has a neighborhood that is homeomorphic to an open set in the .n-dimensional Euclidean space . R n . Thus, a manifold is locally Euclidean, and provides a mechanism for coordinating points in . M. A homeomorphism .φ mapping an open set .U of . M onto an open set n .φ(U ) of . R is called a chart or a local coordinate system. Thus, .φ assigns to each point . p ∈ U , .n coordinates (.x i ) [.i = 1, . . . , n], and .U is a local coordinate neighborhood of . p. Two charts .φ1 and .φ2 on .U1 and .U2 are smoothly compatible if .U1 ∩ U2 = ∅ and the transition function .φ2 oφ1−1 : φ1 (U1 ∩ U2 ) → φ2 (U1 ∩ U2 ), and its inverse are both smooth (i.e. have continuous partial derivatives of all orders). A collection of charts which are smoothly compatible and which cover . M is an atlas. A maximal collection of smoothly compatible charts is called a maximal atlas on . M. We are now ready to define a smooth manifold. Definition: An .n-dimensional smooth manifold is an .n-dimensional manifold . M with a maximal atlas modeled on . R n . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_1

1

2

1 Manifolds and Submanifolds Reviewed

Examples of manifolds are .Rn , .n-dimensional spheres, cylinders, tori, projective spaces and Minkowski space-time (more generally, space-times of general relativity). A smooth manifold with boundary is defined in the same manner as above, replacing . R n by the lower half-plane (.x 1 ≤ 0). The boundary .∂ M of . M is the set of all points whose images lie on the boundary .x 1 = 0. For details, we refer to the standard texts on differential geometry, for example, Kobayashi and Nomizu [1].

1.2 Vector and Tensor Fields A real-valued function . f : M → R is said to be smooth on a smooth manifold . M if, given any local coordinate system .φ, . f oφ −1 is smooth as a real-valued function on an open set of . R n . Obviously, this definition is independent of the choice of .φ. The set of all smooth functions on . M forms a commutative ring under usual sum and product of functions, and is denoted by . F(M). A tangent vector of . M at a point . p is defined as a map . X p : F(M) → R satisfying .

X p (a f + bg) = a X p f + bX p g, X p ( f g) = (X p f )g( p) + f ( p)X p g

where .a, b are real constants, and . f, g ∈ F(M). The set of all tangent vectors at p forms a vector space called the tangent space, under usual addition and scalar multiplication. This space is denoted by .T p (M) and has dimension .= dim M = n. It can be shown (as given in any standard text on differential geometry) that any tangent vector. X p = X p (x i ) ∂∂x i p . Thus,. X p (x i ) are the components of. X p denoted

.

by . X i ( p) with respect to the coordinate basis .( ∂∂x i ) p of .T p (M). A tangent vector . X p can also be looked upon as a smooth curve .C : I → M (. I an open real interval) such i that . p = C(t0 ) for some .t0 ∈ I and . X i ( p) = ddtx |t0 , where .x i (t) are coordinates of the point .C(t) with respect to a local coordinate system containing . p. By a smooth vector field on . M, we mean an assignment . X to each point . p of . M a tangent vector . X p . In general, a vector field may be defined only on an open set of . M. If . f ∈ F(M), and . X is a vector field on . M, then . f X is also a vector field on . M such that .( f X ) p = f ( p)X p . Thus, the set of all vector fields on . M forms a module .X(M) over the ring . F(M) under usual addition and scalar multiplication. We express a vector field locally as . X i ∂∂x i , where . ∂∂x i is the local coordinate basis of .X(M). If . f ∈ F(M) and . X ∈ X(M), then . X f ∈ F(M). The Lie bracket of two vector fields . X, Y is a vector field [. X, Y ] such that [X, Y ] f = X Y f − Y X f

.

for any . f ∈ F(M). Locally, this is expressed as [. X, Y ] .= .(X i ∂i Y j − Y i ∂i X j )∂ j , where .∂i is an abbreviation of . ∂∂x i . The Lie bracket is skew-symmetric and satisfies the following properties:

1.3 Covariant and Exterior Derivatives

3

[a X + bY, Z ] = a[X, Z ] + b[Y, Z ]

.

[X, [Y, Z ]] + [Y, [Z , X ]] + [Z , [X, Y ]] = 0

.

[ f X, gY ] = f g[X, Y ] + f (Xg)Y − g(Y f )X

.

for .a, b ∈ R and . f, g ∈ F(M). One can also verify that [.∂i , ∂ j ] .= 0. Evidently, the set .X forms a Lie algebra under the Lie bracket operation (see the first section of Chap. 2 for the definition of a Lie algebra). A linear map .ω p : T p M → R is called a dual vector, and the set of all dual vectors forms the dual vector space (also known as the cotangent space) .T p∗ M whose dimension is .n. A 1-form .ω on . M assigns a dual vector at each point of . M. A smooth function . f on . M induces a 1-form .d f defined by .(d f )X = X f , where . X is an arbitrary vector field on . M. Locally, .ω = ωi d x i . As a generalization of tangent vectors and dual vectors, we have a tensor of type (.r, s) at . p ∈ M defined as a map ∗ ∗ ∗ . T p : T p M × · · · × T P M × T p M × · · · T p M → R (.r factors of . T p M and .s factors of .T p M), which is (.r + s)-linear, i.e. linear in each factor. A tensor field .T of type (.r, s) on . M is an assignment to each point . p, a tensor .T p of type (.r, s). In terms of a local chart, .T = T i1 ...irj1 ... js ∂i1 ⊗ · · · ∂ir ⊗ d x ji ⊗ · · · ⊗ d x js , where .⊗ stands for the tensor product.

1.3 Covariant and Exterior Derivatives We will denote arbitrary vector fields on . M by . X, Y, Z , W . By an affine connection or a connection on a smooth manifold . M, we mean a map .∇ : X(M) × X(M) → X(M) such that, for . X ∈ X(M), .∇ X defines the operator of covariant differentiation satisfying .∇ X f = X f, ∇ f X +gY Z = f ∇ X Z + g∇Y Z , ∇ X ( f Y + g Z ) = (X f )Y + f ∇ X Y + (Xg)Z + g∇ X Z .

.

The covariant derivative of a vector field .Y along . X is the vector field .∇ X Y which induces a tensor field .∇Y of type (1, 1) such that .(∇Y )(X ) = ∇ X Y . The covariant derivative of a 1-form .ω along . X is a 1-form .∇ X ω defined by (∇ X ω)Y = X (ω(Y )) − ω(∇ X Y ).

.

We follow Einstein’s summation convention: “If an index occurs as a superscript as well as a subscript in an expression, then the expression gets summed over the range of that index, and the same index cannot occur more than twice in the same expression”. With respect to a coordinate system (.x i ), we set

4

1 Manifolds and Submanifolds Reviewed

∇∂i ∂ j = ikj ∂k

(1.1)

.

and call .ikj the connection coefficients. Local expressions for the covariant derivatives are i k i j i .∇ X f = X ∂i f, ∇ X Y = X Y;k ∂i , ∇ X ω = X ωi; j d x where .

Y i;k = ∂k Y i +  ik j Y j

(1.2)

ωi; j = ∂ j ωi − ωk  kji .

.

The covariant derivative of tensors of type (.r, s) can be defined in a generalized manner, e.g. if .T is a tensor field of type (1, 3), then (∇ X T )(ω, Y, Z , U ) = X (T (ω, Y, Z , U )) − T (∇ X ω, Y, Z , U ) − T (ω, ∇ X Y, Z , U ) − T (ω, Y, ∇ X Z , U ) − T (ω, Y, Z , ∇ X U )

.

where . X, Y, Z , U are arbitrary vector fields on . M. Sometimes, we express it by suppressing .ω as (∇ X T )(Y, Z , U ) = ∇ X T (Y, Z , U ) − T (∇ X Y, Z , U ) − T (Y, ∇ X Z , U ) − T (Y, Z , ∇ X U )

.

where each term is a vector field. In local coordinates, .

T ijkl;m = ∂m T ijkl + T

p i jkl  mp

− T ipkl 

p jm

p

p

− T ij pl  km − T ijkp  lm .

The torsion of a connection .∇ is measured by the tensor .T of type (1, 2), defined by . T (X, Y ) = ∇ X Y − ∇Y X − [X, Y ]. (1.3) We say that .∇ is torsion-free or symmetric if .T = 0, i.e. . ijk = ki j . Throughout this book, we will tacitly assume .∇ symmetric. The curvature of . M is measured through a connection .∇, and is defined as a tensor called curvature tensor . R of type (1, 3) such that .

l R(X, Y )Z = ∇ X ∇Y Z − ∇Y ∇ X Z − ∇[X,Y ] Z , R(∂i , ∂ j )∂k = Rki j ∂l .

(1.4)

A straightforward computation shows that the curvature tensor of a symmetric connection satisfies Bianchi’s identities (first and second): .

R(X, Y )Z + R(Y, Z )X + R(Z , X )Y = 0, R lki j + R li jk + R l jki = 0, (∇ X R)(Y, Z , U ) + (∇Y R)(Z , X, U ) + (∇ Z R)(X, Y, U ) = 0,

.

(1.5)

1.3 Covariant and Exterior Derivatives .

R i jkl;m + R i jlm;k + R i jmk;l = 0.

5

(1.6)

A tensor field is said to be parallel with respect to .∇ if its covariant derivative along any vector field vanishes. In particular, a vector field .V is parallel with respect to .∇ if .∇ X V = 0, for any . X ∈ X(M). Suppose .C is a smooth curve in . M with tangent vector field .V . Then .C is said to be a geodesic (auto-parallel curve) if .∇V V = 0. In a local coordinate system (.x i ), a geodesic is given by .

j k d2xi i dx dx =0 +  jk ds 2 ds ds

(1.7)

where .s is an affine parameter. There is a maximal geodesic .C passing through a given point . p, and whose tangent (velocity) vector is equal to a prescribed .V ∈ T p M. If .C is defined for all real values of the parameter .s, then it is complete. An integral curve of a smooth vector field .V on . M is a curve .C whose tangent vector at any point . p of .C is equal to .V p . A vector field .V on . M is geodesic if each of its integral curves are geodesic. Here, we review differential forms and exterior calculus. A tensor field .ω of type (0, . p) on . M is called a differential . p-form if .ω remains invariant under the transpositions of any two of its arguments, i.e. skew-symmetric in any two of its indices. A differential . p-form is simply called a . p-form and is locally expressed as .ω = p!1 ωi1 ...i p d x i1 ∧ · · · ∧ d x i p , where .∧ denotes the exterior or wedge product of two forms. The components .ωi1 ...i p are skew-symmetric between any two indices. The wedge product of a. p-form and a.q-form is a (. p + q)-form. If.α, β, γ are. p, q, r forms respectively, then we know that (1) .(α ∧ β) ∧ γ = α ∧ (β ∧ γ ) (2) .α ∧ β = (−1) pq β ∧ α (3) .(α + β) ∧ γ = α ∧ γ + β ∧ γ . The space of all . p-forms (.0 ≤ p ≤ n) on . M, with the wedge product operation, becomes an algebra known as Exterior algebra. The exterior derivative of a . p-form .ω is a . p + 1-form .dω defined by .(dω)(X 1 , . . . ,

X p) = +

 p+1  1 (−1)i+1 X i ω(X 1 , . . . , Xˆ i , . . . , X p+1 ) p+1 

i=1

(−1)i+ j ω([X i , X j ], X 1 , . . . , Xˆ i , . . . , Xˆ j , . . . , X p+1 )



i< j≤1+ p

in which the hat indicates the omission of the term in the corresponding slot. We illustrate it with a 1-form .α and a 2-form .β as follows:

6

1 Manifolds and Submanifolds Reviewed

1 {X (α(Y )) − Y (α(X )) − α([X, Y ])} 2

(dα)(X, Y ) =

.

1 {X (β(Y, Z )) − Y (β(X, Z )) + Z (β(X, Y )) 3 − β([X, Y ], Z ) + β([X, Z ], Y ) − β([Y, Z ], X )}.

(dβ)(X, Y, Z ) =

.

Locally, we have .dα = where

1 (dα)i j d x i 2!

∧ d x j , and .dβ =

1 (dβ)i jk d x i 3!

∧ dx j ∧ dxk

(dα)i j = ∂i α j − ∂ j αi ,

.

(dβ)i jk = ∂i β jk + ∂ j βki + ∂k βi j .

.

A . p-form .ω is called a closed form if .dω = 0, and an exact form if .ω = dθ for some . p − 1-form .θ . While an exact form is closed, the converse holds in general, only when . M is simply connected. For the exterior derivative operator .d, we know the following facts: 1. Given . f ∈ F(M), .d f is a 1-form such that .(d f )X = X f for any . X ∈ X(M). 2. If .α, β are . p and .q forms, then .d(α ∧ β) = (dα) ∧ β + (−1) p α ∧ dβ. 3. Poincare Lemma: .d 2 = 0. 4. .d(aω1 + bω2 ) = adω1 + bdω2 for .a, b ∈ R and . p forms .ω1 , ω2 . A smooth map .φ from a manifold . M to another manifold . N induces the pull-back differential map .φ ∗ from .r -forms on . N to .r -forms on . M such that .(φ ∗ ω)(X 1 , . . . , X r )| p = ω(φ∗ X 1 , . . . , φ∗ X r )|φ( p) for each . p ∈ M, an .r -form .ω on . N , arbitrary . X 1 , . . . , X p ∈ X(M), and where .(φ∗ X ) f = X ( f ◦ φ) for arbitrary . f ∈ F(N ), X ∈ X(M). The exterior derivative .d commutes with .φ ∗ , i.e. .d ◦ φ ∗ = φ ∗ ◦ d.

1.4 Semi-Riemannian Manifolds Definition: A tensor field .g of type (0, 2) on a smooth manifold . M is called a metric tensor (or, simply a metric) if (1) .g(X, Y ) = g(Y, X ) .∀X, Y ∈ X(M), (2) .g(X, Y ) = 0 .∀X ∈ X(M) ⇒ Y = 0. The smooth manifold . M with a metric tensor .g is called a semi-Riemannian manifold. Given a metric .g, there exists an orthonormal basis (.ei ) of .T p M at each

1.4 Semi-Riemannian Manifolds

7

point . p such that .g(ei , e j ) = 0 for .i = j and .g(ei , ei ) = i = ±1 for each .i. As M is connected, and .g is smooth, the numbers of both positive values and negative values of . i are independent of the choice of an orthonormal basis. The positive and negative signs of . constitute the signature of .g, and the number of negative values is called the index. If the index is zero, . M is said to be Riemannian and the signature is .+ + + · · · +; and if the index is 1, . M is Lorentzian with signature .− + + · · · +. The inner product . of . X, Y ∈ X(M) is .< X, Y >= g(X, Y ). The components of .g in a local coordinate system (.x i ) are expressed in terms of the coordinate basis ij jk .∂i as . gi j = g(∂i , ∂ j ). As . g is non-degenerate, . gi j has inverse . g , i.e. . gi j g = δik . i A tangent vector .V = V ∂i at a point . p ∈ M is said to be timelike, lightlike (null) or spacelike accordingly as .gi j V i V j 0. Thus, the metric tensor .g decomposes the tangent space.T p M into three classes: timelike vectors, null vectors and spacelike vectors. The timelike vectors are separated from spacelike vectors by the null cone at . p defined by the set of all null tangent vectors at . p. The causal character of a tangent vector is determined by the class to which it belongs. We can extend this causal classification to vector fields in a natural way. A smooth curve .C in . M is timelike, null or spacelike depending on whether its tangent vectors are timelike, null or spacelike at each point of .C. In terms of a coordinate basis . ∂∂x i , i j . g = gi j d x ⊗ d x , and the squared length of the infinitesimal arc joining the point i i i 2 i j . x and . x + d x is .ds = gi j d x d x . Riemannian metrics exist on a paracompact manifold (and hence on a smooth manifold) . M; however, the existence of a non-Riemannian non-degenerate metric on . M cannot be guaranteed. It is possible to construct a Lorentz metric on a smooth manifold . M with a Riemannian metric and a non-vanishing line-element field using the following Hawking–Ellis construction [2] as follows. A line-element field assigns to each point . p ∈ M a pair of equal and opposite tangent vectors (.V, −V ) such that . V  = 0. If . g is a Riemannian metric on . M, then we define a Lorentzian metric . g ¯ on . M by g(X, V )g(Y, V ) . g(X, ¯ Y ) = g(X, Y ) − 2 g(V, V ) .

for arbitrary . X, Y ∈ X(M). A fundamental result in semi-Riemannian geometry says that there exists a unique symmetric (torsion-free) connection .∇ on (. M, g) such that (∇ X g)(Y, Z ) = X (g(Y, Z )) − g(∇ X Y, Z ) − g(Y, ∇ X Z ) = 0.

.

Locally, this is expressed as ∇k gi j = ∂k gi j − gi h  h

.

jk

− g j h  hik = 0

where the connection coefficients (also known as Christoffel symbols of second kind) .ihj are given by

8

1 Manifolds and Submanifolds Reviewed

 hi j =

.

1 hk g (∂ j gik + ∂i gk j − ∂k gi j ) 2

and are obviously symmetric in .i, j. This connection .∇ is called the Levi-Civita connection of (. M, g), and is uniquely determined by Koszul’s formula: 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 ).

(1.8)

To the curvature tensor . R of .∇, we associate the curvature tensor of type (0, 4) denoted again by . R and defined by .

R(X, Y, Z , U ) = g(R(X, Y )Z , U ).

The curvature tensor exhibits the following symmetry and skew-symmetry properties: .

R(X, Y, Z , U ) = R(Z , U, X, Y ), R(X, Y, Z , U ) = −R(Y, X, Z , U ), R(X, Y, Z , U ) = −R(X, Y, U, Z ).

Locally, in index notation we have .

R l ki j = ∂i l jk − ∂ j lik +  mk j l mi −  mki l m j Ri jkl = Rkli j Ri jkl = −R jikl Ri jkl = −Ri jlk .

A set of .n orthonormal vector fields (.ei ) on a coordinate neighborhood .U in . M is called a local orthonormal frame on. M. Hence.g(ei , e j ) = i δi j [.i not summed] and n .X = i=1 i g(X, ei )ei where the signature . i = g(ei , ei ). We also have . g(X, Y ) = n i=1 i g(X, ei )g(Y, ei ). Contraction of the curvature tensor yields the Ricci tensor .Ric and the Ricci operator . Q as follows:

.

Ric(X, Y ) =

n 

i g(R(ei , X )Y, ei ), g(Q X, Y ) = Ric(X, Y )

i=1

i.e. in index notation, .

Ri j = R kik j , R ij = g ik Rk j .

The Ricci operator is the tensor . Q of type (1, 1) metrically equivalent to .Ric, i.e. Ric(X, Y ) = g(Q X, Y ).

.

1.4 Semi-Riemannian Manifolds

9

A Riemannian manifold for which .

Ric = kg

(1.9)

for some constant .k is called an Einstein manifold and the corresponding .g an Einstein metric. The constant .k is called the Einstein constant. At this point, we would like to point out that, in dimension 2, .Ric is a function multiple of .g (this multiplier need not be constant). We say that a Riemannian n manifold is Ricci-flat if .Ric = 0. The contraction of .Ric gives rise to a scalar . i=1 i Ric(ei , ei ) called the scalar curvature and denoted by .r . It turns out, therefore, that an Einstein manifold is characterized by r g. (1.10) . Ric = n One can also verify that n  .

i g(R(X, ei )Y, ei ) = − Ric(X, Y ),

i=1

n 

i g(R(X, Y )ei , ei ) = 0.

i=1

Another important differential geometric object is the Weyl conformal tensor .W which is of type (1, 3) and defined by .

1 [Ric(X, Z )Y − Ric(Y, Z )X + g(X, Z )QY n−2 r [g(Y, Z )X − g(X, Z )Y ]. (1.11) − g(Y, Z )Q X ] + (n − 1)(n − 2)

W (X, Y, Z ) = R(X, Y )Z +

In index notation referred to local coordinates, .

1 (δ h Rki − δih Rk j + gki R hj − gk j Rih ) n−2 j r (δ h gk j − δ hj gki ). + (n − 1)(n − 2) i

W hki j = R hki j +

(1.12)

Like the curvature tensor, it enjoys the properties: .

W (X, Y, Z ) = −W (Y, X, Z ), g(W (X, Y, Z ), W ) = g(W (Z , W, X ), Y ), .

W (X, Y, Z ) + W (Y, Z , X ) + W (Z , X, Y ) = 0.

Weyl [3] showed that .W vanishes in dimension 3. We also know, for any dimension, that the Weyl tensor is completely trace-free, i.e.

10

1 Manifolds and Submanifolds Reviewed n  .

i=1 n 

i g(W (ei , X, Y ), ei ) = 0,

n 

i g(W (X, ei , Y ), ei ) = 0,

i=1

i g(W (X, Y, ei ), ei ) = 0.

i=1

At this point, we recall the notions of basic differential operators. First, the gradient of a smooth function . f on . M is a vector field . D f defined by .

g(D f, X ) = X f, D f = g i j ∂i f ∂ j .

(1.13)

The divergence of a vector field .V is a scalar defined as .

div V = i g(∇ei V, ei ) = ∇i V i .

(1.14)

Finally, the Laplacian of an . f ∈ F(M) is the scalar . f = div D f defined as  f = gi j

.

∂2 f ∂f − ikj k i j ∂x ∂x ∂x

.

(1.15)

On . R 3 , the operator .d acts on a .0-form, .1-form and .2-form as if it is acting as the gradient, curl and divergence of a vector field (for details, see Wasserman [4]). The Hessian of . f is defined as the (0, 2)-tensor field .Hess f = ∇∇ f whose local components are.∇i ∇ j f . Obviously,.Hess f is symmetric, i.e..(Hess f )(X, Y ) = g(∇ X D f, Y ) = g(∇Y D f, X ) = (Hess f )(Y, X ), because . D f is gradient and hence closed. By closedness of . D f , we mean the closedness of the metrically equivalent 1-form .d f . TheLaplacian . f can therefore be viewed as the .g-trace (contraction) n

i g(∇ei D f, ei ), and locally expressed as . f = g i j ∇i ∇ j f . of .Hess f , i.e. . i=1 For a smooth function . f on a Riemannian manifold, we have the so-called Bochner–Weitzenbock formula: .

1 (|D f |2 ) = g(D( f ), D f ) + Ric(d f, D f ) + |∇∇ f |2H S , 2

where .|| H S denotes the Hilbert–Schmidt norm. To prove it, we proceed from the left side as follows: . 21 (|D f |2 ) = 21 ∇ i ∇i (∇ j f ∇ j f ) = 21 ∇ i [(∇i ∇ j f )∇ j f + ∇ j f ∇i ∇ j f ] = (∇ i ∇i ∇ j f )∇ j f + |∇∇|2H S . The first term on the right side of this equation is computed as .(∇ i ∇i ∇ j f )∇ j f = (∇ i ∇ j ∇i f )∇ j f = [R k _lk j∇ l f + ∇ j ( f )]∇ j f = [Rl j ∇ l f + ∇ j ( f )]∇ j f = Ric(D f, D f ) + g(D( f ), D f ). This completes the proof. The divergence operator can be applied to higher order tensors, e.g. if .T is a tensor of type (1, 2), then .div T is a tensor of type (0, 2) such that .(div T )(X, Y ) =

i g(∇ei T )(X, Y ), ei ), and locally we express it as .∇i T ijk . Let us contract the Bianchi second identity (1.6) as

1.4 Semi-Riemannian Manifolds

11

(div R)(X, Y, Z ) = (∇ X Ric)(Y, Z ) − (∇Y Ric)(X, Z )

.

(1.16)

whose local version is ∇i R i jkl = ∇k R jl − ∇l R jk .

.

(1.17)

Contracting it with .g jl and noting that contraction commutes with covariant differentiation, we find the important identity: (div Q)X =

.

1 1 Xr, i.e. ∇i R ij = ∇ j r 2 2

(1.18)

which plays a fundamental role in the development of Einstein’s field equations. Equation (1.18) shows that if . Q = λI for some function .λ on a semi-Riemannian manifold . M of dimension .≥ 3, then .λ reduces to a constant, and hence . M becomes Einstein. Now we consider a 2-plane .P in .T p M such that the restriction of .g to .P is nondegenerate. Then the sectional curvature . K (P) of . M with respect to the plane .P is defined as g(R(X, Y )Y, X ) . K (P) = (1.19) g(X, X )g(Y, Y ) − (g(X, Y ))2 where (. X, Y ) is a basis of .P. . K (P) is independent of the choice of the basis. If the sectional curvatures of . M are independent of the choice of 2-planes at each point, then Schur’s theorem asserts that the sectional curvature is constant (say, .c) all over . M. In this case . M is said to have constant curvature .c. The curvature of a semi-Riemannian manifold of constant curvature .c is given by .

R(X, Y )Z = c[g(Y, Z )X − g(X, Z )Y ]

(1.20)

and, in local coordinates, by .

R i jkl = c[δki g jl − δli g jk ].

(1.21)

A complete connected semi-Riemannian manifold of constant curvature is known as a space form. A classical result of Hopf says that a√Riemannian space form n n of constant curvature .c is isometric √ to the sphere . S (1/ c), Euclidean space . E n or the hyperbolic space . H (1/ −c) depending on whether .c is positive, zero or negative, respectively. Its Lorentzian analogue classifies the Lorentzian space form √ √ as the Lorentz sphere . S1n (1/ c) (.n ≥ 3), or . S˜12 (1/ c) (.n = 2), or Minkowski space √ n ˜ 1n (1/ −c) for .c < 0, where the subscripts denote the index and . R1 (.c = 0), or . H tildes the universal covering (see O’Neill [5]).

12

1 Manifolds and Submanifolds Reviewed

1.5 Submanifolds and Hypersurfaces Let . M and . M¯ be manifolds of dimensions .n and .m (.n < m) respectively, and .i : M → M¯ be a smooth map. The differential of .i at a point . p ∈ M is the linear map ¯ defined in the following manner. Pick a .V ∈ T p M. Then there .i ∗ : T p M → Ti( p) M exists a curve .c(t) in . M such that .V is a tangent vector to .c(t) at . p = c(t0 ) for some .t0 in the curve parameter interval. The vector .i ∗ V is then a tangent vector to the curve .i(c(t)) in . M¯ at .i( p) = i(c(t0 )). This can also be interpreted through the condition: .(i ∗ X ) f = X ( f ◦ i) for an arbitrary smooth function . f defined in a neighborhood of .i( p) in .i(M). The map .i : M → M¯ is said to be an immersion if .i ∗ is one-to-one at each point . p ∈ M. In this case, . M is an immersed submanifold ¯ If the immersion is one-to-one, it is called an embedding, and . M is said of . M. ¯ If an open subset . M of a manifold . M¯ is to be an embedded submanifold of . M. considered as a submanifold of . M¯ in a natural manner, then it is an open submanifold ¯ For an embedded submanifold, there is a natural manifold structure on .i(M) of . M. inherited from the manifold structure on . M through the embedding map .i. The tangent space .Ti( p) i(M) is naturally identified with an .n-dimensional subspace of ¯ If the codimension .m − n is 1, i.e. .m = n + 1, . M is called a hypersurface . Ti( p) M. ¯ of . M. An important example of a hypersurface is the boundary .∂ M of a smooth manifold . M with boundary. ¯ Then Suppose .g is a semi-Riemannian metric on . M¯ and . M a submanifold of . M. ∗ ∗ the pull-back map .i induces a metric .i g = γ on . M such that γ (X, Y )| p = g(i ∗ X, i ∗ Y )|i( p)

.

at each point. p ∈ M, and for arbitrary. X, Y ∈ T p M. Note here that the induced metric γ on . M may be non-degenerate or degenerate. In the latter case, . M is called a null or lightlike submanifold. For a comprehensive treatment on such submanifolds, we refer to Kupeli [6], Duggal-Bejancu [7] and Duggal-Sahin [8]. Here, will deal with submanifolds with semi-Riemannian submanifolds, i.e. those having non-degenerate .γ . Identifying . p with .i( p) and . M with .i(M), we define the normal space of . M at . p as ⊥ ¯ : g(V, X ) = 0 ∀X ∈ T p M}. .(T p M) = {V ∈ T p M .

We will now drop the subscript . p in order to describe the induced geometric objects on . M, and denote arbitrary vector fields tangent to . M by . X, Y, Z , U . We denote the ¯ by .∇ Levi-Civita connections of the submanifold (. M, γ ) and the ambient space (. M) ¯ and .∇, respectively. The Gauss formula for the submanifold is ∇¯ X Y = ∇ X Y + B(X, Y )

.

(1.22)

1.5 Submanifolds and Hypersurfaces

13

where . B(X, Y ) is the normal component of .∇¯ X Y , and . B is called the second fundamental form of . M. The curvature tensors of . M and . M¯ are related by the Gauss equation .

¯ R(X, Y, Z , U ) = R(X, Y, Z , U ) − g(B(Y, Z ), B(X, U )) + g(B(X, Z ), B(Y, U )).

One defines the mean curvature vector of . M as . n1 T r.B. The submanifold is called totally geodesic if . B = 0, and totally umbilical if . B(X, Y ) = g(X, Y )N for some normal vector field . N on . M. Now we deal specifically with the case when . M is a ¯ Obviously, for a unit normal vector field . N we see that .∇¯ X N is hypersurface of . M. tangent to . M and so one sets ¯ X N = − AX .∇ (1.23) where . = ±1 is the signature of . N , i.e. .g(N , N ) = , and . is the inner product with respect to either .g or .γ . The (1, 1) tensor . A is called the shape operator and the metrically associated (0, 2) tensor . B (. B(X, Y ) =< AX, Y >) is known as the second fundamental form or extrinsic curvature of. M. The hypersurface. M is totally geodesic in . M¯ if . A = 0. . M is totally umbilical if . AX = f X for some function . f on . M. Sometimes, the total umbilicity is also expressed by saying that the shape operator. A has pure trace. The.n eigenvalues of. A are called the principal curvatures of . M and the corresponding eigenvectors determine the principal directions on . M. The average of all principal curvatures n (taking into account the multiplicities) is the

i < Aei , ei > where .ei is any orthonormal mean curvature and is equal to . n1 i=1 frame on . M, and . i is the signature of .ei . If the mean curvature is constant, then . M is said to be of constant mean curvature (.C MC). The product of the principal curvatures is termed the Gauss–Kronecker curvature of . M. Finally, we write the Gauss and Codazzi–Mainardi equations of the hypersurface as .

¯ g( R(X, Y )Z , U ) = γ (R(X, Y )Z , U ) + (γ (AX, Z )γ (AY, U ) − γ (AY, Z )γ (AX, U )),

(1.24)

¯ g( R(X, Y )Z , N ) = [γ ((∇ X A)Y, Z ) − γ ((∇Y A)X, Z )].

(1.25)

.

Contraction of the above two equations yields the following relation between the Ricci tensors of . M¯ and . M: ¯ , X )Y, N ) = Ric(X, Y ) ¯ Ric(X, Y ) − g( R(N + [γ (AX, AY ) − (T r.A)γ (AX, Y ).

.

(1.26)

Contracting this gives the relation between their scalar curvatures ¯ r¯ − 2 Ric(N , N ) = r + (|A|2 − (T r.A)2 ).

.

(1.27)

14

1 Manifolds and Submanifolds Reviewed

Here, we present a generalization of an integral curve of a vector field. A .ddimensional distribution . D on a smooth manifold . M is a map which assigns to each . p ∈ M a .d-dimensional subspace . D p of .T p M. A vector field .V on . M lies in . D if the tangent vector . V p ∈ D p for each . p ∈ M. The distribution . D is smooth if for each point . p ∈ M, there exist .d independent smooth vector fields . X 1 , . . . , X d spanning . D p in some open neighborhood of . p. A smooth distribution . D is integrable (involutive) if . X, Y ∈ D ⇒ [X, Y ] ∈ D. An integrable distribution . D determines a foliation whose leaves are integral manifolds of . D. More precisely, a submanifold . L of . M is said to be an integral manifold of . D if the tangent space to . L coincides with . D p at each point . p ∈ L. A semi-Riemannian manifold . M is a semi-Riemannian product . M1 × M2 of two semi-Riemannian manifolds . M1 and . M2 when . M1 and . M2 are both totally geodesic submanifolds of . M. A nice generalization of a semi-Riemannian product is the warped product. Given two semi-Riemannian manifolds . B and . F and a positive smooth real-valued function . f on . B, we define their warped product as the semiRiemannian manifold . M = B × f F such that the metric on . M is given by .

g = π ∗ (g B ) + ( f ◦ π )2 σ ∗ (g F )

where .π and .σ stand for the projections of . M on . B and . F respectively, and * on them are the respective pull-backs of the metrics of . B and . F. . B and . F are respectively called the base and fiber. If .t a and .x i are local coordinates on . B and . F, the warped product line-element on . M can be expressed as B (gab )dt a dt b + f 2 (ta )(giFj )d x i d x j .

.

Let .( p, q) be a point in the warped product. Vectors tangent to the leaves . B × q are horizontal and those tangent to fibers . p × F are vertical. Example 1.1 Cone with metric: .dr 2 + r 2 dθ 2 where .r ∈ R + , f (r ) = r and .θ the periodic coordinate on the circle . S 1 . Example 1.2 . R 3 − 0 as warped product . R + ×r S 2 with metric: .dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) where the warping function . f (r ) = r . Example 1.3 Robertson–Walker space-time . I × f  with metric: .−dt 2 + f 2 (t) γi j d x i d x j where .t is the standard time-like coordinate, . f (t) is the warping function and (., γ ) is a space of constant curvature. )dt 2 + Example 1.4 Schwarzschild exterior space-time with metric: .−(1 − 2M r 2M −1 2 2 2 2 2 (1 − r ) dr + r (dθ + sin θ dφ ) where the warping function . f (r ) = r . .r > 2M (. M a constant mass term) and fiber is the unit 2-sphere. We will revisit Examples 1.3 and 1.4 later in Chap. 6. For any horizontal vector fields .U, V, W and vertical vector fields . X, Y, Z the following formulas hold:

References

15

 ∇V X = ∇ X V =

.

(1.30)

R(X, V )Y =

g(X, Y ) ∇V D f, f

g(D f, D f ) [g(Y, Z )X − g(X, Z )Y ], f2

Ric(U, V ) = Ric B (U.V ) − .

.

d (Hess f )(U, V ), f

Ric(V, X ) = 0,

Ric(X, Y ) = Ric F (X.Y ) − f −2 [ f f + (d − 1)g(D f.D f )]g(X, Y ), r = r B + f −2 [r F − 2d f f − d(d − 1)g(D f.D f )].

.

(1.28)

R(U, V )X = 0, R(X, Y )V = 0,

R(X, Y )Z = R F (X, Y )Z −

.

X,

(1.29)

R(U, X )V =

.

.



(Hess f )(U, V ) X, f

.

.

Vf f

(1.31)

(1.32)

(1.33) (1.34) (1.35) (1.36)

For details, we refer to Kobayashi-Nomizu [1], Hicks [9], Bishop-Goldberg [10], O’Neill [5] and Wasserman [4].

References 1. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Interscience Publishers (1963) 2. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973) 3. Weyl, H.: Reine infinitesimal geometrie. Math. Z. 26, 384–411 (1918) 4. Wasserman, R.H.: Tensors and Manifolds with Applications to Mechanics and Relativity. Oxford University Press, New York, Oxford (1992) 5. O’Neill, B.: Semi-Riemanniann Geometry with Applications to Relativity. Academic, New York (1983) 6. Kupeli, D.N.: Singular Semi-Riemannian Geometry, vol. 366. Kluwer Academic Publishers, Dordecht (1996) 7. Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364. Kluwer Academic Publishers, Dordrecht (1996) 8. Duggal, K.L., Sahin, B.: Differential Geometry of Lightlike Submanifolds. Birkhauser-Verlag, Basel, Boston, Berlin (2010) 9. Hicks, N.J.: Notes on Differential Geometry. D. Van Nostrand Company (1965) 10. Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. The Macmillan Company (1968)

Chapter 2

Lie Group and Lie Derivative

2.1 Lie Group and Lie Algebra A non-empty set .G is said to be a group with a binary operation if (i) .a, b ∈ G → ab ∈ G, (ii).a, b, c ∈ G → a(bc) = (ab)c, (iii).G has an element.e (which is unique) satisfying .ea = ae = a .∀ .a ∈ G and (iv) for .a ∈ G, there is an element (inverse) .a −1 such that .aa −1 = a −1 a = e. .G is commutative (abelian) if .ab = ba∀a, b ∈ G. A subset . H of a group .G is called a subgroup of .G if . H is itself a group under the binary operation of .G. A subgroup . H of .G is said to be normal if .aha −1 ∈ H ∀a ∈ G and .h ∈ H . The quotient group of .G by a normal subgroup . H is the group .G/H defined as .{ [a]|a ∈ G}, where .[a] = {c ∈ G|ca −1 ∈ H }, and the group operation .[a][b] = [ab]. A map . f from a group .G 1 to another group .G 2 is said to be a homomorphism if . f (ab) = f (a) f (b) for .a, b ∈ G 1 , and the products taken with respective group operations. Groups .G 1 and .G 2 are said to be isomorphic if . f is injective and surjective. By a Lie group .G, we mean a non-empty set that is a group as well as a smooth manifold such that the group operations .G × G → G : (ab) → ab and .G → G : a → a −1 are smooth. The dimension of .G is the same as the underlying manifold. A local Lie group is a neighborhood of the identity element .e of the group. The .ndimensional vector space . R n is a simple example of a Lie group, with vector addition as the group operation and local coordinates as the components of a vector. The unit circle  iθ  1 .S = e :θ ∈R is another example, and is denoted by .U (1). A Lie subgroup of a Lie group.G is a subgroup of.G and an immersed submanifold of .G. Some examples of Lie groups and Lie subgroups frequently used in mathematics and physics are matrix groups listed below:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_2

17

18

2 Lie Group and Lie Derivative

G L(n, C) General complex linear group: .n × n invertible complex matrices: real dim. .2n 2 .

.

.

.

G L(n, R) General linear group: .n × n invertible real matrices: real dim. .n 2 . S L(n, C) Special complex linear group: .n × n complex matrices with determinant 1: real dim. .2n 2 − 2.

S L(n, R) Special linear group: .n × n real matrices with determinant 1: real dim. n 2 − 1.

. .

.

. O(n) Orthogonal group:.n × n real matrices. A satisfying. A At = I : real dim.. n(n−1) 2

.

S O(n) Special orthogonal group: .{A ∈ O(n) : det (A) = 1}: real dim. . n(n−1) − 1. 2

U (n) Unitary group: .{A ∈ G L(n, C)|A−1 = A¯ t }: real dim. .n 2 .

.

.

SU (n) Special unitary group: .{A ∈ U (n)|det (A) = 1}: with real dim. .n 2 − 1.

We now recall the concept of Lie Algebra. A Lie algebra is a vector space .V with a binary operation, called Lie bracket .[, ]: .V × V → V : .(u, v) → [u, v] satisfying [u, v] = −[v, u]

.

[[u, v], w] + [v, w], u] + [[w, u], v] = 0

.

for any .u, v, w ∈ V . The first property is the skew-symmetry of .[, ] and the second is the Jacobi identity. The set of all vectors in . R 3 forms a Lie algebra under the bracket operation defined as the cross-product of two vectors. In Chap. 1, we noted that the set of all vector fields on a smooth manifold . M is the Lie algebra .X(M), under the bracket of two vector fields. The set of all .n × n real matrices can be made a Lie algebra under the Lie bracket defined by .[A, B] = AB − B A for any two .n × n real matrices. To each Lie group we can associate a Lie algebra in a natural way. This facilitates the study of a Lie group in terms of its Lie algebra. For .a ∈ G, we define the left and right multiplications (translations) as . L a g = ag and . Ra g = ga for all .g ∈ G. As the group operation on .G is smooth, both . L a and . Ra are smooth (actually, diffeomorphisms) and satisfy the following properties: .

L a L b = L ab , Ra Rb = Rba , L a Rb = Rb L a .

The differentials of . L a and . Ra are . L a∗ : Tb (G) → Tab (G) and . Ra∗ : Tb (G) → Tba (G). We say that a vector field .V on .G is left-invariant (respectively, rightinvariant) if. L a∗ Vb = Vab (respectively,. Ra∗ Vb = Vba ). The set.G of all left-invariant vector fields on a Lie group .G forms a Lie algebra under usual vector addition, scalar

2.1 Lie Group and Lie Algebra

19

multiplication and the bracket operation. .G is the Lie algebra of .G and has the same dimension as that of .G. This follows from the fact that the map .G → Te G assigning to each .V ∈ G, the tangent vector .Ve is a linear isomorphism. Linearity is evident. If .Ve = 0, then .Va = L a∗ Ve = 0, which implies injection. For surjection, suppose .v ∈ Te G and define. Va = L a∗ v for every.a ∈ G. It follows that. V is left-invariant such that .Ve = v G. If (. E 1 , . . . , E n ) is a basis of the Lie algebra .G of an .n-dimensional Lie group .G, then, as .G is closed under the Lie bracket operation, we have the structure equations: k .[E i , E j ] = c i j E k (2.1) where .cki j are structure constants. The following relations follow from the properties of the Lie bracket: k k .c i j = −c ji C lim C mjk + C l jm C mki + C lkm C mij = 0.

.

We notice here that, under a change of the basis, .C ki j transform as components of a tensor of type (1, 2). This tensor is known as the structure tensor and is unique for every Lie group and its Lie algebra. Every Lie group .G has its Lie algebra .G. It is known (Warner [1]) that there exists a unique (up to an isomorphism) simply connected (i.e. any closed curve can be continuously shrunk to a point) Lie group for every Lie algebra with structure Eq. (2.1). Thus, though the Lie algebra .G is completely characterized by its structure constants, nevertheless this characterization is local because the structure constants do not, in general, recover a Lie group globally. As a Lie group .G is a smooth manifold, one can endow it with a semi-Riemannian metric. A semi-Riemannian metric on .G is said to be a left-invariant metric if . L a : G → G is an isometry for each .a ∈ G. Thus, a left-invariant metric on . G can be identified with a scalar product on .Te G. If . is the inner product corresponding to a left-invariant metric on .G, then by virtue of the natural isomorphism .V → Ve , .< X, Y > is constant for any . X, Y ∈ G. Given a scalar product . on . Te G, we obtain a left-invariant metric on .G defined by .< v, w >=< L a −1 ∗ v, L a −1 ∗ w > for any .v, w ∈ Ta G, and for all .a ∈ G. A metric on .G is called bi-invariant if it is both left-invariant and right-invariant (i.e. . Ra is also an isometry on .G). Let us briefly touch upon Lie transformation groups. A Lie group .G is said to be a Lie transformation group on a manifold . M, i.e. .G acts smoothly on . M if (i) For every .a ∈ G, there is a transformation of . M given by . p → ap for any . p ∈ M, (ii) .(a, p) → ap is smooth and (iii) .(ab) p = a(bp) for all .a, b ∈ G and . p ∈ M. Sometimes .ap is written as . L a p, and say that .G acts on . M from the left. The condition (iii) shows that the action . L a is one-to-one. Obviously, . L e is the identity transformation of. M..G is said to act freely on. M if. L a p = p for some. p ∈ M implies .a = e. We also say that . G acts effectively on . M if . L a p = p for all . p ∈ M implies .a = e. . G acts transitively on . M if, given . p, q ∈ M, there exists an .a ∈ G such that . L a p = q. A manifold . M is called a homogeneous space of . G if . G acts transitively on . M. For example, the orthogonal group . O(n) acts transitively on . S n−1 , and hence

20 .

2 Lie Group and Lie Derivative

S n−1 is a homogeneous space of . O(n). For another example, consider the isotropy group of a fixed point . p ∈ M defined as . I p = {a ∈ G : ap = p}. Let .G/I p denote the left coset of . I p . The map .G/I p → M is canonically defined by .a I p → ap. It can be shown that .G acts transitively on .G/I p by .a(bI p ) = (ab)I p . Hence .G/I p is a homogeneous space of.G, and the dimension of.G/I p is equal to.dim(G) − dim(I p ). We conclude this section with some examples of Lie transformation groups.

Example 2.1 A simple example is the action of .G = G L(n, R) (the group of all n × n invertible matrices) on . R n defined by .φ : G × R n → R n such that .φ(A, x) = Ax, i.e. multiplication of the .n × 1 column vector .x by the .n × n matrix . A.

.

Example 2.2 Action of .G = G L(n, R) on . R n − {0} with .φ as defined in Exampoint . p= (1, 0, . . . , 0), the isotropic subgroup of .G is the set of ple 2.1. For the  1 A matrices of type. where. B ∈ G L(n − 1, R) and. A ∈ R n−1 and. O is a column O B of .n − 1 zeros. The multiplication is given by .(B, A)(B  , A ) = (B B  , AB  + A ). Here . R n − {0} is a homogeneous space because .G acts transitively on . R n − {0}. Example 2.3 Action of the orthogonal group . O(n) on . R n is smooth and the orbits correspond to concentric spheres having one-to-one correspondence with real numbers .r ≥ 0 defined by mapping each sphere to its radius. Example 2.4 Let. M be the unit sphere. S 2 = {x ∈ R 3 : |x = 1} and.G be the discrete cyclic group . Z 2 = {a, a 2 = e} of order 2. Then .a(x) = −x and .e(x) = x define an action of . Z 2 on . S 2 . One can easily verify that the action .φ : Z 2 × S 2 → S 2 is free and the quotient space . S 2 /Z 2 is the real projective 2-space, i.e. . S 2 with anti-podal points identified. Example 2.5 . O(n) acts transitively on . S n−1 and hence . S n−1 is homogeneous space of the Lie group . O(n).

2.2 Lie Derivative The integral curves of a vector field .V on an .n-dimensional manifold . M are given by the following system of .n ordinary differential equations: .

dxi = V i (x(t)) dt

where (.x i ) is a local coordinate system on . M and .t ∈ I (. I an open interval in . R). The theory of differential equations guarantees the existence of a unique solution of the above system, through a given point . p : x i (0) = x0i , and defined over a real interval

2.2 Lie Derivative

21

containing .0. Each solution is an integral curve (orbit) of .V through . p. Let .U be an open set of . M, containing . p. A local 1-parameter group of local transformations is a map .φ : I × U → M (. I an open interval (.−δ, δ)) satisfying the conditions: (1).φt : p ∈ U → φt ( p) ∈ M is a diffeomorphism of.U onto.φt (U), for every .t ∈ I , (2) for .s, t, s + t ∈ I , and . p, φt ( p) ∈ U, .φs (φt ( p)) = φs+t ( p). The 1-parameter group .φt determines a local flow on . M, and induces a vector field on .U. Conversely, as mentioned before, a vector field .V on . M generates a local flow on . M. In case, each integral curve of .V is defined for all values of the real parameter .t, and . V is called a complete vector field. On a compact manifold, it is known that every vector field is complete. The Lie derivative of a tensor field .T of type (.r, s) with respect to (or, along) a vector field .V is defined at a point . p ∈ M by   (φ ∗ T ) p − T ( p) d ∗ ((φt T ) p = lim t .(L V T ) p = (2.2) t→0 dt t |t=0 where .(φt )∗ is the pull-back along .φt . Thus, the operator .LV of Lie derivation maps a tensor field . R linearly to a tensor of the same type. In [2], the Polish mathematician Slebodzinski introduced a definition of the Lie derivative, though the term Lie derivative occurred first in a paper by Dantzig [3] (see [4]). An algebraic definition is given as follows: The Lie derivative of a tensor field .T with respect to a vector field .V maps a tensor of type (r,s) to a tensor of the same type such that (a) .LV f = V f for any . f ∈ F(M) (b) .LV X = [V, X ] for any . X ∈ X(M) (c) .(LV α)X = V (α(X )) − α([V, X ]) for a 1-form .α on . M (d) .(LV T )(α1 , . . . , αr , X 1 , . . . , X s ) = LV (T (α1 , . . . , αr , X 1 , . . . , X s )) .−T (L V α1 , . . . αr , X 1 , . . . X s ) − · · · − T (α1 , . . . , αr , X 1 , . . . , L V X s ) where .T is a tensor of type (r,s), .α’s are arbitrary 1-forms and . X ’s are arbitrary vector fields on . M. Local components of the Lie derivative of a tensor .T along .V are given by

i 1 ,...,ir k r LV T ij11,...,i ,..., js = V ∂k T j1 ,..., js −

r 

.

+

s  β=1

α=1

i ,...,i

T j11,..., jsα−1

,k,i α+1 ,...,ir

,...,ir k T ji11,..., jβ−1 ,k, jβ+1 ,..., js ∂ jβ V .

∂k V i α (2.3)

22

2 Lie Group and Lie Derivative

2.3 Properties of Lie Derivative The following properties are possessed by the Lie derivative: 1. .LV commutes with contraction of tensors, and also distributes over the sum of tensors. 2. .LV T has the same symmetries and skew-symmetries as those of .T . 3. .LV (S ⊗ T ) = (LV S) ⊗ T + S ⊗ (LV T ). 4. .LV LW − LW LV = L[V,W ] . The Lie derivative acting on a differential form .ω of degree . p satisfies 1. Cartan’s magic formula: .LV = d ◦ i V + i V ◦ d. 2. Lie derivation commutes with exterior differentiation: .LV ◦ d = d ◦ LV . 3. .(LV i X − i X LV )ω = i [V,X ] ω. 4. .L f V ω = f LV ω + (d f ) ∧ (i V ω) for any . f ∈ F(M). 5. .LV ω = (div .V )ω, where .ω is a local volume element on the manifold . M where .i V is the interior product of a form with .V , e.g. .(i V ω)(X 1 , . . . , X p−1 ) = ω(V, X − 1, . . . , X p−1 . If .V is a smooth vector field on a manifold . M and does not vanish at a point . p, then there is a coordinate system (.x i ) on an open neighborhood ∂ .O of . p, such that . V = on .O, and ∂ xi r LV T ij11,...,i ,..., js =

.

∂ i1 ,...,ir T ∂ x 1 j1 ,..., js

which provides a tangible interpretation of the Lie derivative in a local sense. For details, the reader may be referred to the following books: Bishop and Goldberg [5], Wald [6], Hicks [7], Wasserman [8], Kobayashi and Nomizu [9] and Hawking and Ellis [10]. In the following, we compute the Lie derivatives of a function, a vector field and a 1-form as follows.

2.4 Computation of Lie Derivatives Lie derivative of a function:   (φ ∗ f ) − f ) ( p) (LV f ) p = lim t t→0 t   f (φt ( p)) − f ( p)) = lim t→0 t   f (φ p (t)) − f (φ p (0)) = lim t→0 t = (φ p )0 ( f ) = V p ( f ) = (V f )( p).

.

2.4 Computation of Lie Derivatives

23

Hence, .LV f = V f . This shows that .LV LW f − LW LV f = L[ V, W ] f . Lie derivative of a vector field: (φt∗ X f ) − X f ) t→0 t (φt∗ X )(φt∗ f ) − X f ) = lim t→0 t   ∗ φt∗ X − X φ t f − f ∗ f + = lim (φt X ) t→0 t t = X (LV f ) + (LV X ) f.

LV (X f ) = lim

.

Thus, we find that .(LV X ) f = V X f − X V f , and hence that LV X = [V, X ].

.

Lie derivative of a 1-form: In the same manner as above, one could proceed computing the Lie derivative of a 1-form, but we prefer doing the following. Given a 1-form.ω, we operate it on an arbitrary vector field . X in order to get the function .ω(X ). Therefore, .L V (ω(X )) = V (ω(X )). The use of this in the property .L V (ω(X )) = (L V ω)(X ) + ω(LV X ) provides .(LV ω)(X ) = V (ω(X )) − ω[V, X ]. Finally, substituting .∂i for . X we obtain j j .L V ωi = V ∂ j (ωi ) + ∂i (V )ω j . Lie derivatives of higher order tensors and connection: For a tensor of .T of type (1, 1), its Lie derivative along .V is given by (LV T )X = LV T X − T ([V, X ])

.

(2.4)

where . X is an arbitrary vector field, and .T acts linearly on a vector field, yielding another vector field. On a semi-Riemannian manifold . M, the Lie derivative of the metric tensor .g along a vector .V is given by (LV g)(X, Y ) = V (g(X, Y )) − g([V, X ], Y ) − g(X, [V, Y ])

.

= g(∇ X V, Y ) + g(∇Y V, X )

(2.5)

where .∇ denotes the Levi-Civita connection of the metric .g. In local coordinates, we have .L V gi j = ∇i V j + ∇ j Vi (2.6) where .Vi = gi j V j . Even though .∇ is not tensorial, nevertheless we can define its Lie derivative by setting .∇(X, Y ) = ∇ X Y , and taking Lie derivatives on both sides with respect to .V as follows:

24

2 Lie Group and Lie Derivative

(LV ∇)(X, Y ) = LV ∇ X Y − ∇[V,X ] Y − ∇ X [V, Y ]

.

= ∇V ∇ X Y − ∇ X ∇V Y − ∇[V,X ] Y + ∇ X ∇Y V − ∇∇ X Y V = ∇ X ∇Y V − ∇∇ X Y V + R(V, X )Y. (2.7) As .∇∂ j ∂k = i jk ∂i , we may express the above in local coordinates as i m LV i jk = ∇ j ∇k V i + Rkm jV 1 = g im (∇ j (LV gkm ) + ∇k (LV g jm ) − ∇m (LV g jk )). 2

.

(2.8)

2.5 Commutation Formulas for Lie and Covariant Derivatives We will now display formulas showing the deviation from the commutativity of Lie and covariant derivatives acting upon various geometric objects on a Riemannian manifold (. M, g) with Levi-Civita connection .∇. We display them in index-free notation as well as in component form as follows: LV ∇ X Y − ∇ X LV Y − ∇[V,X ] Y = (LV ∇)(X, Y ),

.

(2.9)

j

LV (∇i Y j ) − ∇i (LV Y j ) = (LV ik )Y k ,

.

(LV (∇ X ω) − ∇ X (LV ω) − ∇[V,X ] ω)Y = −ω((LV ∇)(X, Y )),

.

(2.10)

LV (∇i ω j ) − ∇i (LV ω j ) = −(LV ki j )ωk ,

.

((LV (∇ X T ) − ∇ X (LV T ) − ∇[V,X ] T )Y = (LV ∇)(X, T Y ) − T ((LV ∇)(X, Y )),

.

j

j

(2.11)

j

LV (∇i T k ) − ∇i (LV T k ) = (LV j il)Tkl − Tl LV ( lik ),

.

(∇ X (LV ∇))(Y, Z ) − (∇Y (LV ∇))(X, Z ) = (LV R)(X, Y, Z ),

.

j

j

∇i LV kl − ∇k LV il = LV R j

.

(2.12)

lik

where . X, Y, Z are arbitrary vector fields, .ω a 1-form, .T a tensor of type (1, 1) and . R the curvature tensor of type (1, 3) on . M. For details we refer to Yano [11].

References

25

References 1. Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Scott, Foresman, Glenview, Illinois (1971) 2. Slebodzinski, W.: Sur les equations de Hamilton. Bull. Acad. Roy. Belg. 17, 864–870 (1931) 3. van Dantzig, D.: Zur allgemeinen projektiven Differentialgeometrie I, II. Proc. Kon. Akad. Amsterdam 35, 524–542 (1932) 4. Schouten, J.A.: Ricci-Calculus. Springer (1954) 5. Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. The Macmillan Company (1968) 6. Wald, R.M.: General Relativity. University of Chicago Press (1984) 7. Hicks, N.J.: Notes on Differential Geometry. D. Van Nostrand Company (1965) 8. Wasserman, R.H.: Tensors and Manifolds with Applications to Mechanics and Relativity. Oxford University Press, New York, Oxford (1992) 9. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Interscience Publishers (1963) 10. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973) 11. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)

Chapter 3

Conformal Transformations

3.1 Conformal Transformation A conformal map is a map from one domain of the complex plane to another domain, such that the angles between curves through any point are preserved in magnitude and orientation. Thus, conformal maps preserve both angles and shapes of infinitesimally small figures, but not necessarily their size. Actually, conformal maps are nothing but holomorphic functions between two open parts of the complex plane. In higher dimensions, conformal maps are severely restricted. According to classical Liouville theorem [1], a conformal map between two open parts of the 3dimensional Euclidean space is the composition of an isometry, similarity (dilation or contraction by a constant factor) and an inversion (.x → |x|c 2 x for a nonzero constant .c). This was generalized to the case of an .n-dimensional Euclidean space by Lie [2]. It is also known that the conformal maps of semi-Euclidean spaces are those that preserve the sets of hypersurfaces and hyperplanes. Let us now describe the conformal transformation of semi-Riemannian manifolds. Consider a semi-Riemannian metric .g on a .n-dimensional smooth manifold . M. Another metric .g¯ on . M is said to be conformally equivalent (or, related) to .g if there exists a smooth function . that vanishes nowhere on . M, such that .

g¯ = 2 g.

(3.1)

Also, then .g¯ is said to be conformal to .g. The angles between vectors and causal character of a vector (spacelike, null and timelike) remain unchanged under a conformal change of metric. Conversely, if two metrics .g and .g¯ have the same light cones (generated by null vectors) at a point . p, then .g¯ is a non-vanishing function multiple of .g (see Wald [3]). A conformal manifold is a smooth manifold equipped with an equivalence class of semi-Riemannian metrics, in which two metrics .g and .g¯ are equivalent if and only if they are conformally related. The set of metrics conformally related to .g is called the conformal class of .g, and denoted by .[g]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_3

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3 Conformal Transformations

It is important to note in this context that a conformal change of metric need not be associated with a diffeomorphism of . M (i.e. a smooth map .φ : M → M that is one-one, onto and whose inverse is also smooth). A diffeomorphism .φ satisfying the conformal condition .φ ∗ g = 2 g is called a conformal transformation (also known as a conformal isometry) of the semi-Riemannian manifold (. M, g). A conformal transformation is said to be a homothety if . is constant, and an isometry if . = 1. More generally, a diffeomorphism .φ from a semi-Riemannian manifold .(M, g) onto ¯ g) another semi-Riemannian manifold (. M, ¯ is said to be a conformal if there exists a nowhere vanishing function . on . M such that .φ ∗ g¯ = 2 g. Following Kühnel and Rademacher [4], we say that a non-isometric conformal diffeomorphism is a Liouville map if the Ricci tensors of.φ ∗ g¯ and.g are equal pointwise on. M. If. M¯ = M, we call a conformal transformation .g → g¯ a Liouville transformation of the metric if the Ricci tensors of .g¯ and .g are pointwise equal on . M. Kühnel and Rademacher obtained the following generalization of Liouville’s theorem stated earlier. Theorem (Kühnel–Rademacher) Let (. M, g) be a connected semi-Riemannian manifold and . f : (M1 , g) → (M, g) be a Liouville map of class .C 3 for some open . M1 ⊂ M. Assume that the gradient of the induced conformal factor has nonzero . g-norm everywhere. Then (. M1 , g) is isometric to an open subset of a warped product 2 2 .(0, ∞) ×t M∗ with a cone-like metric . g = dt + t g∗ where . = ±1, and (. M∗ , g∗ ) is independent of .t. Up to an isometry, the map . f appears as . f (t, x) = ( 2t , x) and hence is the composition of an isometry and an inversion. If, in addition, . M is Einstein, then it is Ricci flat, and consequently any conformal map is a Liouville map, in which case (. M∗ , g∗ ) is an Einstein metric with the same Ricci curvature as a space of constant sectional curvature .. If, in addition, the dimension of . M is not greater than 4, then .g is flat.

3.2 Conformal Transformations of Geometric Objects We denote arbitrary vector fields on . M by . X, Y, Z . Let .∇ and .∇¯ be the Levi-Civita connections of the conformally related metrics .g and .g¯ = 2 g respectively, on . M. One can easily derive (see Yano [5]) the following formula for the change of the Levi-Civita connection under the conformal change ∇¯ X Y = ∇ X Y + (X log )Y + (Y log )X − g(X, Y )D log 

.

(3.2)

where . D denotes the gradient operator of .g. Remark 3.1 The connection .∇¯ is a special case of Weyl’s connection . E on (. M, g) defined by ∗ . E X Y = ∇ X Y + ω(X )Y + ω(Y )X − g(X, Y )ω where .ω is 1-form, and .ω∗ is the vector field .g-equivalent to .ω. Obviously, . E is symmetric and is characterized by the property

3.2 Conformal Transformations of Geometric Objects

29

(E X g)(Y, Z ) = −2ω(X )g(Y, Z ).

.

A straightforward computation using Eq. (3.2) yields the following conformal transformations formulas for the Riemann curvature tensor . R, Ricci tensor .Ric and the scalar curvature .r : .

¯ R(X, Y )Z = R(X, Y )Z (3.3)   2 . + (Y log )(Z log ) − g(∇Y D log , Z ) − |D log | g(Y, Z ) X − [(X log )(Z log ) − g(∇ X D log , Z ) − |D log |2 g(X, Z )]Y + [g(Y, Z )(X log ) − g(X, Z )(Y log )]D log  + g(X, Z )∇Y D log  − g(Y, Z )∇ X D log 

(3.4)

¯ Ric(X, Y ) = Ric(X, Y ) + (n − 2)(X log )(Y log ) + (2 − n)g(∇ X D log , Y )

.

+ [(2 − n)|D log |2 −  log ]g(X, Y ) and

(3.5)

r¯ = −2 [r + (1 − n)(2 log  + (n − 2)|D log |2 )]

.

(3.6)

where . denotes the Laplacian .∇ i ∇i . The use of Eqs. (3.3), (3.5) and (3.6) in the definition (Chap. 1) of the Weyl conformal tensor .W shows that .

W¯ (X, Y, Z ) = W (X, Y, Z ),

(3.7)

i.e. .W¯ hi jk = W hi jk , and hence .W is conformally invariant. There are equations for certain physical fields that are conformally invariant. We say that a differential equation for a scalar field .u is conformally invariant if there is a real number . p (called the conformal weight) such that .u is a solution corresponding to the metric .g if and only if . p u is a solution corresponding to the conformally related metric .g¯ = 2 g. With the special choice of the weight . p = 1 − n2 , we can show by a straightforward computation that .

¯ 1− n2 u) = −1− n2 Lu L(

n−2 where . L = g i j ∇i ∇ j − 4(n−1) r is the Yamabe operator, and . L¯ is the corresponding ¯ As a result, the second order equation operator for .g.

.

g i j ∇i ∇ j u −

n−2 ru = 0 4(n − 1) n

is invariant under the transformation .g¯ = 2 g and .u¯ = 1− 2 , and hence can be viewed as a conformally invariant generalization of the Laplace and Klein–Gordon

30

3 Conformal Transformations

equations in flat spaces to semi-Riemannian manifolds. For details, we refer the reader to Wald [3] and Choquet-Bruhat et al. [6]. At this point, we exhibit the role played by the Weyl conformal tensor .W in characterizing a class of semi-Riemannian manifolds called conformally flat manifolds. We say that a semi-Riemannian manifold (. M, g) is conformally flat (sometimes also known as locally conformally flat) if, for each point . p ∈ M, there exists a neighborhood .U of . p, such that the metric .2 g is flat on .U (i.e. its curvature tensor vanishes on .U) for a smooth function . that vanishes nowhere on .U. The function need not be defined all over . M. The manifold (. M, g) would be globally conformally flat if there exists a non-vanishing function . on . M such that .2 g is flat on . M, or put another way, if .g is conformal to a globally flat metric on . M. Any 2-dimensional semi-Riemannian manifold is conformally flat because there exists isothermal coordinates .u, v in a neighborhood of any point, such that the line-element takes the form: ds 2 = gi j d x i d x j = λ2 (du 2 ± dv 2 )

.

(3.8)

where .i, j run from 1 to 2, .x 1 = u, .x 2 = v, and .λ is a smooth function of .u and .v. The plus and minus signs correspond to the Riemannian and Lorentzian metrics, respectively. The reason that these coordinates are called isothermal, may be understood for the Riemannian case as follows. If .u, v are isothermal coordinates, then .u and .v are harmonic functions with respect to the Laplace–Beltrami operator of the metric .g, i.e. .u = v = 0 [recall . = g i j ∇i ∇ j ]. The Laplace equation . f = 0 characterizes the stationary states for the heat equation. The level curves for a harmonic function are therefore the isothermal curves for some heat distribution. Thus, the level curves .u = constant and .v = constant are isothermal curves. It is easy to verify that the Weyl conformal tensor .W vanishes in two dimensions.

3.3 Other Tensors Related to Weyl and Ricci Tensors It is known that .W = 0 also for dimension 3, however a 3-dimensional semiRiemannian manifold need not be formally flat. For 3-dimensional case, the conformal flatness is equivalent to the vanishing of the Cotton tensor .C which is a tensor of type (0, 3) defined by C(X, Y, Z ) = (∇ X P)(Y, Z ) − (∇Y P)(X, Z )

.

(3.9)

where . P is the Schouten tensor defined by .

P=

  r 1 Ric − g . n−2 2(n − 1)

(3.10)

3.3 Other Tensors Related to Weyl and Ricci Tensors

31

Thus, the conformal flatness of a 3-dimensional metric can also be expressed by saying that the Schouten tensor . P is Codazzi, i.e. satisfies .(∇ X P)(Y, Z ) = (∇Y P)(X, Z ). One may note here in this context that .C is completely trace-free as well as conformally invariant. In dimensions greater than 3, the conformal flatness is equivalent to the vanishing of .W . These results can be summed up as follows. Weyl–Schouten theorem A semi-Riemannian manifold of dimension .n with .n ≥ 3 is conformally flat if and only if the Schouten tensor . P is a Codazzi tensor (i.e. Cotton tensor .C = 0) for .n = 3, and the Weyl tensor .C = 0 for .n > 3. It is easy to verify the fact that a semi-Riemannian manifold of constant curvature is conformally flat. It is also a well known result of Kuiper [7] that a compact simply connected conformally flat Riemannian manifold is conformally equivalent to the round sphere. Let us point out the fact that the Weyl tensor .W can be reformulated as .

W = R−g P

(3.11)

where . stands for the Kulkarni–Nomizu product of two symmetric tensors .g and h of type (0, 2) by

.

(g h)(X, Y, Z , U ) = g(X, U )h(Y, Z ) + g(Y, Z )h(X, U )

.

− g(X, Z )h(Y, U ) − g(Y, U )h(X, Z ). The Weyl conformal tensor of type (0, 4) also denoted by .W is given by .W (X, Y, Z , U ) = g(W (X, Y, Z ), U ) for arbitrary vector fields . X, Y, Z , U on . M. One can verify, by a straightforward computation, that its divergence at the 4th slot .W is the (0, 3)-tensor .div4 W defined by (div4 W )(X, Y, Z ) =

n 

.

i (∇ei W )(X, Y, Z , ei )

(3.12)

i=1

where .ei is a local orthonormal basis on . M and .i = g(ei , ei )], and is equal to .(n − 3)[(∇ X P)(Y, Z ) − (∇Y P)(X, Z )] = (n − 3)C(X, Y, Z ). Usually, .div4 W is denoted simply by .div ·C. The Weyl tensor is said to be harmonic if .div ·W = 0, i.e. the Cotton tensor vanishes. For a semi-Riemannian manifold of dimension .n > 3, the following two conditions can be shown equivalent: (i) .div ·W = 0, (ii) .⊕ X Y Z (∇ X W )(Y, Z , U ) = 0 (where .⊕ X Y Z denotes cyclic sum over . X, Y, Z ). In order to introduce our next tensor related to .W and .Ric, we recall the concept of a conformally Einstein manifold. A semi-Riemannian manifold (. M, g) is said to be conformally Einstein if each point . p ∈ M has an open neighborhood .U such that the metric .g¯ = ϕ −2 g is Einstein, for a smooth non-vanishing function . : U → R. In view of equation (3.2.4), we find that a semi-Riemannian manifold (. M, g) admits ¯ if and only if a conformal map onto an Einstein manifold (. M, g)

32

3 Conformal Transformations

  r

ϕ g + ϕ Ric − g = 0. (n − 2) ∇∇ϕ − n n

.

(3.13)

This is the so-called conformal Einstein equation. Resetting .ϕ as .eφ , one finds by a straightforward computation that ¯ 4 W¯ )(X, Y, Z ) = (div4 W )(X, Y, Z ) + (3 − n)W (X, Y, Z , Dφ). (div

.

(3.14)

By direct computation using the contracted Bianchi’s second identity, we have  n−3 (∇ X Ric)(Y, Z ) − (∇Y Ric)(X, Z ) n−2  Yr Xr g(X, Z ) g(Y, Z ) + − 2(n − 1) 2(n − 1)

(div4 W )(X, Y, Z ) =

.

¯ 4 W¯ . Thus, if .g¯ is Einstein, then .div ¯ 4 W¯ = 0 and a corresponding expression for .div and so, Eq. (3.14) reduces to (div4 W )(X, Y, Z ) + (3 − n)W (X, Y, Z , Dφ) = 0.

.

(3.15)

Now, suppose .h is a symmetric (0, 2)-tensor. Then we define the (0, 2)-tensor .W [h] by n  . W [h](X, Y ) = i  j W (ei , X, Y, e j )h(ei , e j ). i, j=1

Taking the divergence .div1 of Eq. (3.15) with respect to the first argument, i.e. . X provides the equation .

div1 div4 W +

n−3 W [Ric] − (n − 3)(n − 4)W [dφ ⊗ dφ] = 0. n−2

(3.16)

This equation prompts the following definition. Definition: The Bach tensor . B of an .n-dimensional (.n ≥ 4) semi-Riemannian manifold is defined by n−3 W [Ric]. . B = div1 div4 W + (3.17) n−2 In local coordinates, its components are given by .

Bi j = ∇ l ∇ k Wk jli +

n − 3 lk R Wk jli . n−2

Bach tensor was introduced by Bach [8] in 1920 in the context of conformal relativity.

3.4 Isometries and Killing Vector Fields

33

For dimensions .n ≥ 4, if .g is Einstein, or conformally flat, then its Bach tensor vanishes, i.e. it is Bach-flat. The 4-dimensional case is interesting, for which the Bach tensor is symmetric, divergence-free, traceless and conformally invariant up to a scale factor, i.e. . B¯ i j = −2 Bi j if .g¯ i j = 2 gi j . Thus, . Bi j is an example of a .T T (transverse traceless) tensor which arises in the study of initial value problem in general relativity. Further, for dimension 4, Bach-flat metrics are precisely the critical points (Euler– Lagrange equations) of the conformally invariant Weyl functional  W(g) =

|Wg |2 d Vg

.

M

on the space of metrics on a compact 4-manifold . M. Besides Einstein and conformally flat metrics, other examples of Bach-flat metrics are the 4-dimensional half-conformally flat metrics, i.e. self-dual and anti-self-dual metrics for which .W − and .W + vanish, respectively.

3.4 Isometries and Killing Vector Fields The group of all smooth transformations of a smooth manifold . M is a very large group. This intrigues us to study those transformations that leave a geometric quantity invariant. Let us consider a semi-Riemannian manifold (. M, g) and recall that a diffeomorphism .φ : M → M is called an isometry of . M if it leaves .g invariant, i.e. ∗ .φ g = g which means that g

. φ( p)

((φ∗ )X p , (φ∗ )Y p )) = g p (X p , Y p )

(3.18)

for all. X p , Y p ∈ T p M, and where.φ∗ denotes the differential map of.φ. As.φ∗ is a linear isomorphism of .T p M onto .Tφ( p) M at each point . p, it follows that .φ is an isometry if and only if .φ ∗ is a linear isometry for each . p ∈ M. The set of all isometries of . M forms a group under composition of mappings. It was shown by Myers and Steenrod [9] that the group of all isometries of a Riemannian manifold . M is a Lie group called the isometry group and denoted by . I (M). In a rough sense, the larger . I (M) is, the simpler . M is. For a semi-Riemannian manifold (. M, g), it follows from general results of Palais [10] that. I (M) is a Lie group such that the action. I (M) × M → M is smooth, and a homomorphism.β : R → I (M) is smooth if the. R × M → M sending (.t, p) to.β(t) p is smooth. A semi-Riemannian manifold. M is said to be homogeneous if, given . p, q ∈ M, there is an isometry on . M that maps . p to .q. That means, in order to show that a given semi-Riemannian manifold is homogeneous, one needs to show enough isometries to map a point to any other point. A Riemannian homogeneous manifold is complete. However, a semi-Riemannian homogeneous manifold need not be complete, unless it is compact. For details we refer to O’Neill [11]. We now define an infinitesimal isometry.

34

3 Conformal Transformations

Definition: A vector field .V on a semi-Riemannian manifold (. M, g) is said to be a Killing vector field or an infinitesimal isometry if the local 1-parameter group of local transformations generated by .V in a neighborhood of each point of . M consists of local isometries. Equivalently, .V is said to be Killing if it satisfies the Killing equation (named after the geometer W. Killing): .L V g = 0. (3.19) Operating both sides of this equation on arbitrary vector field. X, Y , we see.V (g(X, Y )) = g(LV X, Y ) + g(X, LV Y ). This simplifies to .

g(∇ X V, Y ) + g(X, ∇Y V ) = 0.

(3.20)

In local coordinates, this equation reads .∇i V j + ∇ j Vi = 0. A non-rigorous but a little more tangible way of obtaining the Killing equation is the following derivation. Let .U be a neighborhood of each . p ∈ M with coordinate system (.x i ). Let the integral curves of .V , through any point .q ∈ U, be defined on an open interval (.−, ) for . > 0. For each .t ∈ (−, ) we define a map .φt on .U such that for .q ∈ U, .φt (q) is the point with parameter value .t on the integral curve of .V through .q. If .φt is an isometry then .V is Killing. As .φt (x i ) = x i + t V i up to the first order in .t and .φt is isometry, we have ∂ (x i + t V i )∂m (x j + t V j )gi j (x + t V ) = gkm .

. k

Expanding .gi j (x + t V ) up to the first order, we reduce the above equation to .

V i ∂i g jk + ∂ j (V i )gik + ∂k (V i )gi j = 0.

We notice that this equation can be reformulated in terms of the Lie derivative as the Killing Eq. (3.20). For a rigorous approach we refer to O’Neill [11]. The Killing equations show that if .V is a Killing vector field then the local geometry remains invariant while moving along the local 1-parameter group of local transformations .φt generated by .V . Thus, a Killing vector preserves the Levi-Civita connection, curvature tensor, Ricci tensor, scalar curvature and Weyl tensor. One can ∂g verify that a coordinate vector field .∂i is Killing if and only if . ∂ xjki = 0 for all . j, k; i i.e. the metric components are independent of .x . The use of Eqs. (2.7) and (2.8) enables us to prove the following conservation lemma. Lemma 3.1 If .V is a Killing vector field on a semi-Riemannian manifold (. M, g) and .γ is a geodesic in . M, then .V restricted to .γ is a Jacobi vector field and .g(V, γ ) is constant along .γ . This lemma can be used to obtain the next lemma.

3.4 Isometries and Killing Vector Fields

35

Lemma 3.2 If a Killing vector field .V on a semi-Riemannian manifold . M and its covariant derivative vanish at some point, then .V vanishes everywhere on . M. Now let us state and prove the following important result: The Lie algebra .i(M) of Killing vector fields on a connected semi-Riemannian manifold . M of dimension .n has dimension at most .n(n + 1)/2. In order to prove it, we pick a point . p ∈ M. Consider the map . E which maps each Killing vector field .V to the pair .(V p , (∇V ) p ). Thus, . E is a linear transformation: .i(M) → T p M × o(T p M). Here .o(T p M) denotes the Lie algebra of skew-adjoint linear operators on .T p M (which can be represented by .n × n skew-symmetric matrices). The previous lemma shows that . E is injective. Denoting the Lie algebra of semi-orthogonal group . O(ν, n − ν) (group of .n × n matrices that preserve the semiEuclidean scalar product of .n-dimensional vector space .Rnν with index .ν) by .oν (n), = n(n+1) , we conclude that.dim(i(M)) ≤ dim(T p (M)) + dim(oν (n)) = n + n(n−1) 2 2 completing the proof. The following result is well known. A complete and connected semi-Riemannian manifold . M admits a maximal isomif and only if . M has constant curvature. etry group of dimension . n(n+1) 2 It is known that a Killing vector field .V on a complete semi-Riemannian manifold . M is complete, i.e. the maximal integral curves of . V are defined for all values of the parameter, i.e. defined on the entire real line. At this point, we consider the following examples of Killing vector fields on certain manifolds. Example 3.1 A Killing vector .V on the 2-dimensional Euclidean space .R2 with coordinates (.x, y) has its components .V1 , V2 governed by Killing equations that reduce to .∂1 V1 = 0, ∂2 V2 = 0, ∂1 V2 + ∂2 V1 = 0. These equations integrate easily to .V1 = −ay + c and .V2 = ax + b which involve three parameters .a, b, c. Hence .(V1 , V2 ) = (−ay + c, ax + b) = (c, b) + a(−y, x) and hence generates a 2-parameter group of translations a 1-parameter group of rotations. Thus, the Euclidean .R2 admits a 3-parameter group of isometries, which = 3. Obviously, the Euclidean .R2 is flat and hence has is maximal because . 2(2+1) 2 constant curvature .0. Example 3.2 The isometries of the .n-dimensional Euclidean space .Rn with Euclidean metric .g = d x 1 ⊗ d x 1 + · · · + d x n ⊗ d x n are the Euclidean transformarotations), and are generated by the Killing vector tions (.n translations and . n(n−1) 2 fields . ∂∂x i and .x j ∂∂x i − x i ∂ ∂x j , where .i, j = 1, . . . , n.

Example 3.3 The isometries of .n-dimensional Minkowski space-time .R1n with metric .g = −d x 0 ⊗ d x 0 + d x 1 ⊗ d x 1 + · · · + d x n−1 ⊗ d x n−1 are the Lorentz transformations generated by Killing vector fields: .

∂ [.n ∂xα

translations].

36

3 Conformal Transformations .

x 0 ∂∂x i + x i ∂∂x 0 [.n − 1 boosts]

.

spatial rotations] x i ∂ ∂x j − x j ∂∂x i [. (n−1)(n−2) 2

where .α = 0, 1, . . . , n, and .i, j = 1, . . . , n − 1. In both Examples 3.2 and 3.3, dimension of the Lie algebra of Killing vector fields . is maximal, i.e. . n(n+1) 2 Example 3.4 The unit 2-sphere . S 2 with its standard metric .g = dθ ⊗ dθ + sin2 θ dφ ⊗ dφ in the local coordinates (.x 1 = θ, x 2 = φ). Its nonzero Christoffel symbols 1 2 = − sin θ cos θ and .12 = cot θ . Hence the Killing equations become are .22 ∂ V1 = 0,

(3.21)

∂ V2 + (sin θ )(cos θ )V1 = 0,

(3.22)

∂ V2 + ∂2 V1 − 2(cot θ )V2 = 0.

(3.23)

. 1

. 2

. 1

Equation (3.21) integrates to .V1 = f (φ). Using this and integrating Eq. (3.22), one gets .V2 = −F(φ) sin θ cos θ + h(θ ), where . F(φ) is the anti-derivative of . f (φ) and .h is a function of .θ . Substituting these values of . V1 and . V2 in (3.23) and separating the variables provides the ordinary differential equations .

dh − 2(cot θ )h(θ ) = k, dθ

df + F(φ) = −k, dφ

where .k is a constant. A straightforward solution of these two equations for .g(θ ) and f (φ) and substitution in the components of .V shows that

.

.

V1 = A sin φ + B cos φ V2 = (A cos φ − B sin φ) sin θ cos θ + C sin2 θ

where . A, B, C are arbitrary constants. Also, we have .V 1 = V1 and .V 2 = Summing up, we obtain .

1 sin2 θ

V2 .

V = (A sin φ + B cos φ)∂θ + (C + A cos φ cot θ − B sin φ cot θ )∂φ

which yields the following basis vectors for the Killing algebra of . S 2 : .

K 1 = − cos φ∂θ + cot θ sin φ∂φ , K 2 = sin φ∂θ + cot θ cos φ∂φ , K 3 = ∂φ .

These Killing vector fields generate rotations about the .x, y and .z axes, and, therefore, generate the Lie algebra of . S O(3). Hence, . S 2 is the homogeneous space . S O(3)/S O(2).

References

37

In higher dimensions, one can verify the fact that . S n = S O(n + 1)/S O(n) is a homogeneous space (actually, a space form, because it has constant curvature) with n(n+1) independent Killing vectors. . 2

References 1. Liouville, J.: Extension au cas des trois dimensions de la question du trace geographique. In: Monge, G. (ed.) Note VI, Applications de l’analyse a la geometrie, pp. 609–917. Bachelier, Paris (1850) 2. Lie, S.: Ueber Complexe, insbesondere Linien- und Kugel-Complexe mit Anwendung auf die Theorie partieller Differentialgleichungen. Math. Ann. 5, 145–246 (1872) 3. Wald, R.M.: General Relativity. University of Chicago Press (1984) 4. K.uhnel, ¨ W., Rademacher, H.-B.: Liouville’s theorem in conformal geometry. Jour. de Mathematiques pures et Appliquees 88, 251–260 (2007) 5. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) 6. Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics. Part I. Elsevier Science Publishers, North-Holland (1982) 7. Kuiper, N.H.: On conformally flat spaces in the large. Ann. Math. 50, 916–924 (1949) 8. Bach, R.: Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungstensorbegriffs. Math. Z. 9, 110–135 (1921) 9. Myers, C.B., Steenrod, N.E.: The group of isometries of a Riemannian manifold. Ann. Math. 40, 400–416 (1939) 10. Palais, R.S.: A global theory of transformation groups. Mem. Amer. Math. Soc. 22, 1–23 (1957) 11. O’Neill, B.: Semi-Riemanniann Geometry with Applications to Relativity. Academic, New York (1983)

Chapter 4

Conformal Vector Fields

4.1 Conformal Vector Fields A conformal Killing vector field .V on a semi-Riemannian manifold (. M, g) is a natural generalization of a Killing vector field, and is defined by LV g = 2σ g

.

(4.1)

for a smooth function .σ on . M. A conformal Killing vector field (abbreviated as C K V ) is also called a conformal vector field and sometimes a conformal motion, and determines a conformal symmetry of the semi-Riemannian manifold (. M, g). It preserves the metric.g up to a general point-dependent scale factor.2σ . For.σ constant, . V is a homothetic vector field, and for .σ = 0, . V reduces to a Killing vector field. The .g-trace of the conformal Killing equation (4.1) provides the relationship .

σ =

.

1 div V n

(4.2)

between the conformal scale function .σ and the divergence of the .C K V . The local one-parameter group of local transformations generated by a vector field . V is conformal if and only if . V is a .C K V . The set of all .C K V fields on . M form a Lie algebra under usual addition and Lie bracket of vector fields. To see this, let us have two .C K V fields .V and .W on (. M, g) with the .C K V equations: .LV g = 2σ g and .LW g = 2τ g, respectively. Then .L[V,W ] g = L V LW g − LW L V g = 2(V τ − W σ )g. Here, we would like to state the following important result. Theorem 4.1 The group of conformal transformations of a connected.n-dimensional (.n ≥ 3) semi-Riemannian manifold is a Lie transformation group (conformal group) of dimension .≤ 21 (n + 1)(n + 2).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_4

39

40

4 Conformal Vector Fields

The upper bound on the dimension of the conformal group, as mentioned in this theorem, can be verified by the fact that the integrability conditions of the .C K V equations (4.1) imply that the Lie algebra of.C K V ’s is of dimension.≤ 21 (n + 1)(n + 2) (as explained on page 285 of Eisenhart’s book [1]). The maximum dimension of the conformal algebra occurs if the manifold (. M, g) is conformally flat (see, for example, Yano [2]). Maartens-Maharaj [3] obtained the full conformal algebra .G 15 generating the 15-parameter group of conformal motions in Robertson–Walker space-time that is known to be conformally flat. At this point, we recall the following result from [1]: On an .n-dimensional semiRiemannian manifold . M of constant curvature .k, there exists a coordinate system (.x i ) such that the distance element of . M is ds 2 = (a1 + ... + an )−2

n 

.

i (d x i )2

(4.3)

i=1

where each .i = ±1, according to the signature of . M, and .ai =i [a(x i )22 bi x i + ci ], n (aci − bi2 ) = k. and the constants .ai , bi , ci are tied up with the condition: .4 i=1 Obviously, from Eq. (4.3), a space (. M, g) of constant curvature is conformally flat and hence admits a maximal group of conformal motions, of dimension. 21 (n + 1)(n + 2). We have the following integrability conditions for a .C K V field as defined by Eq. (4.1) (see [2]): (LV ∇)(X, Y ) = (X σ )Y + (Y σ )X − g(X, Y )Dσ,

(4.4)

LV ikj = (∇i σ )δ kj + (∇ j σ )δik − gi j ∇ k σ

(4.5)

(LV R)(X, Y, Z ) = −(∇∇σ )(Y, Z )X + (∇∇σ )(X, Z )Y − g(Y, Z )∇ X Dσ + g(X, Z )∇Y Dσ,

(4.6)

LV R i jkl = −δki ∇l ∇ j σ + δli ∇k ∇ j σ − g jl ∇k ∇ i σ + g jk ∇l ∇ i σ

(4.7)

(LV Ric)(X, Y ) = −(n − 2)(∇∇σ )(X, Y ) − (σ )g(X, Y ),

(4.8)

LV Ri j = −(n − 2)∇i ∇ j σ − (σ )gi j

(4.9)

LV r = −2(n − 1)σ − 2r σ

(4.10)

.

i.e. .

.

i.e. .

.

i.e. .

.

4.2 Zeros of Conformal Vector Fields

41

.

LV W = 0,

(4.11)

LV W i jkl = 0

(4.12)

i.e. .

where . D denotes the gradient operator of .g, and . denotes the Laplacian .div ·D = g i j ∇i ∇ j .

4.2 Zeros of Conformal Vector Fields Recall that a conformal manifold (. M, [g]) is a smooth manifold. M with a Riemannian metric .g and the set .[g] of all Riemannian metrics conformally related to .g. If it admits a complete and essential conformal vector field (whose global flow acts by conformal transformations, but not by isometries with respect to any compatible metric), then it is conformally flat [4]. For a given .C K V field .V (whose flow is not globally defined in general) there might exist local metrics in the conformal class, preserved by the local flow of .V . The union of the domains of these .V -invariant local metrics is the open set of non-essential points of .V . A point . p ∈ M is said to be essential for the .C K V field .V if there exists no local metric in the conformal class .[g], preserved by the local flow of . V around . p. If .V does not vanish at some point . p, then . p is not essential. To show this, let .V be a .C K V field satisfying (4.1) and .g¯ = 2 g for a smooth positive function .. As . V  = 0 at. p, it is non-vanishing on some open neighborhood.U of. p. A straightforward computation provides the equation LV g¯ = 2(V log  + σ )g. ¯

.

If we choose . such that .−2 = g(V, V ), then the above equation shows that .V is Killing with respect to .g¯ on .U . One may note that this argument holds also on a semi-Riemannian manifold . M with a .C K V field .V as long as .g(V, V ) = 0 on . M. In [5], Kobayashi proved that, for any Killing vector field.V on a Riemannian manifold, the connected components of the zero set of .V are mutually isolated totally geodesic submanifolds of even codimensions. Assuming compactness, Blair [6] generalized Kobayashi’s result for.C K V fields, in which the word “geodesic” is replaced by “umbilical” and the codimension clause is relaxed for one-point connected components. Belgun, Moroianu and Ornea [7] showed that the set of all essential points of a .C K V field .V on a Riemannian manifold of dimension .n ≥ 2 consists of isolated zeros of .V (this is therefore, an extension to higher dimensions, of the 2-dimensional result that the zeros of a non-constant holomorphic function are isolated). As an application of this result, they generalized Blair’s result by waiving the compactness condition. This generalized result also follows from a theorem of Frances [8], which states that a .C K V field on a Riemannian manifold is linearizable at any given zero

42

4 Conformal Vector Fields

p unless some neighborhood of . p is conformally flat. We recall the fact that a vector field .V is said to be linearizable about a point . p if, in some coordinate neighborhood of . p, the components of .V are linear functions of the coordinates.

.

4.3 Zeros of Conformal Vector Fields on Semi-Riemannian Manifolds We point out that the aforementioned result of Belgum, Moroianu and Ornea does not hold for indefinite metrics. The connected components of the zero set of any .C K V field.V in a semi-Riemannian manifold (. M, g) of arbitrary signature are of two types. The first type consists of points at which .V is essential, that is, cannot be turned into a Killing vector field by a local conformal change of the metric. In a component of the second type, points at which .V is non-essential form a relatively open dense subset that is a totally umbilical submanifold of (. M, g), and so is the set of those points in a non-essential component at which .V is essential (unless this set, consisting of all the singular points of the component, is empty). Both kinds of null totally geodesic submanifolds arising here carry a 1-form, defined up to multiplications by functions without zeros, which satisfies a projective version of the Killing equation. In [9], Derdzinski studied the zeros of a .C K V field for a metric of arbitrary signature and obtained the following result. Theorem 4.2 (Derdzinski) Let . Z denote the set of zeros of a .C K V field .V on a semi-Riemannian manifold (. M, g) of dimension .n ≥ 3. Then every point .z ∈ Z has a neighborhood .U  in . M such that, for some star-shaped neighborhood .U of 0 in    . Tz M, and some metric . g on .U conformal to . g, the exponential map .ex pz of . g at . z is   defined on .U and maps .U diffeomorphically onto .U , while . Z ∩ U = ex pz [E ∩ U ] for . E ⊆ Tz M which is (a) a vector subspace of .Tz M, or (b) the set of all null vectors in a vector subspace . H ⊆ Tz M. The singular subset . of . Z ∩ U  equals .ex pz [H ∩ H ⊥ ∩U ] in case (b), if the metric restricted to . H is not semi-definite, and  . = ∅ otherwise. The connected components of .(Z ∩ U )\ are totally umbilical submanifolds of (. M, g), and their codimensions are even unless . − ∅ and . Z ∩ U  is a null totally geodesic submanifold of (. M, g). In addition, .div V is constant along each connected component of . Z . In particular, for the Lorentzian case, the above-mentioned theorem can be rephrased so as to reflect the fact that null submanifolds can be at most 1-dimensional, while the only singularities of the zero set that may occur are those associated with null cones in Lorentzian subspaces of the tangent space. We now recall the following two results of Beig [10] and Derdzinski [9] on the zeros of a .C K V field on a semi-Riemannian manifold. Theorem 4.3 (Beig) Let . Z be the zero set of a .C K V field .V on a semi-Riemannian manifold (. M, g) of dimension .n ≥ 3. A point . p ∈ Z is non-essential if and only if

4.4 Essential Conformal Vector Fields

σ ( p) = 0 and (∇σ ) p ∈ (∇V ) p (T p M)

.

43

(4.13)

for the conformal scale function .σ as defined by (4.1). In other words, . p ∈ Z is essential if and only if either σ ( p) = 0, or σ ( p) = 0 and (∇σ ) p ∈ / (∇V ) p (T p M).

.

(4.14)

Theorem 4.4 (Derdzinski) Suppose that .V is a .C K V field on a semi-Riemannian manifold (. M, g) of dimension .n ≥ 3. If . Z is the set of zeros of .V , while . p ∈ Z satisfies Eq. (1.14), and .C = {W ∈ T p M : g p (W, W ) = 0} denotes the null cone, then, with .U, U  as mentioned before and . H = K er (∇V ) p ∩ K er (dσ ) p , . Z ∩ U  = ex p p [C ∩ H ∩ U ]. In addition, the conformal scale function .σ is constant along each connected component of . Z . Many researchers have studied semi-Riemannian manifolds with conformal vector fields in terms of the structure of their zero sets. It is noteworthy that a zero . p of the .C K V field .V is a fixed point of the local diffeomorphisms .φt generated by . V , i.e. .φt ( p) = p. The differential map .(φt ) p : T p M → T p M is represented with respect to the basis .( ∂∂x i ) p of .T p M for some coordinate neighborhood around . p with i coordinates .x i , by the transpose of the matrix .exp t A where . Ai j = ( ∂∂ Vx j ) p (see Hall [11]). Kühnel and Rademacher [4] have provided a classification of complete semiRiemannian manifolds with a constant scalar curvature and a non-trivial gradient conformal vector field, and Hall et al. [12] have obtained a sufficient condition in terms of the Weyl curvature tensor, conformal scale function .σ and conformal bivector . Fi j , for the existence of local conformal changes of a space-time metric around a zero of the .C K V that turns the .C K V field into a homothetic vector field.

4.4 Essential Conformal Vector Fields A .C K V field which becomes Killing after a suitable conformal change of the metric is called inessential. In the case of a Riemannian manifold any .C K V without zeros is inessential. As much is known about Killing vector fields, it is more interesting to study essential .C K V fields, i.e. those .C K V fields which cannot be reduced to Killing vector fields by a global conformal change of the metric. We mention the following two results on essential .C K V fields in the Riemannian case. Theorem 4.5 (Alekseevskii [13], Ferrand [14], Yoshimatsu [15]) An .n-dimensional Riemannian manifold admitting a complete and essential .C K V field is conformally diffeomorphic to either the stand sphere . S n or the Euclidean space . E n . Theorem 4.6 (Obata [16], Lelong-Ferrand [17], Lafontaine [18]) A compact .ndimensional Riemannian manifold admitting an essential .C K V field is conformally diffeomorphic to the standard sphere . S n .

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4 Conformal Vector Fields

This result resolved the following conjecture (Lichnerowicz conjecture) raised by Lichnerowicz: “Among compact Riemannian manifolds, the standard sphere is the only essential conformal structure”. So, the following natural question arises “Suppose that the group of conformal transformations of a compact pseudo-Riemannian (also known as semi-Riemannian) manifold (. M, g) of dimension .n ≥ 3 is essential. Is then (. M, g) necessarily conformally flat?” That the answer to this question is in the negative was shown by the following result of Frances [19]. Theorem 4.7 (Frances) For every . p ≥ 2, and .q ≥ p, one can construct on the product . S 1 × S p+q−1 a 2-parameter family of distinct type (. p, q) analytic pseudoRiemannian conformal structures, which are not conformally flat, and with an essential conformal group. The following result of Ejiri [20] is worth mentioning. Theorem 4.8 (Ejiri) For any .n, there is a compact Riemannian .n-manifold of constant scalar curvature admitting a .C K V field without zeros. For example, . S 1 × S 3 with the warped product metric .g = dt 2 + (2 + cos t)g S , . S. For this metric, the scalar where .g S is the standard metric on the unit sphere √ curvature is constant and the vector field .V = 2 + cos t∂t has no zeros and is an inessential .C K V .

4.5 Lichnerowicz Conjecture in CR Geometry A CR structure on (.2n + 1)-dimensional manifold . M is a couple (. D, J ) where (i) D is a .2n-dimensional subbundle of .T M, (ii) . J is pseudocomplex (almost complex) operator on . D such that . J : D → D, . J 2 = −I and (iii) for all vector fields . X, Y tangent to . D, the vector field .[J X, Y ] + [X, J Y ] is tangent to . D and . J is integrable, i.e.. J ([J X, Y ] + [X, J Y ]) = [J X, J Y ] − [X, Y ]. A differentiable map between two .C R manifolds is called a .C R map if it conjugates the hyperplane distributions and the corresponding pseudocomplex operators. An automorphism of a .C R manifold is a diffeomorphism of . M that is a .C R map. The group of all automorphisms is denoted by . Aut (M) and its identity component by . Aut0 (M). A 1-form (possibly local) .η on a .C R manifold . M such that . D = ker (η) is called a (local) calibration. For orientable . M, we can find a global calibration.. For a given calibration .η, Levi form . L η is defined on . D by .

.

L η (., .) = dη(., J.).

A .C R structure is called strictly pseudoconvex when its Levi form is definite (we can choose the calibration so that it is positive definite). Hence .dη is non-degenerate on .ker (η), i.e. .η is a contact form. For a given calibration .η, there exists a unique vector field (Reeb vector field) .ξ such that .η(ξ ) = 1 and .(dη)(ξ, .) = 0. If the Levi form is positive definite, then the Webster metric on . M is defined as the Riemannian

References

45

metric . L η oπ + η2 , where .π is the linear projection .π : T M → D. There is also a natural metric connection (which need not be torsion free) .∇η on .T M and hence its curvature tensor . Rη . There are two flat .C R models, (i) the .C R sphere .S 2n+1 which is the unit sphere n S = {(z 0 , ...., z n ) ∈ Cn+1 : k=0 |z k |2 = 1}

.

in .Cn+1 with its strictly pseudo-convex .C R structure. (ii) The Heisenberg group is the .C R non-compact manifold .H obtained by removing one point of .S, and hence diffeomorphic to the Euclidean space. We now state the following results obtained by Webster [21]. Theorem 4.9 (Webster) If a .2n + 1-dimensional strictly pseudoconvex .C R manifold . M is compact and . Aut0 (M) is non-compact, then . M is flat. Theorem 4.10 (Webster) If a .2n + 1-dimensional strictly pseudoconvex .C R manifold . M is compact, has finite fundamental group and . Aut0 (M) is non-compact, then . M is globally equivalent to the standard sphere. Theorem 4.11 (Webster) If a .2n + 1-dimensional strictly pseudoconvex .C R manifold . M is compact and . Aut0 (M) admits a non-compact one-parameter Lie subgroup that has a fixed point . p0 , then . M is globally equivalent to the standard sphere .S. Finally, we conclude with the following result of Schoen [22] in which he gave a proof in the conformal case adaptable to .C R geometry. Theorem 4.12 (Schoen–Webster) Let . M be a strictly pseudoconvex .C R manifold, not necessarily compact. If its automorphism group . Aut (M) acts non-properly, then . M is either the standard .C R sphere .S or .S with one point deleted. We note here that an action of a topological group .G is proper if for any compact subset . K of . M, the subset .G K = {g ∈ G : g(K ) ∩ K = ∅} of .G is compact. For . M compact, . Aut (M) acts properly if and only if it is compact. For details, we refer to Kloeckner and Minerbe [23].

References 1. Eisenhart, L.P.: Riemannian Geometry, 2nd edn. Princeton University Press (1949) 2. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) 3. Maartens, R., Maharaj, S.D.: Conformal Killing vectors in Robertson-Walker spacetimes. Class. Quant. Grav. 3, 1005–1011 (1986) 4. Liouville’s theorem in conformal geometry: K.uhnel, ¨ W., Rademacher, H-B. Jour. de Mathematiques pures et Appliquees 88, 251–260 (2007) 5. Kobayashi, S.: Fixed points of isometries. Nagoya Math. J. 13, 63–68 (1958) 6. Blair, D.E.: On the zeros of a conformal vector field. Nagoya Math. J. 55, 1–3 (1974) 7. Belgum, F., Moroianu, A., Ornea, L.: Essential points of conformal vector fields. J. Geom. Phys. 61, 589–593 (2011)

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8. Frances, C.: Essential conformal structures in Riemannian and Lorentzian geometry. In: D.V. Alekseevsky, H. Baum (eds.) Recent Developments in Pseudo-Riemannian Geometry. ESI Lecture Mathematics Physics, pp. 231–260. EMS, Zurich (2008) 9. Derdzinski, A.: Zeros of conformal fields in any signature, Class. Quant. Grav. 28, 075011 (23 pages) (2011) 10. Beig, R.: Conformal Killing Vectors Near a Fixed Point. Institut fur Theoretische Physik, Universitat Wien, Unpublished manuscript 11. Hall, G.S.: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore (2004) 12. Hall, G.S., Capocci, M., Beig, R.: Zeros of conformal vector fields. Class. Quant. Grav. 14, L49 (1997) 13. Alekseevskii, D.V.: Groups of conformal transformations of Riemannian spaces. Math. USSR Sbornik 18, 285–301 (1972) 14. Ferrand, J.: Sur une lemme d’Alekseevskii relatif aux transformations conformes. C.R. Acad. Sci. Paris, Ser. A 284, 121–123 (1977) 15. Yoshimatsu.: On a theorem of Alekseevskii concerning conformal transformations. J. Math. Soc. Japan 28, 278–289 (1976) 16. Obata, M.: The conjectures about conformal transformations. J. Differ. Geom. 6, 247–258 (1971) 17. Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des varietes riemanniennes. Acad. Roy. Belg. Sci. Mem. Coll. 8(2), 39 (1971) 18. Lafontaine, J.: The theorem of Lelong-Ferrand and Obata. In: Kulkarni, R.S., Pinkall, U. (eds.) Conformal Geometry. Aspects of Mathematics E, vol. 12, pp. 65–92. Vieweg Verlag Braunschweig, Weisbaden (1988) 19. Frances, C.: About pseudo-Riemannian Lichnerowicz conjecture. Trans. Groups 20, 1015– 1022 (2015) 20. Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J. Math. Soc. Japan 23, 261–266 (1981) 21. Webster, S.M.: On the transformation group of a real hypersurface. Trans. Am. Math. Soc. 231, 179–190 (1977) 22. Schoen, R.: On the compact and CR automorphism groups. Geom. Funct. Anal. 5, 464–481 (1995) 23. Kloeckner, B., Minerbe, V.: Rigidity in CR geometry: the Schoen-Webster theorem. Differ. Geom. Appl. 27, 399–411 (2009)

Chapter 5

Integral Formulas and Conformal Vector Fields

5.1 Integration Theorems First, we recall the Stokes’ theorem. Theorem 5.1 (Stokes’ Theorem) If . M is a compact orientable .m-dimensional manifold . M with boundary .∂ M and .ω is an (.m − 1)-form on . M, then 



.

∂M

ω=

dω.

(5.1)

M

For a Riemannian manifold (. M, g), if . N denotes a unit outward vector field normal to .∂ M, Stokes’ theorem becomes Gauss’ divergence Theorem. Theorem 5.2 If (. M, g) is a compact orientable Riemannian manifold with boundary ∂ M, and .V is a smooth vector field on . M, then

.



 (div V )dv =

.

M

∂M

g(N , V )d S

(5.2)

where .dv denotes the volume element of . M and .d S is the surface element of .∂ M. If . M is semi-Riemannian, its boundary may have some or all of its points where the metric becomes degenerate. Thus, at those points, the outward normal is not well defined. This issue was raised by Duggal [1] and then Unal [2], Garcia-Rio and Kupeli [3] succeeded in resolving this issue in part by assuming some conditions on the degenerate part of .∂ M. Let .∂ M+ , ∂ M− and .∂ M0 denote the pairwise disjoint subsets of.∂ M on which the nonzero vectors orthogonal to.∂ M are spacelike, timelike and null respectively, such that ∂ M = ∂ M+ ∪ ∂ M− ∪ ∂ M0 .

.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_5

47

48

5 Integral Formulas and Conformal Vector Fields

One can show that .∂ M+ , .∂ M− and .∂ M  = (∂ M+ ) ∪ (∂ M− ) are open submanifolds of .∂ M. We use the same symbol . N for the unit normal vector field to .∂ M induced on .∂ M  . We also denote by .∂ M  = i N  the induced volume element on .∂ M+ and .∂ M− , when restricted to these submanifolds, respectively. We now state the following divergence theorems due to Unal [2]. Theorem 5.3 (Unal) Suppose (. M, g) is an oriented semi-Riemannian manifold with √ boundary .∂ M and semi-Riemannian volume element . = |g|d x 1 ∧ ... ∧ d x m . If .V is a vector field on . M, with compact support, and .∂ M0 has measure zero in .∂ M, then 

 (div V ) =

.

 ∂ M+

M

g(N , V )∂ M  −

∂ M−

g(N , V )∂ M  .

Theorem 5.4 (Unal) Suppose (. M, g) is an oriented semi-Riemannian manifold with boundary .∂ M and semi-Riemannian volume element .. If .V is a vector field on . M, with compact support, such that .V is tangent to .∂ M at the points of .∂ M0 , then 

 (div V ) =

.

 ∂ M+

M

g(N , V )∂ M  −

∂ M−

g(N , V )∂ M  .

5.2 Integral Formulas Let us assume (. M, g) to be a compact orientable.n-dimensional Riemannian manifold without boundary. If .dv denotes the volume element of . M, then the divergence theorem becomes  . (div V )dv = 0 (5.3) M

for a smooth vector field .V on . M. If .V is a gradient vector field, i.e. .V = D f , then we have  . ( f )dv = 0 (5.4) M

where . denotes the Laplacian .∇i ∇ i . Computing the Laplacian of . f 2 , we have . f 2 = ∇i ∇ i ( f 2 ) = ∇i (2 f ∇ i f ) = 2[(∇i f )∇ i f + f ∇i ∇ i f ]. Thus we find the relation  f 2 = 2(|D f |2 + f  f ).

.

(5.5)

Its integral over . M and the use of (5.4) with . f 2 in place of . f provides the integral formula  . (|D f |2 + f  f )dv = 0. (5.6) M

An immediate consequence of the above formula is Hopf’s lemma.

5.2 Integral Formulas

49

Lemma 5.1 (Hopf) If a smooth function . f on a compact orientable Riemannian manifold . M without boundary satisfies . f = 0, then it is constant on . M. A generalization of the above lemma is Bochner’s lemma which considers . f ≥ 0 or . f ≤ 0. Application of the formula (5.4) shows . f = 0. Thus, by virtue of Hopf’s lemma, we have the following. Lemma 5.2 (Bochner) If a smooth function . f on a compact orientable Riemannian manifold . M without boundary satisfies . f ≥ 0 or . f ≥ 0, then it is constant on . M. Let us now consider an eigenfunction . f of ., i.e. a non-constant function . f satisfying . f = c f for a nonzero constant .c on . M. Substituting this in the formula (5.6) we find . M (c f 2 + |D f |2 )dv = 0. This shows that .c < 0. We state this consequence as follows. Proposition 5.1 If a non-constant function . f on a compact orientable Riemannian manifold without boundary satisfies . f = c f , and .c is a nonzero constant, then .c is negative. Remark 5.1 If we had defined the Laplacian as . = −∇i ∇ i , then the eigenvalue .c would have been positive, i.e. the spectrum of . would be positive. In order to obtain the next formula, we consider a smooth vector field .V on . M and set . f = |V |2 . Hence  f = (∇ i ∇i )(V j V j ) = ∇ i [(∇i V j )V j + V j (∇i V j )]

.

= 2[(∇ i ∇i V j )V j + (∇i V j )(∇ i V j )]. Integrating it over the compact . M without boundary and applying formula (5.4) provides  . (|∇V |2 + < V, V >)dv = 0 (5.7) M

where acting on a vector field .V as .V = n . denotes the rough i Laplacian j ∇ ∇ V − ∇ V = (∇ ∇ V )∂ denotes and . is the inner product with ∇ei ei i j i=1 ei i respect to the Riemannian metric .g on . M. The foregoing integral formula implies the following result of Bochner [4]. Proposition 5.2 If the second covariant derivative of a vector field .V vanishes on a compact orientable Riemannian manifold . M without boundary, then .V is parallel on . M. Next, for a vector field .V on a Riemannian manifold . M, let us compute the divergence:

50

5 Integral Formulas and Conformal Vector Fields .

div(∇V V − (div V )V ) = ∇i (V j ∇ j V i − (∇ j V j )V i ) = (∇i V j )(∇ j V i ) + V j ∇i ∇ j V i − (∇i ∇ j V j )V i − (∇i V i )2 = (∇i V j )(∇ j V i ) − (∇i V i )2 + V i (∇ j ∇i V j − ∇i ∇ j V j ) = (∇i V j )(∇ j V i ) − (∇i V i )2 + Rki V k V i .

Its integral over a compact. M without boundary and the use of (5.2) give the following integral formula:  [Ri j V i V j + (∇i V j )(∇ j V i ) − (div V )2 ]dv = 0.

.

(5.8)

M

Assuming .V ∗ to be the 1-form metrically equivalent to .V , i.e. .V ∗ (X ) = g(V, X ), we have |d V ∗ |2 = (∇i V j − ∇ j Vi )(∇ i V j − ∇ j V i ) = 2[|∇V |2 − (∇i V j )(∇ j V i )].

.

The use of the above equation in (5.8) provides the formula  .

1 [Ric(V, V ) + |∇V |2 − |d V ∗ |2 − (div V )2 ]dv = 0. 2 M

(5.9)

Let us compute the squared norm of the strain tensor .LV g as |LV g|2 = (∇i V j + ∇ j Vi )(∇ i V j + ∇ j V i )

.

= 2[|∇V |2 + (∇i V j )(∇ j V i )]. Using it in (5.8) gives the formula 

1 (5.10) [Ric(V, V ) − |∇V |2 + |LV g|2 − (div V )2 ]dv = 0. 2 M  If we take.V as a Killing vector field, then the above equation reduces to. M [Ric(V, V ) − |∇V |2 ]dv = 0. This leads to the following result. .

Theorem 5.5 (Bochner [4]) If the Ricci curvature of a compact Riemannian manifold . M is negative semi-definite, then a Killing vector field on . M is parallel. On the other hand, if the Ricci tensor is negative definite, then a Killing vector field other than zero does not exist on . M.

5.2 Integral Formulas

51

In view of (5.7), we see that the formula (5.10) assumes the form  1 . [< QV + V, V > + |LV g|2 − (div V )2 ]dv = 0. 2 M

(5.11)

We also note for a smooth function . f and a vector field .V on . M that div( f V ) = ∇i ( f V i ) = (∇i f )V i + f (∇i V i ).

.

Its integral over . M yields the following generalization of divergence theorem  [< V, D f > + f (div V )]dv = 0.

.

(5.12)

M

Using this with . f = div V and the identity |LV g|2 =

.

2 4 (div V )2 + |LV g − (div V )g|2 , n n

we see that formula (5.11) becomes  1 2 n−2 D(div V ), V > + |LV g − (div V )g|2 ]dv = 0 . [< V + 2 n n M

(5.13)

where . denotes the Yano operator . : X(M) → X(M) such that .X = X + Q X for all . X ∈ X(M). For a Killing vector field .V , we know that .LV  i jk = ∇ j ∇k V i + R ikm j V m = 0. Contracting it with .g jk gives .V = 0. Further, a Killing vector field has zero divergence. Conversely, for a compact Riemannian manifold satisfying .V = 0, .div V = 0, formula (5.13) shows that .V is Killing. Thus we have (Yano [5]) the following. Theorem 5.6 A vector field .V on a compact Riemannian manifold is Killing if and only if .V = 0 and .div V = 0. At this point, we will relate the Hodge Laplacian acting on vector fields with the Yano operator. Let . X be a vector field and .ξ the 1-form metrically equivalent to . X . Then, the Hodge Laplacian . = dδ + δd where .δ is the co-differential operator 1 ∇ i θi j1 ..... jr d x j1 .... defined by .δξ = −∇ i ξi [in general, for an .r form .θ as .δθ = − (r −1) jr d x ]. Just like the exterior derivative operator .d, we have .δδ = 0. Hodge Laplacian . on .r -forms is defined as .dδ + δd. By a straightforward computation, we can show j for a 1-form .ξ that .ξ = (∇ j ∇ j ξi + Ri ξ j ) ∂∂x i . So, for a vector field .X metrically associated with .δξ , we have .X = (−∇ j ∇ j X i + R ij X j ) ∂∂x i . The following relation is evident: .X + X = 2Q X.

52

5 Integral Formulas and Conformal Vector Fields

5.3 Some Classical Results on Conformal Vector Fields Let us consider a conformal vector field .V on a Riemannian manifold, defined by LV g = 2σ g. Its .g-trace gives .σ = divn V . Inserting these equations in (5.10), one obtains  n−2 (div V )2 − Ric(V, V )]dv = 0. . [|∇V |2 + n M

.

An immediate consequence of the above formula is the following result of Yano [6]. Theorem 5.7 If the Ricci curvature .Ric(V, V ) of a compact Riemannian manifold along a conformal vector field .V is non-positive, then .V is parallel. If .Ric(V, V ) is negative definite, then .V = 0 on . M. Let us recall from Chap. 4 that a conformal vector field.V defined as above satisfies the equation .(L V ∇)(X, Y ) = (X σ )Y + (Y σ )X − g(X, Y )Dσ, for any . X, Y ∈ X(M); contracting this equation at . X and .Y provides the relation V +

.

n−2 D(div V ) = 0. n

(5.14)

The other way around, if (5.14) holds for a vector field .V on a compact Riemannian manifold . M, then the formula (5.13) shows that .V is conformal. Summing up, we have the following result (Lichnerowicz [7], Sato [8], Yano [6]). Theorem 5.8 A vector field .V on a compact Riemannian manifold of dimension .n D(div V ) = 0. is conformal if and only if .V + n−2 n We close this section with the following result for conformal vector fields on a compact Riemannian manifold with boundary. Theorem 5.9 (Schoen [9]) Let (. M, g) be an .n-dimensional (.n > 2) compact Riemannian manifold with boundary .∂ M. Let .r denote the scalar curvature of . M and . V a conformal vector field on . M. Then  (LV r )dv =

.

M

2n n−2



r (Ric − g)(V, N )ds, n ∂M

where . N is the unit outward normal vector field to .∂ M, .dv and .ds denote volume and surface measures on . M and .∂ M, respectively. Let us now consider an Einstein manifold of dimension.n > 2 (hence.r is constant) admitting a conformal vector field .V . The Lie derivative of the Einstein condition r g along .V , along with the combined use of equations (4.8) and (4.10) of .Ric = n Chap. 4, provides the following equation:

5.3 Some Classical Results on Conformal Vector Fields

∇∇σ = −

.

r g. n(n − 1)

53

(5.15)

So, if .r > 0 (which is true for compact case), then we invoke the following theorem of Obata [10]. Theorem 5.10 (Obata) A complete Riemannian manifold . M of dimension .n ≥ 2 admits a non-trivial solution .σ of the system of partial differential equations .∇∇σ = −c2 σ g for a constant .c > 0 if and only if . M is isometric to a sphere of radius . 1c in the .n + 1-dimensional Euclidean space. Thus, for .r > 0, we conclude that a complete Einstein . M with a non-homothetic conformal vector field is isometric to a round sphere . S n . Yano and Nagano [11] waived the positivity of .r by assuming the completeness of .V and established the following result. Theorem 5.11 (Yano–Nagano) A complete Einstein manifold of dimension .n ≥ 3 admitting a complete non-homothetic conformal vector field is isometric to a round sphere. The completeness of the conformal vector field was unfortunately not included for the above theorem stated in [5]. This requirement cannot be omitted from the hypothesis of the above them, as indicated in the following two examples. (i) Let . f be a non-trivial function on a Riemannian manifold (. M, g) satisfying .∇∇ f = −k f g with a negative constant .k. Then . D f is non-homothetic and conformal, but is not complete, even if (. M, g) is complete. (ii) The vector field .V = x n (x 1 ∂∂x 1 + ...x n−1 ∂ x∂n−1 ) + 21 (x n )2 ∂ ∂x n on the Euclidean space .Rn is non-homothetic and conformal, but is not complete. The aforementioned result of Yano and Nagano was further generalized by Kanai [12] waiving the completeness of .V as follows. Theorem 5.12 (Kanai) An .n (.≥ 3)-dimensional complete Einstein manifold (. M, g) with constant scalar curvature .n(n − 1)k admits a (not necessarily complete) nonhomothetic conformal vector field if and only if one of the following conditions holds: (i) .k > 0 and (. M, g) is isometric to the round sphere of radius . √1k , (ii) .k = 0 and (. M, g) is isometric to the .n-dimensional Euclidean space, (iii) .k < 0 and (. M, g) is isometric to the warped product . R × f (M ∗ , g ∗ ) of the real line . R with a complete curvature.4(n − 1)(n − 2)kc1 c2 by the warping Einstein manifold (. M√∗ , g ∗ ) of scalar √ function . f (t) = c1 e −kt + c2 e− −k t, where .c1 and .c2 are non-negative constants. Let us state some more results as follows. Theorem 5.13 (Lichnerowicz [13], Yano-Obata [14], Bishop–Goldberg [15]) Let a compact Riemannian manifold . M admit a non-Killing conformal vector field .V , with conformal scale function .σ such that one of the following conditions holds. (1) The 1-form associated with .V is exact, (ii) . Dσ is an eigenvector of the Ricci tensor with constant eigenvalues and (iii) .LV Ric is a smooth function multiple of .g. Then . M is isometric to a sphere.

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5 Integral Formulas and Conformal Vector Fields

Theorem 5.14 (Goldberg [16]. Yano [17]) A compact Riemannian manifold . M of dimension .n > 2 with constant scalar curvature .r and admitting a non-Killing conformal vector field with conformal scale function .σ is isometric to a sphere if and only if . M (Ric − nr g)(Dσ, Dσ )dv = 0. Theorem 5.15 (Goldberg [16]. Yano [17]) If a complete Riemannian manifold . M of dimension .n > 2 has constant scalar curvature .r > 0 and admits a non-Killing conformal vector field with conformal scale function .σ , then .σ 2 r 2 ≤ n(n − 1)2 |∇∇σ |2 , and equality holds if and only if . M is isometric to a sphere. Theorem 5.16 (Yano [5]) If a complete Riemannian manifold. M of dimension.n > 2 has constant scalar curvature .r and admits a non-Killing conformal vector field .V that leaves the norm of the Ricci tensor invariant, i.e. .V (| Ric |) = 0, then . M is isometric to a sphere. A corollary of the above stated Yano’s result for constant .r and constant .| Ric | was obtained by Lichnerowicz [13], and another corollary, for . M homogeneous, was obtained by Goldberg and Kobayashi [18]. We note that Yano’s theorem also holds if the Ricci tensor is replaced with the curvature tensor. Several conditions for a compact Riemannian manifold admitting a non-Killing conformal vector field to be conformal and isometric to a sphere were obtained under some curvature conditions by Hsuing and Stern [19], Yano and Sawaki [20], Amur and Hegde [21] and Amur and Pujar [22]. Here, we recall that the group .C(M, g) of all conformal diffeomorphisms of a Riemannian manifold . M is a Lie group with respect to the compact open topology. Let  .C 0 (M, g) be the connected component of the identity of .C(M, g). If . g and . g are  conformal, then .C(M, g) = C(M, g ). The group . I (M, g) of all isometries of . M is a closed subgroup of .C(M, g). A subgroup .G of .C(M, g) is said to be essential if .G is not contained in . I (M, e2 f g) for any smooth function . f on . M, and is inessential otherwise. For a connected compact smooth . M of dimension .> 2, Obata [23] established the following results: (1) .C0 (M, g) is essential if and only if . M is conformal to a Euclidean sphere . S n . (2) If . M has constant scalar curvature .k, then .C0 (M, g) is essential if and only if .k is positive and . M is isometric to a Euclidean sphere . S n of radius . √1k .

5.4 More Results on Conformal Vector Fields In particular, if the 1-form metrically equivalent to a conformal vector field .V is closed, then .V is called a closed conformal vector field. More particularly, if .V is the gradient of a smooth function, then .V is said to be a gradient conformal vector field. Obviously a gradient conformal vector field is closed. If .V is a gradient conformal vector field with .V = D f for a smooth function . f on the Riemannian manifold .(M, g), then we get the Poisson equation . f = nσ . Thus, the geometry of gradient conformal vector fields on a Riemannian manifold is related to the Poisson

5.4 More Results on Conformal Vector Fields

55

equation on the Riemannian manifold. The role of differential equations in studying the geometry of a Riemannian manifold was initiated by the pioneering work of Obata (Theorem 14 stated earlier) that characterizes a sphere. Then Tashiro [24] has shown that the Euclidean spaces .Rn are characterized by the differential equation .Hess f = cg (where .Hess f is the Hessian of a smooth function . f and .c is a nonzero constant). The geometry of conformal vector fields is naturally divided in two classes, (i) that of closed conformal vector fields and (ii) that of non-closed conformal vector fields. The initial work on the conformal vector fields was originated with the geometry of closed (in particular, gradient) conformal vector fields. Riemannian manifolds admitting such vector fields have been investigated in Caminha [25], Castro et al. [26], Ros and Urbano [27], Tanno and Weber [28], Deshmukh and Al-Solamy [29], Fischer and Marsden [30], Ishan et al. [31], Kühnel and Rademacher [32], and Ranjan and Santhanam [33]. There are many examples of gradient conformal vector fields, on the.n-dimensional sphere. S n (c). If. N is the unit normal vector field on. S n (c), in the Euclidean space.Rn+1 with Euclidean metric. , , then for any constant vector field. Z on the Euclidean space n+1 .R its restriction to . S n (c) can be expressed as . Z = V + f N , where . f = Z , N is a smooth function and .V is a vector field on . S n (c). Then it is straightforward to show n that √.V is a gradient conformal vector field on . S (c) with conformal scale function .− c f . Other classes of conformal gradient vector fields are provided by warped product spaces. For instance, consider an .(n − 1)-dimensional Riemannian manifold .(M, g) and an open interval. I ⊂ R and set. M = I × M with projections.π1 : M → I and .π2 → M. Then for a positive function . f : I → R, we get the warped product .(M, g), with metric . g given by .

g(X, Y ) = (dπ1 X, dπ1 Y ) + ( f ◦ π1 )2 g(dπ2 X, dπ2 Y ).

If .∂t is the unit vector field on . I , then .V = ( f ◦ π1 )∂t is a gradient conformal vector  field on the warped product space .(M, g) with conformal scale function . f ◦ π1 (cf. [36], Proposition 35, p.206). The Euclidean space .(Rn , , ) provides many examples of conformal vector fields, a trivial example being the position vector field .ψ which is a gradient conformal vector field. Similarly, on the Euclidean space .(Rn , , ) with Euclidean coordinates .x 1 , .., x n , the vector field     ∂ ∂ ∂ ∂ + ψ, , . V = ψ − ψ, i j j ∂x ∂x ∂x ∂xi where .i, j are two fixed indices with .i = j, is a conformal vector field that is not closed. These examples suggest that there are many conformal vector fields on the sphere . S n (c) and the Euclidean space .(Rn , , ). Therefore, a natural question arises “In how many ways can we characterize these spaces using the conformal vector fields?” Though we have already mentioned some results earlier answering this question, nevertheless let us report more results answering this question.

56

5 Integral Formulas and Conformal Vector Fields

Writing the conformal Killing vector field equation as .

g(∇ X V, Y ) + g(∇Y V, X ) = 2σ g(X, Y ),

(5.16)

we can easily show that ∇ X V = σ X + F X , X ∈ X(M),

.

(5.17)

where . F is a skew-symmetric tensor field of type .(1, 1) defined by (dv)(X, Y ) = 2g(F X, Y ), X, Y ∈ X(M)

.

and .v is the .1-form metrically equivalent to .V . The skew-symmetric tensor field . F in the above equation is called the associated tensor field of the conformal vector field. Note that the smooth .2-form metrically equivalent to . F is closed, and hence .

g ((∇ F)(X, Y ), Z ) + g ((∇ F)(Y, Z ), X ) + g ((∇ F)(Z , X ), Y ) = 0.

(5.18)

Using Eq. (2.4) one gets .

R(X, Y )V = X (σ )Y − Y (σ )X + (∇ F)(X, Y ) − (∇ F)(Y, X ).

Using the above equation in conjunction with Eq. (5.18) gives (∇ F)(X, Y ) = R(X, V )Y + Y (σ )X − g(X, Y )Dσ.

.

(5.19)

It follows from this equation that n  .

(∇ F)(ei , ei ) = −Q(V ) − (n − 1)Dσ.

(5.20)

i=1

We now state the following results. Theorem 5.17 (Tanno and Weber [28]) Let .(M, g) be a connected compact Riemannian manifold with constant scalar curvature .r > 0.Then . M is isometric to a sphere if it admits a closed conformal vector field .V that vanishes at some point of . M. Theorem 5.18 (Deshmukh–Al-Solamy [29]) Let.V be a conformal vector field on an n-dimensional compact Riemannian manifold .(M, g) with conformal scale function .σ . Then,   . g(Dσ, V ) = −n σ 2. .

M

M

5.4 More Results on Conformal Vector Fields

57

Theorem 5.19 (Deshmukh [34], Deshmukh–Al-Solamy [35]) Let.V be a conformal vector field on a compact Riemannian manifold.(M, g) with conformal scale function .σ . Then    n−2 2 r r σ + g(Dσ, V ) . . Ric(Dσ, V ) = 2 2 M M Theorem 5.20 (Deshmukh–Al-Solamy [29]) Let .V be a conformal vector field on a compact Riemannian manifold .(M, g) with conformal scale function .σ . Then 

 Ric(V, V ) − n(n − 1)σ 2 − F2 = 0.

.

M

Theorem 5.21 (Deshmukh–Al-Solamy [29]) Let .(M, g) be an .n-dimensional compact and connected Riemannian manifold whose Ricci curvature satisfies  nc c 0 < Ric ≤ (n − 1) 2 − λ1

.

for a constant .c and the first nonzero eigenvalue .λ1 of the Laplace operator. Then (M, g) admits a nonzero gradient conformal vector field if and only if it is isometric to . S n (c).

.

Theorem 5.22 (Deshmukh–Al-Solamy [29]) Let .(M, g) be an .n-dimensional compact and connected Einstein manifold with Einstein constant .λ = (n − 1)c. Then .(M, g) admits a nonzero gradient conformal vector field if and only if .c > 0 and it is isometric to . S n (c). Corollary. On a compact and connected Riemannian manifold of constant nonpositive scalar curvature, there does not exist a nonzero gradient conformal vector field. Theorem 5.23 (Deshmukh–Al-Solamy [36]) Let .(M, g) be an .n-dimensional compact and connected Riemannian manifold of constant scalar curvature .r with .r ≤ (n − 1)λ1 , .λ1 being the first nonzero eigenvalue of the Laplace operator on . M. Then .(M, g) admits a nonzero closed conformal vector field . V with conformal scale function .σ which satisfies 

r r V V, Dσ + . Ric Dσ + n(n − 1) n(n − 1)

≥ 0,

if and only if it is isometric to . S n (c) for a constant .c. On a compact Riemannian manifold .(M, g), the energy .e(X ) of a smooth vector field . X is defined by  1 X 2 . .e(X ) = 2 M

58

5 Integral Formulas and Conformal Vector Fields

For the sphere . S n (c) of constant curvature .c in Euclidean space .Rn+1 , if we denote by n+1 . V the tangential component of a nonzero constant vector field. Z on.R and by.√ N the √ n = cV , unit normal vector field on . S (c), then we have .∇ X V = − cρ X and .∇ρ √ that is .ξ is a conformal vector field with conformal scale function .σ = − cρ, where .ρ is the normal component of the constant vector field . Z . Moreover, we have e(V ) = c−2 e(∇σ ).

.

This example motivates the following question: Is a compact Riemannian manifold .(M, g) that admits a non-trivial conformal vector field .V satisfying .e(V ) = c−2 e(∇σ ) for a positive constant .c necessarily isometric to . S n (c)? The following result shows that the answer to this question is in the affirmative for compact Riemannian manifolds of constant scalar curvature. Theorem 5.24 (Deshmukh [34]) Let .(M, g) be an .n-dimensional compact and connected Riemannian manifold of constant scalar curvature .r = n(n − 1)c. Then the Riemannian manifold .(M, g) admits a nonzero closed conformal vector field .V with conformal scale function .σ satisfying e(V ) ≤ c−2 e(∇σ ),

.

if and only if it is isometric to . S n (c). The next result studies the geometry of a compact Riemannian manifold of nonconstant scalar curvature that admits a non-trivial conformal vector field, under the mild condition that the scalar curvature is constant along the integral curves of the conformal vector field. Such a condition together with an upper bound on the scalar curvature and a lower bound on the Ricci curvature in a certain direction gives the following characterization of a sphere. Theorem 5.25 (Deshmukh–Al-Solamy [35]) Let .(M, g) be an .n-dimensional compact and connected Riemannian manifold .(M, g) with scalar curvature .r and first nonzero eigenvalue of the Laplace operator .λ1 satisfying .r ≤ (n − 1)λ1 . Then . M admits a non-trivial conformal vector field .V with conformal scale function .σ that satisfies .V (r ) = 0 and the Ricci curvature in the direction of the vector field .∇σ is bounded below by .n −1r if and only if . M is isometric to . S n (c) for a constant .c.

References 1. Duggal, K.L.: Affine conformal vector fields in semi-Riemannian manifolds. Acta Appl. Math. 23, 275–294 (1991) 2. Unal, B.: Divergence theorems in semi-Riemannian geometry. Acta Appl. Math. 40, 173–178 (1995) 3. Garcia-Rio, E. and Kupeli, D.N.: Divergence theorems in semi-Riemannian geometry. In: Proceedings of the Workshop on Recent Topics in Differential Geometry, Santiago de Compostela (Spain), vol. 89, pp. 131–140 (1998)

References 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

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Bochner, S.: Curvature and Betti numbers. Ann. Math. 49, 379–390 (1948) Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) Yano, K.: On harmonic and Killing vector fields. Ann. Math. 55, 38–45 (1952) Lichnerowicz, A.: Transformations infinitesimales conformes de certaines varietes riemanniennes compacts. Compt. Rend. 241, 726–729 (1955) Sato, I.: On conformal Killing tensor fields. Bull. Yamagata Univ. 3, 175–180 (1956) Schoen, R.M.: The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation. Commun. Pure Appl. Math. 41, 317–392 (1988) Obata, M.: Certain conditions for a Riemannian manifold to be isometric to a sphere. J. Math. Soc. Japan 14, 333–340 (1962) Yano, K., Nagano, T.: Einstein spaces admitting a one-parameter group of conformal transformations. Ann. Math. 69, 451–461 (1959) Kanai, M.: On a differential equation characterizing a Riemannian structure of a manifold. Tokyo J. Math. 6, 143–151 (1983) Lichnerowicz, A.: Sur les transformations conformes d’une variete riemannienne compacte. Compt. Rend. 259, 697–700 (1964) Yano, K., Obata, M.: Sur le groupe de transformations conformes d’une variete de Riemann dont le scalaire de courbure est constant. Compt. Rend. 260, 2698–2700 (1965) Bishop, R.L., Goldberg, S.I.: A characterization of the Euclidean sphere. Bull. Am. Math. Soc. 72, 122–124 (1966) Goldberg, S.I.: Curvature and Homology. Academic, New York (1962) Yano, K.: Riemannian manifolds admitting a conformal transformation group. Proc. Nat. Acad. Sci. U.S. 62, 314–319 (1969) Goldberg, S.I., Kobayashi, S.: The conformal transformation group of a compact homogeneous Riemannian manifold. Bull. Am. Math. Soc. 68, 378–381 (1962) Hsiung, C.C., Stern, L.W.: Conformality and isometry of Riemannian manifolds to spheres. Trans. Am. Math. Soc. 163, 65–73 (1972) Yano, K., Sawaki, S.: Riemannian manifolds admitting a conformal transformation group. J. Differ. Geom. 2, 161–184 (1968) Amur, K., Hegde, V.S.: Conformality of Riemannian manifolds to spheres. J. Differ. Geom. 9, 571–576 (1974) Amur, K., Pujar, S.S.: Conformality and isometry of Riemannian manifolds to spheres. J. Differ. Geom. 13, 243–250 (1978) Obata, M.: The conjectures about conformal transformations. J. Differ. Geom. 6, 247–258 (1971) Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965) Caminha, A.: The geometry of closed conformal vector fields on Riemannian spaces. Bull. Braz. Math. Soc. 42, 277–300 (2011) Castro, I., Montealegre, C.R., Urbano, F.: Closed conformal vector fields and Lagragian submanifolds. Pacific J. Math. Complex Space Forms. 199, 269–302 (2001) Ros, A., Urbano, F.: Lagrangian submanifolds of .C n with confornal Maslov form and the Whitney sphere. J. Math. Soc. Japan 50, 203–226 (1998) Tanno, S., Weber, W.: Closed conformal vector fields. J. Differ. Geom. 3, 361–366 (1969) Deshmukh, S., Al-Solamy, F.: Gradient conformal vector fields on a compact Riemannian manifold. Colloquium Math. 112, 157–161 (2008) Fischer, A.E., Marsden, J.E.: Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Am. Math. Soc. 80(3), 479–484 (1974) Ishan, A., Deshmukh, S., Vilcu, G.: Conformal vector fields and the De-Rham Laplacian on a Riemannian manifold. Mathematics 9, art no. 863 (2021) K.uhnel, ¨ W., Rademacher, H.B.: Conformal diffeomorphisms preserving the Ricci tensor. Proc. Am. Math. Soc. 123, 2841–2848 (1995) Ranjan, A., Santhanam, G.: A generalization of Obata’s theorem. J. Geom. Anal. 7, 357–375 (1997)

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34. Deshmukh, S.: Characterizing spheres by conformal vector fields. Ann. Univ. Ferrara 56(2), 231–236 (2010) 35. Deshmukh, S., Alsolamy, F.: A note on conformal vector fields on a Riemannian manifold. Colloq. Math. 136, 65–73 (2014) 36. Deshmukh, S., Al-Solamy, F.: Conformal vector fields and conformal transformations on a Riemannian manifold. Balkan J. Geom. Appl. 17, 9–16 (2012)

Chapter 6

Conformal Vector Fields on Lorentzian Manifolds

6.1 Space-Times and Kinematics Let (. M, g) denote a space-time manifold (usually 4-dimensional, but in general of any finite dimension), i.e. a smooth manifold with a time orientable Lorentz metric of normal hyperbolic signature (.− + ++). The space-time coordinates are denoted by (.x 0 , x 1 , x 2 , x 3 ) = (.x a ), where .x 0 is the time coordinate .t, and indices .a, b, c, d run over .0, 1, 2, 3. The set of all integral curves of a unit timelike (unit spacelike or null) vector field.u is known as the congruence of timelike (spacelike or null) curves. The acceleration of the flow lines along .u is .∇u u whose components are .u a;b u b , where the semi-colon .; denotes covariant differentiation. The tensor .h ab = gab + u a u b is called the projection tensor and projects a vector tangent to . M to its component orthogonal to .u. The rate of change of the separation of flow lines from a timelike curve tangent to .u is called the expansion tensor given by .θab = h ac h db u (c;d) where 1 .u (c;d) is the symmetrization of .u c;d with respect to .c, d, i.e. .u (c;d) = (u c;d + u d;c ). 2 Now we provide the definitions of the following kinematical quantities: the volume expansion .θ , the shear tensor .σab , the vorticity tensor .ωab and the vorticity vector a .ω , as follows: ab .θ = div(u) = θab h σ

. ab

= θab −

θ h ab 3

ωab = h ac h db u [c;d]

.

ωa =

.

1 abcd η u b ωcd 2

where .u [c;d] is the anti-symmetrization of .u c;d with respect to .c, d (i.e. .u [c;d] = √ 1 0 1 2 3 δ f δg δh] is the Levi(u − u d;c )),.ηabcd = g ac g b f g cg g dh ηe f gh ,.ηabcd = (4!) −gδ[a 2 c;d Civita volume form and .g is the determinant of the metric tensor .gab . The covariant derivative of .u is decomposable kinematically as

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_6

61

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6 Conformal Vector Fields on Lorentzian Manifolds

u

. a;b

= ωab + σab +

θ h ab − u b (u c u a:c ). 3

(6.1)

The rate of change of the expansion.θ along.u is given by the Raychaudhuri equation [1]: dθ 1 = −Rab u a u b + 2ω2 − 2σ 2 − θ 2 + div(∇u u) (6.2) .uθ = ds 3 where .ω2 = 21 ωab ωab and .σ 2 = 21 σab σ ab and .s is the arc length parameter of an integral curve of .u. Next we consider a congruence of null geodesics given by a null vector field .l. As the arc length parametrization is not possible for a null curve .C generated by .l, we d = l, k, U, V ) with respect to a distinguished parameter use a Frenet frame . F = ( dp ˙ = −τ U − βV, U˙ = −τl + κk, V˙ = −βl, where . p whose Frenet equations are .k .κ is the curvature function of .C. Let us recall that . g(l, l) = g(k, k) = 0, g(l, k) = −1, g(U, U ) = g(V, V ) = 1. For the congruence of null geodesic, .l˙ = κU = 0 and hence .κ = 0. We also use the 2-dimensional screen distribution . S which is complementary to the tangent space .T (C) in .(T (C))⊥ at each point of .C. This enables us to define a projection tensor .hˆ ab = Ua Ub + Va V − b. As .hˆ is positive definite on ˆ ab = hˆ cd g ca g db . Then we have the following decomposition: . S, its inverse .h lˆ

. a;b

θˆ ˆ h ab + σˆ ab + ωˆ ab 2

=

where .θˆ , .σˆ ab and .ωˆ ab are respectively called expansion, shear and twist of the null congruence and are given by ˆ ab lˆa;b .θˆ = h σˆ

. ab

θˆ = lˆ(a;b) − hˆ ab 2 ωˆ ab = lˆ[a;b]

.

and satisfy (Hawking–Ellis [2]) l θˆ =

.

1 2 d θˆ ˆ + 2(ω) = − (θ) ˆ 2 + 2(σˆ )2 − Rab l a l b dp 2

where .2ωˆ 2 = ωˆ ab ωˆ ab and .2σˆ 2 = ωˆ ab ωˆ ab . This equation is the analogue of the Raychaudhuri equation for a null congruence.

6.2 Einstein’s Field Equations

63

6.2 Einstein’s Field Equations In 1915, Einstein [3] published his general theory of relativity in which he expressed gravity in terms of the metric tensor relating the space-time geometry with the matter. More precisely, he accomplished it by means of his equations called Einstein’s field equations: 8π G r gab = 4 Tab . Rab − 2 c where .G and .c denote the gravitational constant and speed of light respectively, and T denotes the stress–energy tensor (also known as energy–momentum tensor) of the space-time. Choosing units in which .c = 1 and .G = 1, we can re-write Einstein’s equations in index-free form as

. ab

.

r Ric − g = 8π T. 2

Einstein added the term .g (. as the cosmological constant) to the left side of the above equation in order to obtain a static cosmological model, however with Hubble’s discovery of an expanding universe in 1931, abandoned this constant, saying that it was the biggest blunder of his life. But, now . has turned out to be pretty significant, especially in describing the enigmatic dark energy. Incorporating the cosmological term, we write Einstein’s field equations as .

r Ric − g + g = 8π T. 2

(6.3)

When .gab is the Minkowski metric .ηab , we have the special theory of relativity. In the Newtonian limit, Einstein field equations reduce to the Poisson equation . φ = 4πρ, where the gravitational potential .φ is related to the metric through . g00 = −(1 + 2φ). As the divergence of the left side of Eq. (6.3) is zero (this follows from the twice contracted Bianchi second identity:.div Q = 21 dr ), it follows that.Tab is divergence-free, i.e. the stress–energy tensor is conserved. Einstein’s field equations are formally of least action on the Einstein–Hilbert  1 derived by applying√the principle (r − 2) + Lmatter ] −gd 4 x wherein .Lmatter is the matter Lagrangian. action . [ 16 For details we refer to [2]. If .T = 0, then the space-time is called a vacuum solution. Next, we combine the electric and magnetic fields in a way that gives a skew-symmetric tensor . F with components . Fab , called the electromagnetic field which satisfies the Maxwell equations .d F = 0, i.e. F[ab;c] = 0, (6.4) .

div F = 4π J, i.e. F ab;b = 4π J a ,

(6.5)

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6 Conformal Vector Fields on Lorentzian Manifolds

where . F ab = g ac g bd Fcd . It follows from (6.5) that .div J = 0, i.e. . J;aa = 0 (the continuity equation). The stress–energy tensor is T

. ab

=

1 Fcd F cd gab − Fac F c b . 4

(6.6)

Next, we describe a perfect fluid. It is the fluid matter with mass density .ρ and a congruence of timelike curves, called flow lines generated by a unit timelike vector field .u. The fluid current vector . j = ρu, such that . j is conserved, i.e. .div j = 0. The flow of . j provides the internal energy . as a function of .ρ. The function 2 d is the pressure of the fluid. The .μ = ρ(1 + ) is the energy density and . p = ρ dρ stress–energy of the perfect fluid is T

. ab

= (μ + p)u a u b + pgab

(6.7)

where .μ + p = 0 and .μ > o. A perfect fluid satisfying a barotropic equation of state .μ = μ( p) is said to be isentropic. Special cases of interest are the dust matter for which . p = 0, the radiation field for which . p = μ3 and the stiff matter for which . p = μ. The conservation equation .div T = 0 gives the energy equation (μ + p)u a ;b u b = − p;b (g ab + u a u b ).

.

(6.8)

For a charged fluid with a conserved electric charge, i.e. .div J = 0 where . J = eu is the electric current, we have T

. ab

1 = (μ + p)u a u b + pgab + (Fcd F cd )gab − Fac Fbc . 4

For a physically reasonable solution of Einstein’s equations, stress–energy tensor must satisfy the following energy conditions: 1. Weak Energy Condition. At each point . p of the space-time . M, .Tab V a V b ≥ 0 for any non-spacelike vector .V ∈ T p M. For a perfect fluid with .V = u, .Tab u a u b = μ which is the non-negative local energy density, and hence the weak energy condition is satisfied. 2. Dominant Energy Condition. In terms of any orthonormal frame (.ea ), the dominant energy condition requires .T00 ≥ |Tab | for .a, b = 0, 1, 2, 3. So, this condition is the weak energy condition plus the condition that the pressure should not exceed the energy density. 3. Strong Energy  Condition. A space-time . M satisfies the strong energy condition if .Tab V a V b ≥ T2 V a Va (.T = T race(T )) for any timelike vector field .V .

6.3 Some Important Space-Times

65

6.3 Some Important Space-Times A space-time (. M, g) is said to be globally hyperbolic if there exists a spacelike hypersurface . S such that every endless causal (timelike or null) curve intersects . S once and exactly once. Such a hypersurface is called a Cauchy surface. If . M is globally hyperbolic, then (a) . M is homeomorphic to . R × S where . S is a hypersurface of . M, and for each .t, .{t} × S is a Cauchy surface, (b) if . S  is any compact hypersurface without boundary, of . M, then . S  must be a Cauchy surface (see Eardley et al. [4] and Beem et al. [5]). The simplest example of a globally hyperbolic space-time is the Minkowski space-time (.R41 , η) with line-element a b 0 2 1 2 2 2 3 2 0 .ηab d x d x = −(d x ) + (d x ) + (d x ) + (d x ) where . x = t, the time coordinate. The spacelike hypersurfaces (.t constant) form a family of Cauchy hypersurfaces that cover the entire .R14 . Hence, .R14 = R × N , η = −dt 2 + G, where (. N , G) is a 3-dimensional Euclidean space. A space-time (. M, g) of nonzero constant curvature .k are given by .

Rabcd = k(gbd gac − gbc gad ).

Contracting it with .g ac shows that .

Rbd = 3kgbd .

Thus (. M, g) is Einstein. Contracting the above equation with .g bd provides .r = 12k. Plugging these equations in the expression for Weyl tensor .W (Chap. 1) shows that . W = 0, i.e. . M is conformally flat. A space-time of constant positive curvature is called the de Sitter space-time which is globally hyperbolic and topologically .R1 × S 3 with metric   t 2 2 2 2 [dr 2 + sin2 r (dθ 2 + sin2 θ dφ 2 )] .ds = −dt + a cosh a where .a is a nonzero constant. The spatial slices (.t = constant) are 3-spheres and belong to a family of Cauchy surfaces. The space-time of constant negative curvature is known as anti-de Sitter space-time which is topologically . S 1 × R3 but not globally hyperbolic. Its universal covering space of constant curvature .k = −1 (constructed by unwrapping the circle . S 1 ) has the metric ds 2 = − cosh2 r dt 2 + dr 2 + sinh2 r (dθ 2 + sin2 θ dφ 2 )]

.

where the coordinates (.t, r, θ, φ) cover the entire space-time. It can been shown by suitable coordinate transformations that de Sitter and anti-de Sitter space-times are locally conformal to the Einstein static universe.ds 2 = −dt 2 + dr 2 + sin2 r (dθ 2 + sin2 θ dφ 2 ).

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6 Conformal Vector Fields on Lorentzian Manifolds

As the Einstein field equations are a complicated system of nonlinear partial differential equations, one often assumes some relevant symmetry conditions for a satisfactory representation of our universe. Through extragalactic observations, we know that the universe is approximately spherically symmetric about an observer. So, it would be reasonable to assume that the universe is isotropic, i.e. spherically symmetric about each point. This means that the universe is spatially homogeneous, i.e. admits a 6-parameter group.G 6 of isometries whose surfaces of transitivity are spacelike hypersurfaces of constant curvature. Such a space-time is called Robertson– Walker space-time which is globally hyperbolic and whose metric is the warped product metric 2 2 2 2 .ds = −dt + a (t)d , (6.9) where .d 2 is the metric of a spacelike hypersurface . with spherical symmetry and constant curvature .k = 1, −1 or .0. With respect to a local spherical coordinate system (.r, θ, φ), this metric is given by d 2 = dr 2 + f 2 (r )(dθ 2 + sin2 θ dφ 2 ),

.

(6.10)

where . f (r ) = sin(r ), sinh(r ) or .r according as .k = 1, −1 or .0, respectively. It is noteworthy that any two slices .t = t1 and .t = t2 are homothetic, each slice is totally umbilical and has constant mean curvature. Also, the Robertson–Walker space-time is conformally flat. For .k = 1, . is diffeomorphic to . S 3 and hence compact. For .k = −1, 0, we can compactify . by identifying suitable points. However, it follows from a result of Yano and Bochner (Theorem 5.12 of Chap. 5) that for .k = −1, the compact . would not admit any global Killing vector field even though it can have Killing vectors at each point. Also, for .k = 0, a compact . can have only 3parameter group .G 3 of isometries. So, in both cases .k = −1 and .k = 0, compact . cannot be isotropic. Thus, for a Robertson–Walker space-time . cannot be compact for .k = −1, 0. The isotropy suggests that the stress–energy tensor of the Robertson– Walker space-time is of the form of a perfect fluid. Evidently, the energy density and pressure depend only on .t, the space coordinates are comoving and the fluid is made up of galaxies as particles. The function .a(t) can be taken as the distance between two nearby galaxies. The use of Einstein’s field equations (6.3) along with the energy Eq. (6.8) and Raychaudhuri equation (6.2) provides the following ordinary differential equations: a˙ .μ ˙ = −3(μ + p) , (6.11) a a¨ 4π(μ + 3 p) −  = −3 , a

(6.12)

 2 3k a˙ = 8π μ +  − 2 , .3 a a

(6.13)

.

6.3 Some Important Space-Times

67

with the reasonable assumption that .μ > 0 and . p ≥ 0. Present day experiments confirm that the universe is now going through the stage of negligible pressure and so assume . p = 0. So, taking . p = 0 and also . = 0, we obtain .

M 4π μ= 3 3 a

3(a) ˙ 2−6

.

E M = −3k = M a

(6.14)

where . E is the sum of kinetic and potential energies and . M a constant. Equation (6.14) is called the Friedmann equation, and its solution is called the Friedmann– dt and integrate Robertson–Walker (FRW) universe. We rescale .t by .T as .dT = a(t) Friedmann’s equation so as to obtain the solutions: a=

.

E E (cosh T − 1), t = (sinh T − T ), 3 3 a = T 2, t =

.

a=−

.

T3 , 3

f or k = −1

f or k = 0

E E (1 − cos T ), t = − (T − sin T ), 3 3

for k = 1.

Analyzing these solutions leads us to conclude that there was an initial singularity a = 0 at .t = 0, and that the universe expands forever if .k = 0 or 1; and collapses to .a = 0 at a later time, if .k = 1. A higher dimensional extension and generalization of Robertson–Walker spacetime can be stated as follows. Let (. M, g) be a (.n + 1)-dimensional space-time with warped product metric .

ds 2 = gab d x a d x b = −dt 2 + a 2 (t)γi j d x i d x j

.

where.t is the time coordinate.x0 , and.i, j are coordinate indices for an.n-dimensional arbitrary Riemannian manifold (., γi j ) (not necessarily of constant curvature). Following Alias, Romero and Sanchez [6], let us call it a generalized Robertson– Walker (.G RW ) space-time. If .γ is 3-dimensional and Einstein, then it has constant curvature. This is not true in higher dimensions. This raises the following question: Under what condition on the geometry of the .G RW space-time of dimension greater than 4 would the spatial metric .γ be (a) Einstein, (b) of constant curvature? The answer was provided by Sharma and Duggal [7] in the form of the following result. Theorem 6.1 Let (. M, g) be a .G RW space-time and denote the Weyl conformal tensors of .g and .γ by .W and .W ∗ , respectively. Similarly, the quantities with an asterisk refer to .γ . Then the electric part .W (∂t , ∂i , ∂t , ∂ j ) = W0i0 j of .W vanishes if and only if .γ is Einstein, (b) the purely spatial component .W (∂i , ∂ j , ∂k , ∂l ) = Wi jkl

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6 Conformal Vector Fields on Lorentzian Manifolds

of .W vanishes if and only if .γ has constant curvature, in which case (. M, g) is forced to become conformally flat. The proof is based on the basic equations .∇∂0 ∂0 = 0, ∇∂0 ∂i = ∇∂i ∂0 = aa˙ ∂i , the Gauss and Codazzi equations, and computing the following Weyl tensor components: .

W0i j0 =

  1 r∗ Rici∗j − γi j , n n−1 .

.

Wi jkl = Wi∗jkl +

Wi jk0 = 0,

1 ∗0 ∗0 ∗0 [Ric∗0 jk gil − Ricik g jl + Ril g jk − Ric jl gik ], (n − 1)(n − 2) r∗ . n

where . Ric∗0 = Ric∗ −

The aforementioned result was generalized by Sharma in [8] on synchronous space-times with pure trace extrinsic curvature slices, i.e. space-times with metric ds 2 = −dt 2 + a 2 (t, x k )γi j d x i d x j .

.

One of the important areas of research in general relativity is the study of isolated systems, such as the Sun and a host of stars in our universe. In order to understand these systems, we first define stationary and static space-times. A space-time is said to be stationary if it has a timelike Killing vector field .V , and is said to be static if, furthermore, .V is hypersurface orthogonal, i.e. there exist spacelike hypersurfaces orthogonal to .V at each point. A space-time is called spherically symmetric if its isometry group contains a subgroup isomorphic to the group . S O(3) whose orbits are 2-dimensional spheres. Let (. M, g) be a static and spherically symmetric space-time. One can choose coordinates (.t, r, θ, φ) such that the metric .g can be written as ds 2 = −eλ dt 2 + eν dr 2 + Adr dt + Br 2 (dθ 2 + sin2 θ dφ 2 )

.

where .λ, ν, A, B are smooth functions of .t and .r only. The inherent freedom in choosing some of the coefficients allows us to consider a Lorentz transformation such that . A = 0 and . B = 1. As the star is isolated, the space-time outside it is vacuum (empty) and so we have . Rab = 0. By standard computations, we can verify that the metric assumes the form 

2m .ds = − 1 − r 2





2m dt + 1 − r 2

−1

dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 )

where.m is the mass. This is called the exterior (.r > 2m) Schwarzschild space-time. We note that this metric is asymptotically flat, i.e. as .r → ∞, the metric approaches the Minkowski (flat) metric.

6.3 Some Important Space-Times

69

The .t = constant spatial slices . are totally geodesic in the Schwarzschild exterior space-time. It follows by a straightforward computation that the only nonzero com, R22 = rm3 , R33 = rm3 . Hence the scalar ponents of the Ricci tensor of . are . R11 = −2m r3 curvature is zero and . are asymptotically flat. Let us now consider the space-time (. M, g) outside a spherically symmetric body having an electric charge .e but no spin or magnetic dipole. The line-element of this space-time is given by Reissner–Nordstrom solution:     e2 e2 −1 2 2m 2m 2 + 2 dt + 1 − + 2 dr + r 2 (dθ 2 + sin2 θ dφ 2 ). .ds = − 1 − r r r r 2

Finally, we define a pp-wave space-time as a Lorentzian manifold whose metric is given, with respect to Brinkmann coordinates (Brinkmann [9]), by the form ds 2 = H (u, x, y)du 2 + 2dudv + d x 2 + dy 2

.

(6.15)

where . H is any smooth function of .u, x, y. Of course, .v is the null coordinate ∂ ∂ , ∂v ) = 0). A more standard definition (Ehlers and Kundt [10]) states that a (.g( ∂v pp-wave space-time is a Lorentzian manifold which admits a parallel (covariantly constant) null vector field .k, i.e. a null vector field .k satisfying .∇k = 0. .k is related to the Brinkmann coordinate .v by .k = ∂∂v . A . pp-wave space-time is a vacuum spacetime if and only if . H is harmonic with respect to spatial coordinates .x, y. A special case of . pp-space-time is a plane wave for which . H (u, x, y) = a(u)(x 2 − y 2 ) + 2b(u)x y + c(u)(x 2 + y 2 ) where .a, b, c are any smooth functions of .u. In particular, for .c = 0 the plane wave is vacuum, and is called the plane gravitational wave. The functions .a and .b determine the amplitude and polarization of the wave. These space-times admit no Cauchy surfaces. Here, we briefly describe a classification of space-times, known as the Petrov classification (Petrov [11]). By virtue of its symmetries, the Weyl tensor .Wabcd can be viewed as a real .6 × 6 matrix .Wi j , where .i, j run from 1 to 6. The tracelessness of .Wabcd shows that   A B . Wi j = B T −A where . A and . B are .3 × 3 matrices such that . A is symmetric and both . A and . B are trace-free. The matrix.Wi j can also be viewed as a complex.3 × 3 matrix. D = A + i B for which there are the following three possible Jordan forms: Petrov type I ⎛ ⎞ α 0 0 . ⎝0 β 0 ⎠ 0 0γ

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6 Conformal Vector Fields on Lorentzian Manifolds

Petrov type II ⎛ ⎞ α1 0 0 ⎠ . ⎝0 α 0 0 −2α Petrov type III ⎛

⎞ 010 . ⎝0 0 1 ⎠ 000 where the trace free conditions on . A and . B (and hence on . D) are used. Following are subspaces of Petrov types I and II. Petrov type D Type I with .α = β Petrov type O Type I with .α = β = γ = 0, equivalently, .W = 0 Petrov type N Type II with .α = 0 Another useful criteria for Petrov classification is due to Bel [12]. The complex + + ∗ is defined as .Wabcd = Wabcd + i Wabcd where .W ∗ is self-dual Weyl tensor .Wabcd ef ∗ the dual of .W defined as .Wabcd = 21 Wab cde f where .abcd √ is the Levi-Civita pseudotensor which is completely anti-symmetric and .0123 = −det (gab ). Let .k be a null direction satisfying the condition k Wa]bc[d k f ] k b k c = 0.

. [e

The Bel criteria are given below (1) .W is Petrov type I if there are exactly 4 distinct null vectors (principal null directions) .k satisfying the above condition. (2) .W is Petrov type II if there are two coincident null directions .k satisfying the condition. (3) .W is Petrov type III if there are three coincident null directions .k satisfying the + k a k c = 0. condition, equivalently .Wabcd (4) .W is Petrov type N if there are all four coincident null directions .k satisfying + k d = 0. the condition, equivalently, .Wabcd (5) .W is of Petrov type D if its principal null directions are coincident pairs. (6) The Petrov type O does not single out any null direction, equivalently, the Weyl tensor vanishes identically. For details on space-time manifolds, we refer to Hawking and Ellis [2], Wald [13], Wasserman [14], O’Neill [15], Stephani et al. [16] and d’Inverno [17].

6.4 Homothetic Vector Fields in General Relativity

71

6.4 Homothetic Vector Fields in General Relativity Let .V be a homothetic vector field on a 4-dimensional space-time manifold. Then we have .L V gab = 2cgab (6.16) where the conformal scale function .c is a constant. This implies the following integrability conditions: LV  abc = 0,

LV R abcd = 0, LV W abcd = 0.

.

Therefore we also have LV Rab = 0,

.

LV r = −2cr.

It also follows from Einstein field equation (6.3) that .LV Tab = 0, i.e. the stress– energy tensor is left-invariant by a homothetic vector field. For a local orthonormal frame (.e0 , e1 , e2 , e3 ) on (. M, g), where .e0 is unit timelike, Newman–Penrose null ¯ is defined by tetrad (.l, k, m, m) 1 1 1 1 l = √ (e0 + e1 ), k = √ (e0 − e1 ), m = √ (e2 + ie3 ), m¯ = √ (e2 − ie3 ). 2 2 2 2

.

The only surviving inner products of these 4 null vector fields are .g(l, k) = −1 and g(m, m) ¯ = 1. Let us follow the work of McIntosh [18], on the kinematic properties of a homothetic vector field .V . The projection tensor . P for a non-null .V is given by

.

.

Pab = gab −

Va Vb , Pab V b = 0. g(V, V )

Decomposing .Va:b into its symmetric and anti-symmetric parts, we have .

Va;b = cgab + Fab

(6.17)

where . Fab = V[a;b] is called the homothetic bivector. In addition, .V satisfies .

Va;bc − Va;cb = R dabc Vd .

(6.18)

We state and prove the following result. Theorem 6.2 (McIntosh [18]) If a homothetic vector field .V is non-null, then it is shear free and its expansion is constant .= 3c. To prove it, let us recall that the relative velocities of neighboring particles is given by

72

6 Conformal Vector Fields on Lorentzian Manifolds .

Pac Pbd V c;d = θab + ωab .

The use of Eq. (6.17) in the preceding equation gives θ

. ab

1 = θ(ab) = σab + θ Pab = c Pab . 3

Its trace with .g ab and the fact .g ab σab = 0 yields that .θ = 3c and thus .σab = 0, completing the proof. c cb Va )V The vorticity tensor turns out to be .ωab = Fab + (Fca VbV−F . The vorticity dV d vector .ω is given by 1 abcd a η .ω = Vd Vc;d . (6.19) 2 Next, we consider a null homothetic vector field .V , i.e. .V a Va = 0. Its derivative along .V and use of (6.16) lead to .Va;b V b = 2cVa , i.e. .V is tangent to a null geodesic. That is, .V a = λl a and .la;b l b = 0 where .l is a null geodesic congruence. Contraction of (6.16) with the projection tensor.hˆ ab (as defined in the first Sect. 6.1 of this chapter) and .hˆ ac hˆ bd − 21 hˆ ab hˆ cd lead us to the following result. Theorem 6.3 If .V is a null homothetic vector field, then it is parallel to a principal null vector, say .l (.V = λl, λ > 0) which is geodesic, shear free and has constant . expansion .θ = 2c λ Following McIntosh [18], we take . Fab as a test electromagnetic field of any homothetic vector field. By virtue of Eq. (6.5), we have .

F ab;b = −V b;a;b = R ab Vb = 4π J a

(6.20)

where . J is the 4-current vector of the test electromagnetic field generated by . F. In a space-time with . Rab V a = 0, . F is source-free, i.e. . J = 0. On the basis of Eq. (6.20), we can write Eq. (6.18) as . Fab;c − Fac;b = R dabc Vd . This equation, in conjunction with Maxwell equations (6.4), provides .

Va;bc = Fab;c = Rabcd V d .

In view of the integrability condition .LV r = −2cr , we find that . J satisfies the continuity equation . J;aa = 0. It also follows from . Rabcd V c V d = 0 that . F is parallel along . V . We state the next results without proof [we refer to [18] for proofs]. Theorem 6.4 (McIntosh [18]) If a non-null homothetic vector field .V is parallel to its source vector . J , then .V is Killing. Theorem 6.5 (McIntosh [18]) An electrovac space-time (i.e. electromagnetic field with with zero current vector) does not admit any proper null homothetic vector field.

6.5 Homothetic Symmetry and Cauchy Surfaces

73

Theorem 6.6 (McIntosh [18]) If a space-time (. M, g) which contains a perfect fluid with 4-velocity .u, density .μ and pressure . p admits a non-null homothetic vector field .V with corresponding source vector . J , then one of the following cases occurs: (a) .μ = p, J = 0. Any of the conditions .g(u, V ) = 0, u V, g(u, J ) = 0 or .u J gives .V J and thus .V is Killing; except in the special case .g(u, J ) = 0, .3 p + μ = 0, .g(u, V ) = 0. (b). J = 0 : V = g(u.V )u and .3 p + μ = 0 or .g(u, V ) = 0 and .μ = p. (c) .μ = p, J = 4μg(u, V )u. Theorem 6.7 (McIntosh [18]) If a space-time (. M, g) which contains a perfect fluid with 4-velocity .u, density .μ and pressure . p admits a proper null homothetic vector field .V with corresponding source vector . J , then .V cannot be orthogonal to .u. We now present some examples. Example 6.1 (Minkowski space-time).ds 2 = −dt 2 + δi j d x i d x j has the homothetic vector field .V = t∂t + x i ∂i which is unique up to spatial translations .V → V + a i ∂i and time translation .V → Va ∂t where .a i and .a are constants. Example 6.2 (Robertson–Walker space-times) Equation (6.9) with spatial curvature k = 0 and .t aa˙ = constant. Here .V = t∂t + (1 − t aa˙ x i ∂i ) is homothetic. In this case, b .a is proportional to some constant power of .t, say .t . For .b ≥ 2/3, this solution is a 2 perfect fluid with equation of state . p = ( 3b − 1)μ.

.

Example 6.3 (Robertson–Walker space-times) Equation (6.9) with spatial curvature k = 0 admits a proper homothetic vector field only in the unphysical case.μ + 3 p = 0 (see Eardley [19]).

2 pi

2 2 i 2 Example

2 6.4 (Kasner Vacuum space-times).ds = −dt + i t (d x ) , i pi = 1, i pi = 1, where . pi are constants. Any homothetic vector field is given by i . V = t∂t + i (1 − pi )x ∂i . For details on the above example and some other examples, we refer to Hsu and Wainwright [20] and also some other references given in that paper. We note here that the maximum dimension of the homothetic algebra in 4-dimensional space-time 4(4+1) . M is . + 1 = 11, and this occurs when . M is flat. 2 .

6.5 Homothetic Symmetry and Cauchy Surfaces Let us assume that (. M, g) is a globally hyperbolic space-time with a Cauchy surface . Then, (i) . M is homotopic to . R × , (ii) if .  is any compact spacelike hypersurface without boundary and embedded in . M, then it must be a Cauchy surface and (iii) if (. M, g) satisfies the vacuum Einstein equations, then .g may be completely determined from a set of Cauchy data (metric and second fundamental form) specified on . or if . M satisfies the Einstein equations coupled to a well-posed hyperbolic system of matter equations, then the coupled system has the same property. (iii) allows us to do essentially all of the analysis on a Cauchy surface which is Riemannian with .

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6 Conformal Vector Fields on Lorentzian Manifolds

the pull-back-induced metric .γ = i ∗ g, where .i ∗ is the differential map induced by the embedding .i :  → M. Here, let the indices.i, j run from.0 to.3, whereas.a, b run from.1 to.3. Let. N denote the unit timelike normal vector field and .∂a , ∂b coordinate vector fields tangent to .i() (briefly .), then the second fundamental form . B of . is .

Bab = −g(∇∂a N , ∂b ).

The pair (.γab , Bab ) is called a Cauchy data for the induced gravitational field on .. If the matter fields are present, they would need their own Cauchy data. The mean curvature of . is . 13 T race(B) = 13 Bab γ ab . It is convenient to define the traceless tensor . Aab = Bba − 13 τ γba , where .τ = T race(B). As . M is globally hyperbolic, we may decompose the homothetic vector field (defined by (6.16)) as .

V = f N + X.

where . X is tangent to .. The Einstein field equations (6.3) with . = 0 yield the constraint equations: 2 r + τ 2 − |A|2 = 16π T00 , T00 = T (N , N ), 3

.

∇a Aab = ∇b τ − 8π Tab , T0b = T (N , ∂b ),

.

where .r is the scalar curvature of . and .∇a is the surface (.)-intrinsic covariant derivative compatible with .γ . As .V is homothetic, we have .

L V γab = 2cγab , L V Aab = −c Aab , .

L V τ = −cτ,

where . L V denotes the surface-projected Lie derivative along .V . Using Berger’s trick [21], i.e. setting the evolution vector field .∂t = V , we obtain the following homothetic evolution equations from Einstein field equations:  1 .2cγab = L X γab − 2 f Aab + τ γab , 3     1 a 1 d a a a a a . − c Ab = f Rb − γb τ + τ Ab − 8π Tb − Td γb 3 3   1 − ∇ a ∇b f − γba f + L X Aab , 3 

.

  1 − cτ = f |A|2 + 4π(T00 + Tdd ) + τ 2 + L X τ − f, 3

(6.21)

(6.22)

(6.23)

6.5 Homothetic Symmetry and Cauchy Surfaces

75

where . = ∇ a ∇a and .|A|2 = Aab Aab . According to Eardley et al. [4], a space-time is said to obey the mixed energy condition if, at any point . p ∈ , (a) the strong energy condition (.T00 + Taa ≥ 0) holds, and (b) equality in (a) implies that all components of .T are zero. For a perfect fluid, this means .3 p + μ > 0 everywhere. We now state and prove the following nice result. Theorem 6.8 (Eardley et al. [4]) Suppose (. M, g) is a globally hyperbolic space-time which (i) satisfies the Einstein field equations for a stress–energy tensor .T obeying the mixed and strong energy conditions, (ii) admits a homothetic vector field .V of .g and (iii) admits a compact hypersurface of constant mean curvature. Then, either . M is an expanding hyperbolic model with metric .ds 2 = eλt (−dt 2 + h ab d x a d x b ) with a b .h ab d x d x a 3-dimensional Riemannian metric of constant negative curvature on a compact manifold and .T vanishing, or .V is Killing. If .c = 0, then .V is Killing, and we are done. So, let .c = 0. By our assumption, .τ is constant. Tracing (6.21) gives 3c = − f τ + div X

.

where divergence has been taken with respect to .γ . Integrating it over the compact  and using the divergence theorem provides

.

 3c = −τ

f,

.

(6.24)



 being the volume of . with respect to .γ . As .c = 0, the preceding equation implies that .τ = 0. Since .τ is constant, Eq. (6.23) can be expressed as the following elliptic equation in . f : .(− + U ) f = −cτ, (6.25)

.

where .U = |A|2 + 4π(T00 + Tdd ) + 13 τ 2 . By mixed energy condition, .U > 0 on .. As the elliptic operator .− + U is positive and the right-hand side of (6.25) is constant, . f is strictly bounded away from zero on .. To see this, let .cτ < 0. At a point . p ∈ , where . f has global minimum, .( f )( p) ≥ 0. Then (6.25) shows that .(U f )( p) > 0. But .U ( p) > 0, so . f ( p) > 0 at its global minimum and hence . f > 0 on .. Now, the integral of (6.25) over . is  .

− cτ  =

U f. 

Adding the above equation to . τ3 times Eq. (6.24), we obtain  .



[|A|2 + 4π(T00 + Taa )] = 0.

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6 Conformal Vector Fields on Lorentzian Manifolds

The integrand is either .≥ 0 or .≤ 0 everywhere, so it vanishes everywhere. But .c = 0, therefore, . Aab = 0 and .T00 + Taa = 0 everywhere. Thus, the mixed energy condition 2 implies that .Ti j = 0 on .. Thus, .U = τ3 = constant and hence . f is uniquely determined as c = constant. . f = −3 τ Further, . Aab = 0 implies that . Bab = 13 τ γab , i.e. . is totally umbilical in . M. Plugging all these consequences in Eq. (6.22), we find that .

2 2 Rab = − τ 2 γab , τ = − τ 2 9 3

(6.26)

which shows that the metric .γ has constant negative curvature. Only the standard hyperbolic metric satisfies the condition (6.26) and is specified by the choice of global topology and the choice of a single scale factor. It also follows immediately from Eq. (6.21) that . X is Killing on (., γ ). But we know from Bochner [22] that the standard hyperbolic metrics do not admit any global Killing vector field. Thus,. X = 0. Consequently, the admissible initial data turns out to be.γab = τ92 h ab , Bab = τ3 γ for a nonzero constant .τ and .h ab a metric of constant negative curvature .−6. By dominant energy condition, .T vanishes throughout the space-time. One can verify that the vacuum space-time development of the initial data is of the form mentioned within the theorem, and this completes the proof. Next, let us assume that (. M, g) is an .n-dimensional semi-Riemannian orientable manifold other than compact and without boundary. For indefinite hypersurfaces, the eigenvalues and eigenvectors of the second fundamental form need not be all real, possibly complex and null. Following Fialkow [23], we say that an indefinite hypersurface is proper if the eigenvalues of its second fundamental form are real and no eigenvector is null at every point. Based on this idea, we state the following result whose Riemannian analogue was proved by Yano [24]. Theorem 6.9 Let (. M, g) be an orientable semi-Riemannian manifold other than compact and without boundary. Let it admit a homothetic vector field .V , and a compact proper constant mean curvature hypersurface . without boundary. If the inner product of .V with the unit normal vector . N has constant sign on . and the Ricci curvature of . M along . N is non-negative on ., then . is totally umbilical and Ricci curvature of . M vanishes on .. A special case of the above theorem is the following result. Theorem 6.10 (Alias et al. [6]) Let . M be a space-time admitting a timelike homothetic vector field .V and satisfying timelike convergence condition, i.e. Ricci curvature is non-negative along all timelike vectors tangent to . M. Then, every compact spacelike hypersurface of constant mean curvature in . M is totally umbilical. Remark. The timelike convergence condition is also known as the strong energy condition described earlier in this chapter.

6.6 Conformal Vector Fields on Space-Times

77

6.6 Conformal Vector Fields on Space-Times Let us recall from Sect. 4.1 of Chap. 4 that a conformal vector field .V is defined by LV gab = 2σ gab

.

(6.27)

and satisfies some integrability conditions mentioned there. A special conformal vector field is a conformal vector field for which . Dσ is parallel, i.e. .σa;b = 0. Examples are Killing (.σ = 0), homothetic (.σ is constant). For a conformal vector field .V on the 4-dimensional space-time, one can verify by a straightforward computation that .L V G ab = 2[( σ )gab − σa;b ] where .G ab = Rab − r2 gab is the Einstein tensor. Hence, a conformal vector field .V is special if and only if the Einstein tensor is left-invariant by .V . Virtually all explicit space-time solutions of Einstein’s equations admit some isometry group. This is not surprising since Einstein’s equations are very difficult to solve, and isometries simplify them considerably. Therefore, it would be useful to examine solutions replacing isometries with conformal symmetries defined by conformal vector fields. We do have cosmological space-times admitting non-Killing and non-homothetic conformal vector fields. Robertson–Walker space-time is conformally flat and possesses a 15-parameter group .G 15 of conformal transformations. The Lie algebra of this full . G 15 was completely determined by Maartens and Maharaj [25]. Out of . G 15 , there are .G 6 of Killing vector fields tangential to the spatial slices, one normal to them, and the 8 neither tangential nor normal (oblique). Nevertheless, the existence of a conformal vector field places severe restrictions on the space-times, as indicated by the following results. Theorem 6.11 (Collinson–French [26]) A non-flat 4-dimensional vacuum spacetime with zero cosmological constant and a non-homothetic conformal vector field . V is of type N, and represents a plane-fronted gravitational wave with parallel rays. Its proof is based on taking the Lie derivative of the vacuum field equation. Rab = 0 along the conformal vector field .V and using the integrability condition (4.9) [Chap. d σ c = 0, 4]. This gives .∇a ∇b σ = 0 (so .g(Dσ, Dσ ) is constant), which implies . Rcab d c c c a where.σ = ∇ σ and hence.Wcab σ = 0. As.σ = 0, this shows (by the lemma stated d = 0 (and hence the space-time is flat which contradicts the below) that either .Wcab a hypothesis) or .σ is null and as it is parallel, we have a plane-fronted gravitational wave with parallel rays (pp-wave). Lemma 6.1 (Eardley et al. [4]) Let (. M, g) be a 4-dimensional space-time with a non-vanishing non-null vector field vector field .ξ which is annihilated by the Weyl tensor .W . Then . M is conformally flat. If we include the cosmological term . in the above theorem, then we have the following.

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Theorem 6.12 (Garfinkle–Tian [27]) Suppose (. M, g) is a 4-dimensional vacuum space-time with cosmological constant ., and .V a non-Killing conformal vector field. Then . M is locally isometric to the de Sitter space-time if . > 0 or anti- de Sitter space-time if . < 0. .

To prove this, we first note that the Einstein field equations can be written as Rab = gab . Lie-differentiating it and using the formula (4.9) [Chap. 4] gives σ

. a;b

σ = − gab . 3

This equation provides σ b R abcd =

.

(6.28)

σ a (δ − δba ). 3 c

The combined use of the above two equations and the definition (1.12) [Chap. 1] of the Weyl tensor yields a .σa W bcd = 0. (6.29) So, lemma 12 is applicable, and hence either (i) .W = 0, i.e. . M is conformally flat, or (ii) .σ a is null. To rule out (ii), we see that this implies .σ a (σa;b ) = 0. The use of this consequence in (6.28) shows that .σ = 0, contradicting our hypothesis. Thus .W = 0, which, in conjunction with . Rab = gab , leads to .

Rabcd =

 (gac gbd − gad gbc ). 3

Therefore, . M has constant curvature ., and hence locally isometric to de Sitter space-time for . > 0 and anti- de Sitter space-time for . < 0. The proof is completed. We state the next result for asymptotically flat space-time as follows. Theorem 6.13 (Eardley et al. [4]) Let (. M, g) be a globally hyperbolic space-time which (i) is spatially asymptotically flat, (ii) satisfies Einstein’s field equations for a stress–energy tensor .T obeying the dominant energy condition, with .T asymptotically of order . O(r −4 ) and (iii) admits a conformal vector field which asymptotically approaches the dilation vector field .x i ∂i . Then (. M, g) is a Minkowski space-time. Its proof dwells on setting .gab = ηab + O(r −1 ) in a coordinate system (.x i ) (.i = )δi j + O(r −1 ) 0, 1, 2, 3) where .η is the Minkowski metric such that .gi j = (1 + 2m r k and .∂k gi j = −2mx δi j + O(r −3 ), .∂0 gi j = O(r −2 ). It follows by a series of computar3 tions (see [4] for details) that .m = 0 and then the positive energy theorem (a deep result) implies that the space-time is Minkowski, completing the proof. Let us turn our attention back to Theorem 8 which motivated Sharma [28] to study the corresponding case of a proper (non-homothetic) conformal vector field .V under certain geometric conditions, and establish the following result.

6.6 Conformal Vector Fields on Space-Times

79

Theorem 6.14 (Sharma [28]) Let (. M, g) be a space-time solution of Einstein’s field equations admitting a conformal vector field .V and be evolved by a complete spacelike hypersurface . such that (a) . is totally umbilical in . M, (b) the normal component of .V is non-constant on ., and (c) the normal sectional curvature of . M is independent of the choice of the tangential direction at each point of .. Then . is conformally diffeomorphic to (i) a 3-sphere . S 3 , or (ii) Euclidean space . E 3 , or (iii) hyperbolic space . H 3 , or (iv) the product of a complete 2-dimensional manifold and an open real interval. If . is compact, then only (i) occurs. The normal sectional curvature condition is a weakening of constant curvature condition that holds for Minkowski, de Sitter and anti-de Sitter and Robertson– Walker space-times. The proof uses the following result and conformal evolution equations similar to the homothetic evolution Eqs. (6.21), (6.22) and (6.23). Theorem 6.15 (Kühnel [29]) Let (., g) be an .n-dimensional complete connected g. Then the Riemannian manifold admitting a non-constant solution .ρ of .∇∇ρ = ρ n number of critical points of .ρ is . N ≤ 2, and . is conformally diffeomorphic to (i) a 3-sphere . S 3 if . N = 2, or (ii) Euclidean space . E 3 , or hyperbolic space . H 3 if . N = 1, or (iii) the product of a complete 2-dimensional manifold and an open real interval, if . N = 0. If . is compact, then only (i) occurs. Duggal and Sharma [30] developed this study based on the following ADM .3 + 1 splitting representation (Arnowitt, Deser and Misner [31]) of the space-time metric as a b 2 i 2 a b i j . gab d x d x = (−λ + S Si )dt + 2γi j S d x dt + γi j d x d x where .λ is the lapse function, . S is the shift vector, .x 0 = t, .x i (.i = 1, 2, 3) are spatial coordinates and.γi j is the 3-metric on space-like slices. (.t=constant). The first result of [30] is as follows. Theorem 6.16 (Duggal-Sharma1) Let (. M, g) be a space-time foliated by a 1parameter family of space-like hypersurfaces .t , which admit a conformal vector field .V tangential to .t . If the lapse function is constant along .V , then .V is Killing on (. M, g). This result generalizes the following known result (see [25]): “The conformal vector fields on a Robertson–Walker space-time, that are tangential to the space-like slices, are Killing”. Their next three results are given below. Theorem 6.17 (Duggal-Sharma2) Let (. M, g) be a space-time evolved out of a complete initial hypersurface . that is totally umbilical and has non-vanishing constant curvature. If (. M, g) admits a closed conformal vector field .V non-vanishing on ., then either .V is orthogonal to . and the lapse function is constant over ., or . is conformally diffeomorphic to . E 3 or . S 3 or . H 3 or the product of an open interval with a 2-dimensional Riemannian manifold.

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Theorem 6.18 (Duggal-Sharma3) Let a conformally flat perfect fluid solution (. M, g) of Einstein’s field equations be evolved out of an initial space-like hypersurface . that is compact, has constant mean curvature and is orthogonal to the 4-velocity. If (. M, g) has a non-vanishing non-Killing conformal vector field .V which is nowhere tangential to ., then . is totally umbilical in . M, and is of constant curvature. In the case when . M is of constant negative curvature, .V is orthogonal to .. Theorem 6.19 (Duggal-Sharma4) Let .V be a non-null conformal vector field on a space-time. If .V is also a geodesic vector field, then it is either homothetic or locally closed is Killing on (. M, g).

.

In the above-mentioned results, a closed vector field .V means that the 1-form V ∗ metrically equivalent to .V is closed, i.e. .d V ∗ = 0. A further generalization of generalized Robertson–Walker space-time (that was defined right after Robertson– Walker space-times in this chapter) is a synchronous space-time whose metric is expressed in terms of Gaussian normal coordinates as ds 2 = −dt 2 + γi j d x i d x j

.

where .γi j depend on all coordinates (see Misner–Thorne–Wheeler [32] and Coley– Tupper [33]). Sharma and Duggal [7] obtained the following result. Theorem 6.20 (Sharma-Duggal [7]) Let (. M, g) be a spatially complete synchronous space-time solution of Einstein’s field equations and .t be a spatial slice (.t = constant) with the induced metric .γt . If . M admits a conformal vector field whose normal component is non-constant and tangential component is conformal on .t , then .t is conformally diffeomorphic to a (i) Euclidean space, or (ii) a sphere, or (iii) hyperbolic space, or (iv) the product of an open real interval and a 2-dimensional Riemannia manifold. In case .t is compact, only (ii) occurs. Proofs of some of the above-mentioned results use the formulas developed by Fischer and Marsden [34], for the geometry of .t = constant slices in a space-time manifold. In what follows, we mention some results (without giving their proof) that indicate the restrictions on space-times placed by the existence of a conformal vector field. Garfinkle [35]. A conformal vector field on an asymptotically Minkowskian, vacuum space-time with positive Bondi energy is necessarily Killing. Sharma [36]. A space-time (. M, g) with parallel Weyl tensor and a non-homothetic conformal vector field preserving the Einstein tensor is either conformally flat, or of Petrov type . N in case it represents plane-fronted gravitational waves with parallel rays. Sharma [37]. A space-time (. M, g) with divergence-free (i.e. harmonic) Weyl tensor W and a non-homothetic conformal vector field is either (i) conformally flat, or (ii)

.

6.6 Conformal Vector Fields on Space-Times

81

locally of Petrov type . N in which case the quadruply repeated principal null direction of .W is given by the gradient vector field . Dσ of the conformal scale function .σ . In case (ii), the null sectional curvature with respect to . Dσ is non-negative if . M satisfies the null convergence condition mentioned in [15]. Kerckhove [38]. Let (. M, g) be an .n-dimensional Einstein manifold of arbitrary signature with . Ric = k(n − 1)g, k = 0. Let .C( p) denote the set of all closed conformal vector fields at . p ∈ M. If each subspace .C( p) is a non-degenerate subspace of .T p M, whose dimension .d is independent of . p, then . M is locally isometric to a warped product . B × f F. The base (. B, g B ) is a .d-dimensional space of constant sectional curvature .k; the fiber (. F, g F ) is an (.n − d)-dimensional Einstein manifold with . Ric F = c(n − d − 1)g F for some constant .c; and the gradient of the warping function . f is a closed conformal vector field on . B satisfying .g(D f, D f ) = c − k f 2 . In spite of the severe restrictions placed on space-times by a conformal vector field, a considerable amount of work has been done which shows that conformal vector fields play a considerable role in exploring deep physical insights on some astrophysical and cosmological issues, as described below. Kramer [39, 40] studied perfect fluid stationary and axisymmetric exact solution rigid rotation. The first paper investigated conformal vector commuting with two Killing vectors and showed that the only solution is a static conformally flat Schwarzschild interior solution. In the second paper, Kramer showed that there is no solution of Einstein’s equations if a conformal vector field whose commutation with each Killing vector is an arbitrary linear combination of the Killing vectors. Mars–Senovilla [41, 42]. For an axially symmetric space-time with a conformal vector field .V and the axial Killing vector . K they showed: (1) Exactly one .V commutes with . K , so if they do not commute, there is at least one 3-dimensional conformal group. (2) If .V is time-like and non-commuting with . K , then either .V and . K are the only symmetries, or else there is at least one 4-dimensional conformal group. (3) Given one .V and only one more Killing vector, . K commutes with both. (4) If . M is also stationary with one .V , then . K commutes with both. Herrera et al. [43–45] They found solutions for spherically symmetric anisotropic matter with a special conformal vector field and matched them with the Schwarzschild exterior metric on the boundary of the matter. For more work on conformal vector fields, we refer to the following: Kramer– Carot [46], Capocci–Hall [47], Hall [48], Maartens–Mason–Tsamparlis [49], Dafterdar– Dadhich [50], Castejon–Amenedo–Coley [51], Dyer–McVittie–Oattes, [52], Maartens– Maharaj–Tupper [53, 54], Maharaj–Maharaj [55, 56], Sussman [57], Sussman– Lake [58], Rahman–Maharaj–Sardar–Chakraborty [59] and others.

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6.7 Inheriting Conformal Vector Fields In order to study kinematic and dynamic properties of a fluid space-time with a conformal vector, we recall the following relations [49]: LV u a = −σ u a + Y a , LV u a = σ u a + Ya , Ya u a = 0,

(6.30)

LV n a = −σ n a + W a , LV n a = σ n a + Wa , Wa n a = 0,

(6.31)

.

.

LV l = −σ k +

.

Y +W Y −W √ , LV k = −σ l + √ , 2 2

(6.32)

such that .< W, u > + < Y, n >= 0, .< l, Y + W >= 0, .< k, Y − W >= 0, and where .u is the fluid 4-vector (.< u, u >= −1), .n is the unit space-like vector .⊥ u, and .k, l are null vectors .⊥ to both .u and .n. Thus, we see that a conformal vector field does not, in general, map 4-velocity vector .u a conformally to .u a . Let . M be a solution of Einstein’s field equations with imperfect matter tensor (i.e. viscous, heat conducting fluids), defined by T

. ab

= μu a u b + ph ab − 2ησab + qa yb + qb u a

where .μ, . p, .q a , .η (.≥ 0), .h ab , .σab are energy density, the isotropic pressure, the heat flux vector relative to the fluid velocity vector .u, the shear viscosity coefficient, the projection tensor and the shear tensor, respectively. If .V is a homothetic vector field on . M, then Eardley [19] showed that the following symmetry inheritance equations hold: .

LV u a = σ u a , LV μ = −2σ μ, LV p = −2σ p LV qa = −σ qa , LV σab = σ σab , LV η = −σ η.

For a general conformal vector field, the above equations need not hold. Therefore, Coley and Tupper [33] defined the inheriting conformal vector field as follows. Definition The space-time symmetries of a conformal vector field .V are said to be inherited if fluid flow lines are mapped conformally by .V . They studied synchronous space-times whose metric is given by .ds 2 = −dt 2 + Hi j d x i d x j (where . Hi j depend on all 4 coordinates and .u is comoving, i.e. .u a = δ0a ), and showed that synchronous perfect fluid space-times other than Robertson–Walker space-times do not admit any inheriting conformal vector field. In the following we briefly point out the physical significance of such vector fields as mentioned and cited in [33].

References

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1. Israel (1972) has shown for a distribution of massless particles in equilibrium that the inverse temperature function vector .V = T1 u a is an inheriting conformal vector field, where .T is the temperature. Therefore, .V is closely related to the relativistic thermodynamics of fluids. 2. Tauber and Weinberg (1961) have argued that the isotropy of the cosmic microwave background can be used to imply the existence of an inheriting conformal vector field, since if the cosmic microwave background is isotropic about each observer moving with the timelike velocity congruence, then there exists (locally) a conformal vector field parallel to .u and hence inheriting.

References 1. Raychaudhuri, A.K.: Relativistic cosmology. Phys. Rev. 98, 1123–1126 (1955) 2. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973) 3. Einstein, A.: Die Feldgleichungen der Gravitation. Sitzungsberichle der Preussischen Akademie der Wissenschaften zu Berlin, pp. 844–847 (1915) 4. Eardley, D.M., Isenberg, J., Marsden, J., Moncrief, V.: Homothetic and conformal symmetries of solutions to Einstein’s equations. Commun. Math. Phys. 106, 137–158 (1986) 5. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, 2nd edn. Marcel Dekker, Inc., New York (1996) 6. Alias, L.J., Romero, M., Sanchez, M.: Spacelike hypersurfaces of constant mean curvature in certain spacetimes. Nonlinear Anal. 30, 655–661 (1997) 7. Sharma, R., Duggal, K.L.: Solutions of Einstein’s equations with conformal symmetries and a contact slice. Nonlinear Anal. 71, e701-705 (2009) 8. Sharma, R.: Synchronous space-times with pure trace extrinsic curvature slices. Class. Quant. Grav. 28, 195021 (7 pp) (2011) 9. Brinkmann, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 18, 119–145 (1925) 10. Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: Witten, L. (eds.) Gravitation: An Introduction to Current Research, pp. 49–101 (1962) 11. Petrov, A.Z.: Einstein Spaces. Pergamon Press, Oxford (1969) 12. Bel, L.: Quelques remarques sur le classification de Petrov. C.R. Acad. Sci. (Paris) 148, 2561– 2564 (1959) 13. Wald, R.M.: General Relativity. University of Chicago Press (1984) 14. Wasserman, R.H.: Tensors and Manifolds with Applications to Mechanics and Relativity. Oxford University Press, New York, Oxford (1992) 15. O’Neill, B.: Semi-Riemanniann Geometry with Applications to Relativity. Academic, New York (1983) 16. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2009) 17. d’Inverno, R.: Introducing Einstein’s Relativity. Clarendon Press, Oxford (1992) 18. McIntosh, C.B.G.: Homothetic motions in general relativity Gen. Rel. Grav. 7, 199–213 (1972) 19. Eardley, D.M.: Self-similar space-times: geometry and dynamics. Commun. Math. Phys. 37, 287–309 (1974) 20. Hsu, L., Wainwright, J.: Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions. Class. Quant. Gravit. 3, 1105–1124 (1986)

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21. Berger, B.: Homothetic and conformal motions in spacelike slices of solutions of Einstein equations. J. Math. Phys. 17, 1268–1273 (1976) 22. Bochner, S.: Curvature and Betti numbers. Ann. Math. 49, 379–390 (1948) 23. Fialkow, A.: Hypersurfaces of a space of constant curvature. Ann. Math. 39, 762–785 (1938) 24. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) 25. Maartens, R., Maharaj, S.D.: Conformal Killing vectors in Robertson-Walker spacetimes. Class. Quant. Grav. 3, 1005–1011 (1986) 26. Collinson, C.D., French, D.C.: Null tetrad approach to motions in empty spacetime. J. Math. Phys. 8, 701–708 (1967) 27. Garfinkle, D., Tian, Q.: Spacetimes with cosmological constant and a conformal Killing field have constant curvature. Class. Quant. grav. 49, 137–139 (1987) 28. Sharma, R.: Conformal symmetries of Einstein’s field equations and initial data. J. Math. Phys. 46, 042502 (1–8) (2005) 29. K.uhnel, ¨ W.: Conformal transformations between Einstein spaces. In: Kulkarni, R.S., Pinkall, U. (eds.), Conformal Geometry. Wiesbaden, Vieweg Braunschweig (1988) 30. Duggal, K.L., Sharma, R.: Conformal Killing vector fields on spacetime solutions of Einstein’s equations and initial data. Nonlinear Anal. 63, e447-454 (2005) 31. Arnowitt, A., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. Wiley (1962) 32. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman and Company, San Francisco (1973) 33. Coley, A.A., Tupper, B.O.J.: Spacetimes admitting inheriting conformal Killing vector fields. Class. Quantum Grav. 7, 1961–1981 (1990) 34. Fischer, A.E., Marsden, J.A.: The Einstein equations of evolution-a geometric approach. J. Math. Phys. 13, 546–568 (1972) 35. Garfinkle, D.: Asymptotically flat space-times have no conformal Killing fields. J. Math. Phys. 28, 28–32 (1987) 36. Sharma, R.: Proper conformal symmetries of conformal symmetric spacetimes. J. Math. Phys. 29, 2421–2422 (1988) 37. Sharma, R.: Proper conformal symmetries of spacetimes with divergence-free conformal tensor. J. Math. Phys. 34, 3582–3587 (1993) 38. Kerckhove, M.: The structure of Einstein spaces admitting conformal motions. Class. quant. Grav. 8, 819–825 (1991) 39. Kramer, D.: Rigidly rotating perfect fluids admitting a conformal Killing vector. In: Proceedings of the 3rd Hungarian Relativity Workshop, Tihany (1989) 40. Kramer, D.: Perfect fluids with conformal motion. Gen. Rel. Grav. 22, 1157–1162 (1990) 41. Mars, M., Senovilla, J.M.M.: Axial symmetry and conformal Killing vectors. Class. Quant. Grav. 10, 1633–1647 (1993) 42. Mars, M., Senovilla, J.M.M.: Stationary and axisymmetric perfect fluid solutions with conformal motion. Class. Quant. Grav. 11, 3049–3068 (1994) 43. Herrera, L., Jimene, J, Leal, L., Ponce de Leon, J.: Anisotropic fluids and conformal motions in general relativity. J. Math. Phys.25, 3274–3278 (1984) 44. Herrera, L., Ponce de Leon, J.: Anisotropic spheres admitting a 1-parameter group of conformal motions. J. Math. Phys. 26, 2018–2023 (1985) 45. Herrera, L., Ponce de Leon, J.: Isotropic and anisotropic charged spheres admitting a 1parameter group of conformal motions. J. Math. Phys. 26, 2302–2307 (1985) 46. Kramer, D., Carot, J.: Conformal symmetry of perfect fluids in general relativity. J. Math. Phys. 32, 1857–1860 (1991) 47. Capocci, M.S., Hall, G.S.: Conformal vector fields on decomposable space-times. Grav. Cosmol. 3, 001–010 (1997) 48. Hall, G.S.: Symmetries and Curvature Structure in General Relativity. World Scientific, Singapore (2004) 49. Maartens, R., Mason, D.P., Tsamparlis, M.: Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids. J. Math. Phys. 27, 2987–2994 (1986)

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50. Daftardar, V., Dadhich, N.: Gradient conformal Killing vectors and exact solutions. Gen. Rel. Grav. 26, 850–868 (1994) 51. Castejon-Amenedo, J., Coley, A.A.: Exact solutions with conformal Killing vector fields. Class. Quant. Grav. 9, 2203–2215 (1992) 52. Dyer, C.C., McVittie, G.C., Oattes, L.M.: A class of sphrerically symmetric solutions with conformal Killing vectors. Gen. Rel. Grav. 19, 887–898 (1987) 53. Maartens, R., Maharaj, S.D., Tupper, B.O.J.: General solution and classification of conformal motions in static spherical spacetimes. Class. Quant. Grav. 12, 2577–2586 (1995) 54. Maartens, R., Maharaj, S.D., Tupper, B.O.J.: Addendum-conformal motions in static spherical spacetimes. Class. Quant. Grav. 13, 317–318 (1996) 55. Maharaj, S.D., Maharaj, M.S.: Conformal symmetries of pp-waves. Class. Quant. Grav. 8, 503–514 (1991) 56. Maharaj, S.D., Maharaj, M.S.: A conformal vector in shearing spacetimes. Il Nuovo Cimento 109, 983–988 (1994) 57. Sussman, R.A.: Radial conformal Killing vectors in spherically symmetric shear-free spacetimes. Gen. Rel. Grav. 21, 1281–1301 (1989) 58. Sussman, R.A., Lake, K.: On spherically symmetric solutions admitting a spacelike conformal motion. Gen. Rel. Grav. 21, 1281–1301 (1989) 59. Rahman, F., Maharaj, S.D., Sardar, I.H. and Chakraborty, K.: Conformally symmetric relativistic star. Modern Phys. Lett. A 32, 1750053 (15 p.) (2017)

Chapter 7

Ricci Solitons

7.1 Preliminaries A soliton (solitary wave) is basically a self-reinforcing wave packet that maintains its shape while propagating at a constant velocity. The soliton phenomenon was first described in 1834 by Scott Russell [1] who observed a solitary wave in the Union Canal in Scotland. He replicated the phenomenon in a wave tank and named it the wave of translation. Later, Korteweg and de Vries [2] represented this phenomenon by means of a partial differential equation known as KdV equation. The solitons occur as solutions of nonlinear Schrodinger equation, Sine-Gordon equation and also in nonlinear optics, fluid mechanics, biophysics, DNA and other areas of science. For a conservative (non-dissipative) system, a soliton is a solitary wave whose amplitude, shape and velocity are conserved after a collision with another soliton.

7.2 Ricci Flow In 1904, Poincaré made the following conjecture: “A closed simply connected .3manifold is homeomorphic to the .3-dimensional sphere . S 3 .” A more general conjecture than Poincaré Conjecture is Thurston’s geometrization conjecture which says that any closed .3-manifold can be decomposed into pieces such that each piece has a locally homogeneous metric, and are . S 3 , .R3 , . H 3 , 2 2 R), .nil 3 and .sol 3 . . S × R, . H × R, . S L(2, With the aim of proving the geometrization conjecture, Hamilton [3] initiated a program in 1982, called Ricci flow that starts with a given Riemannian metric .g on a smooth .n-dimensional manifold . M and evolves it as a 1-parameter family of metrics . g(t) satisfying .∂t gi j = −2Ri j , g(0) = g0 . (7.1) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_7

87

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7 Ricci Solitons

The idea is to try to evolve the metric in some way that will make the manifold rounder and rounder. The minus sign makes the Ricci flow a heat type equation, and so it is expected to average out the curvature. Due to the minus sign before the Ricci tensor, the solution metric shrinks in positive Ricci curvature direction and expands in the negative Ricci curvature direction. We will see later that if the initial metric .g0 has strictly positive Ricci curvature, then the manifold . M 3 will shrink to a point in finite time under the Ricci flow. But, if we dilate the metric by a time-dependent factor so that the volume remains constant, the problem of shrinking to a point is removed, and we can show that the rescaled metric converges uniformly to the desired metric of constant positive curvature on . M 3 . In 2002 and 2003, Grisha Perelman posted three papers [4–6] on arXiv.org, claiming to have completed Hamilton’s program toward using Ricci flow with surgery to prove Geometrization conjecture (in particular, Poincaré conjecture). The Ricci flow does not preserve the volume. The volume preserved by normalized Ricci flow on a closed manifold is given by ∂ gi j = −2Ri j +

. t

2ρ gi j n

(7.2)

  where .ρ = ( M r dv/ M dv)(t) is the average scalar curvature and .r the scalar curvature of .g(t). Under the unnormalized Ricci flow, the following evolution equations hold: ij .∂t g = 2R i j , (7.3) ∂ ikj = −g kl (∇i R jl + ∇ j Ril − ∇l Ri j ),

(7.4)

∂ Ri j = −g pq (∇q ∇i R j p + ∇q ∇ j Ri p − ∇q ∇ p Ri j − ∇i ∇ j Rq p ),

(7.5)

∂ r = r + 2|Ric|2 ,

(7.6)

∂ dv = −r dv

(7.7)

. t

. t

. t

. t

where .dv is the volume element with respect to .g(t). We now state the following existence theorems for Ricci flow. Theorem 7.1 The Ricci flow on a closed Riemannian manifold (. M, g0 ) has a unique solution .g(t) on some positive time interval .[0, T ) such that .g(0) = g0 . If (. M, g(t)), .t ∈ (0, T ), where . T < ∞ is a solution to the Ricci flow on a closed manifold with .sup M×(0,T ) |Ric| < ∞, then the solution . g(t) can be uniquely extended past . T . Its proof is sophisticated. In order to simplify the proof, DeTurck [7] showed that the Ricci flow (which is weakly parabolic) is equivalent to a strictly parabolic system called the Ricci–DeTurck flow:

7.2 Ricci Flow

89

∂ gi j = −2Ri j + LW gi j

. t

where .W is a vector field with covariant components .

Wi = gi j g kl ( kl − ˜ kl ) j

j

where .˜ kl are the connection coefficients of a fixed background metric. As the difference between two connections is a tensor,.W is a globally well-defined vector field. j

In the passing, we note that the Ricci flow was initiated by the following famous result of Hamilton [3]. Theorem 7.2 Let .(M 3 , g0 ) be a closed Riemannian .3-manifold of strictly positive Ricci curvature. Then a unique solution.g(t) of the normalized Ricci flow with.g(0) = g0 exists for all positive time; and as.t → ∞, the metrics.g(t) converge exponentially fast in every .C k -norm, .k ∈ N , to a metric .g∞ of constant positive curvature. Let us view the Ricci flow in harmonic coordinates. The harmonic coordinates (.x i ) are such that each coordinate function is harmonic, i.e. .x i = 0 and hence ij k . g i j = 0. Such coordinates exist around any point of a Riemannian manifold. In harmonic coordinates, the Ricci flow takes on the form ∂ gi j = gi j + Q i j (g −1 , ∂g)

. t

where . Q is a quadratic form in .g −1 and .∂g. Thus, the Ricci flow resembles the heat flow, and one hopes that the Ricci curvature becomes uniform after some time during the flow. We also note that there is a reaction term. Q(g −1 , ∂g) which is of lower order in .∂g and hence the above equation is a reaction–diffusion equation. Let us try to understand Ricci flow through the following example. Consider an 2 .n-dimensional sphere of radius .a = a(t) and metric . g = a gcan where . gcan is the n canonical metric on the unit sphere . S (1). As .g and .gcan are homothetic, their Ricci tensors are same and hence. Ric = (n − 1)gcan . Applying the Ricci flow gives. dtd a 2 =  −2(n − 1) which readily integrates to .a = a02 − 2(n − 1)t where .a0 = a(0) the a2

0 ; consequently, the initial radius of the sphere. This shows that .a → 0 as .t → 2(n−1) sphere shrinks to a point. If we had taken a multiple of the hyperbolic space . H n of constant curvature.−1 by a time-dependent function.a, and applied the Ricci flow, we

would have gotten .a = a02 + 2(n − 1)t, and so the solution expands out to infinity. The Euclidean metric on . E n has zero Ricci tensor, so it does not evolve at all under the Ricci flow. More generally, a Ricci flat metric is a fixed point of the Ricci flow. For more details on the Ricci flow and its singularity types, we refer to Chow, Lu and Ni [8].

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7 Ricci Solitons

Singular solutions of Ricci flow are classified as TypeI when .sup M×[0,T ) (T − t)|Rm| < ∞ TypeIIa when .sup M×[0,T ) (T − t)|Rm| = ∞ III when .T = ∞ and .sup M×[0,T ) t|Rm| < ∞ TypeIIb when .sup M×[0,T ) t|Rm| = ∞. Shrinking spherical space forms and neckpinches (where a region of the manifold asymptotically approaches a shrinking round cylinder) are Type I singularities.

7.3 Ricci Soliton Let us consider generalized fixed points of the Ricci flow as those manifolds that change by a diffeomorphism and a rescaling under the Ricci flow. More precisely, let . M be an.n-dimensional smooth manifold and (. M, g(t)) be a solution of the Ricci flow such that .g(0) = g. Let .ϕt : M → M be a 1-parameter family of diffeomorphisms generated by the family of vector fields .V (t) and .σ (t) be a time-dependent scale factor. Then a solution of the Ricci flow, of the form: .

g(t) = σ (t)ϕt∗ g

(7.8)

is called a Ricci soliton. Thus, a Ricci soliton is a generalized fixed point of the Ricci flow, viewed as a dynamical system on the space of Riemannian metrics modulo diffeomorphisms and scalings. Taking the derivative of the above equation with respect to .t, substituting .t = 0 and assuming .σ˙ (0) = −2λ, σ (0) = 1, ϕ0 = identity, one obtains .L V g + 2Ric = 2λg (7.9) where .V = V (0). So, we define a Ricci soliton as a Riemannian manifold (. M, g) with a vector field .V and a real number .λ such that Eq. (7.9) is satisfied. A Ricci soliton is said to be shrinking, steady, or expanding according as.λ is.> 0,.= 0, or.< 0, respectively. Naturally, a Ricci soliton is a generalization of an Einstein manifold for which .V is zero (more generally, a Killing vector field). In case .V = D f , i.e. the gradient of a smooth function . f on . M, Eq. (7.9) becomes .

H ess f + Ric = λg

(7.10)

where . H ess f = ∇∇ f is the . H essian of . f , and then (7.10) defines a gradient Ricci soliton. The function . f is called the potential function. For a unique Ricci soliton, a canonical choice for .σ (t) is .1 − 2λt. By rescaling one can assume .λ = 1/2, 0, −1/2 for a shrinking, steady, expanding Ricci soliton, respectively. We note that the term .L V gi j can be expressed as .∇i V j + ∇ j Vi . Gradient shrinking Ricci solitons play an important role in the Ricci flow, as they correspond to self-similar solutions, and often arise as blow-up limits of the Ricci flow when singularities develop. Ricci solitons

7.3 Ricci Soliton

91

are of interest to physicists as well and are called quasi-Einstein metrics (see Friedan [9]). Let us consider two simple examples. Example 7.1 Let (. M, g) be an Einstein manifold, i.e. . Ric = λg. Then .g(t) = (1 − 2λt)g satisfies the Ricci flow .∂t g(t) = 2Ric(g(t)) because . Ric(g(t)) = Ric(g) = λg. Since a complete Einstein manifold with .λ > 0 is compact, by Myers’ theorem, 1 . and as the metric approaches zero, the manifold shrinks to a point in time . 2λ Example 7.2 Let (. E n , gcan ) be the Euclidean .n-space with its canonical metric λ|x|2 . gcan . Obviously . Ric = 0. So, if we set . f (x) = , where .|x|2 = xk xk , then one 2 can verify by a straightforward calculation that .∇ j f = λx j and .∇i ∇ j f = λδi j , i.e. (. E n , gcan ) is a gradient Ricci soliton with potential function . f . This is called the Gaussian soliton. Example 7.3 A 2-dimensional example is .R2 with metric .g((x, y), t) having line2 2 element .ds 2 = ed4tx+x+dy 2 +y 2 . Direct calculations show that it satisfies the Ricci flow 4t

equation, has positive Gaussian curvature . (e4t +x2e2 +y 2 )2 . Then .g((x, y), 0) with lineelement d x 2 + dy 2 2 .ds = 1 + x 2 + y2 is called the cigar soliton. Its potential function . f is given by . f (x, y) = −log(1 + x 2 + y 2 ). We can express it in polar coordinates as the warped product ds 2 = dr 2 +

.

1 tanh 2 (αr )dθ 2 α2

where .α is a positive constant. Example 7.4 The cigar metric may be generalized to a rotationally symmetric non-compact steady gradient Ricci soliton (Bryant soliton) in higher dimensions as a warped product . I ×w S n−1 with some warping function .w. This has positive sectional curvature, and the volume of a ball . Bo (ρ) grows on the order of .ρ (n+1)/2 and curvature .→ 0 like .1/s as .s → ∞. Example 7.5 Baird and Danielo [10] and independently, Lott [11] constructed 3 3-dimensional Ricci solitons. Consider the 3-dimensional Heisenberg group ⎡ ⎤ .nil 1x y as the multiplicative group of upper triangular matrices . A = ⎣ 0 1 z ⎦, where 001 3 . x, y, z ∈ R. This is simply connected and diffeomorphic to .R . Taking . x, y, z as coordinates, we define the frame field .

e = 2∂x , e2 = 2(∂ y − x∂ y ), e3 = 2∂z .

. 1

92

7 Ricci Solitons

All Lie brackets vanish except .[e1 , e2 ] = −2e3 . The dual frame is e1 =

.

1 1 1 d x, e2 = dy, e3 = (xdy + dz). 2 2 2

If we define a left-invariant metric by .g = 4(e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 ) then Ric = 2(−e1 ⊗ e1 − e2 ⊗ e2 + e3 ⊗ e3 ). Taking the vector field.V = (−1/2)(xe1 + ye2 + (x y + 2z)e3 ), performing a straightforward calculation shows that .LV g + 2Ric = −3g, and hence (.nil 3 , g, V ) is an expanding Ricci soliton, however it is not gradient. Similarly, a Ricci non-gradient expanding soliton can be constructed on 4 .nil . .

Example 7.6 The 3-dimensional solvable Lie group .sol 3 = R  R2 such that .x ∈ R sends.(y, z) ∈ R2 to.(e x y, e−x z). This is diffeomorphic to.R3 , and can be viewed as the group of rigid motions of Minkowski 2-space. With standard coordinates .x, y, z, we define the frame .e1 = 2∂x , e2 = 2(e−x ∂ y + e x ∂z ), e3 = 2(e−x ∂ y − e x ∂z ). Their Lie brackets are .[e1 , e2 ] = −2e3 , [e2 , e3 ] = 0, [e3 , e1 ] = 2e2 , and hence define a 1 x 1 −x 2 3 1 dz), e3 = 41 (e x dy − .sol frame. Taking the dual field .e = d x, e = (e dy + e 4 2 −x 1 1 2 2 e dz) and the left-invariant metric .g = 4(e ⊗ e + 2e ⊗ e + 2e3 ⊗ e3 ) and a straightforward calculation shows that . Ric = −8(e1 ⊗ e1 ). Now, taking the vector field .V = c(−e1 − e−x ze2 + e−x ze3 ) + (1 − c)(e1 − e x ye2 − e x ye3 ), one can verify (exercise) that .LV g + 2Ric = −4g. Hence it is is an expanding Ricci soliton which is not gradient. 3 3 .nil and .sol are examples of Type III solutions of Ricci flow. At this point, let us study some properties of Ricci solitons. The .g-trace of Ricci soliton equation is . div(V ) + r = nλ. (7.11) Next, writing (7.9) as .∇i V j + ∇ j Vi + 2Ri j = 2λgi j , operating it by .∇ i and using Schur’s lemma: .∇ i Ri j = 21 ∇ j r gives .∇ i ∇i V j + Rk j V k = 0, where we also used (7.11). The preceding equation can also be written as V = 0

.

(7.12)

where . is the Yano operator .−(g i j ∇i ∇ j + Ri j ) as defined in Chap. 5. Equation (7.12) is equivalent to .g jk LV  ijk = 0 whose solution vector field .V is called an infinitesimal harmonic transformation (see Stepanov and Shandra [12]). Hence, the Ricci soliton vector field .V is an example of an infinitesimal harmonic transformation. Let us denote the inverse of .gi j by .g i j . Taking the Lie derivative of the relation .gi j g jk = δik along .V , using Eq. (7.9) and subsequently operating the resulting equation by .g il , we immediately get LV g kl = 2R kl − 2λg kl .

.

(7.13)

7.4 Some Results on Ricci Solitons

93

Next, the use of Eq. (7.9) in the formula (see Chap. 5): LV ihj =

.

1 ht g [∇ j (L X git ) + ∇i (L X g jt ) − ∇t (L X gi j )] 2

yields the evolution equation LV ihj = ∇ h Ri j − ∇ j Rih − ∇i R hj .

.

(7.14)

Following our convention:.∇k ∇ j Z h − ∇ j ∇k Z h = R hik j Z i , where. Z i are components of an arbitrary vector field and using the above Eq. (7.14) in the following commutation formula (see Chap. 5): h ∇k (LV ihj ) − ∇ j (LV ik ) = LV R hik j ,

.

we obtain the evolution equation: LV R hik j = ∇ j ∇k Rih − ∇k ∇ j Rih + ∇ j ∇i Rkh − ∇k ∇i R hj

.

+ ∇k ∇ h Ri j − ∇ j ∇ h Rik .

(7.15)

Contracting this equation with.g hk and using Schur’s lemma:.∇i R ij = 21 ∇ j r , we have LV R ji = ∇ j ∇i r − ∇h ∇ j Rih − ∇h ∇i R hj + Ri j .

.

(7.16)

Lie-differentiating .r = Ri j g i j along .V , and using the above equation and Eq. (7.13) provides the evolution equation for the scalar curvature: LV r = 2|Ric|2 + r − 2λr.

.

(7.17)

From the last equation, we find that, if the scalar curvature .r is zero, then . Ric = 0.

7.4 Some Results on Ricci Solitons We will recall some results on Ricci solitons. However, our list is not exhaustive. Let us state the following important result of Perelman [4]. Theorem 7.3 (Perelman) A compact Ricci soliton is necessarily gradient. In [13] Eminenti, Nave and Mantegazza raised the question of proving this result directly from Eq. (7.9). This was answered by Stepanov [14] more generally, by showing that an infinitesimal harmonic transformation defined earlier by (7.12), on a compact Riemannian manifold, is orthogonally decomposed as the sum of a gradient

94

7 Ricci Solitons

infinitesimal transformation and a Killing vector field. As we saw through Eq. (7.12), a Ricci soliton vector field .V is an infinitesimal harmonic transformation, Perelman’s result follows by the aforementioned result as a special case. We note that a complete shrinking Ricci soliton was shown to have finite fundamental group, by Wylie [15] (the compact case was proved by Fernández-López and García-Río in [16]). Naber [17] showed that a complete shrinking Ricci soliton can be made into a gradient Ricci soliton by adding an appropriate Killing vector field to . V . Sharma, Balasubramanian and Uday Kiran [18] showed that a non-steady Ricci soliton is gradient if and only if (.δ + i V )dv is exact, where .v is the 1-form metrically equivalent to .V , .δ is the co-differential operator acting on a . p-form .ω such that .δω is a . p − 1 form given by .(δω)i2 ...i p = −∇ i1 ωi1 ...i p and .i V is the interior product with . V . In particular, a non-steady Ricci soliton is gradient if .v is closed. Proposition 7.1 For a gradient Ricci soliton, the following formulas hold: 1. . f + r = nλ, 2. . R(X, Y )D f = (∇Y Q)X − (∇ X Q)Y , 3. . Q(D f ) = 21 Dr , 4. .|D f |2 + r − 2λ f is constant, where .r is the scalar curvature, . Q the Ricci operator and . D the gradient operator. These formulas were derived by Hamilton in [19]. (1) is simply the trace of (7.9). (2) follows from (7.10). (3) follows by contracting (2) at . X and using Schur’s lemma: .div Q = 21 dr . For (4), we use (7.10) and the above-mentioned result (3) to proceed as follows. . X |D f |2 = 2g(∇ X d f, D f ) = 2g(−Q X + λX, D f ) = −2g(X, Q D f ) + 2λX f = −Xr + 2λX f . This leads to (4). For compact Ricci soliton, we have the following result. Proposition 7.2 (i) A compact expanding or steady Ricci soliton is Einstein. (ii) A compact shrinking Ricci soliton has positive scalar curvature. Proof The scalar curvature evolution Eq. (7.17) for the Ricci soliton can be written as 2 .(r − nλ) − V (r − nλ) + 2|Ric − λg| + 2λ(r − nλ) = 0. Suppose .r takes its minimum at a point . p ∈ M. Then, the above equation shows that |Ric − nλ|2 + λ(r − nλ) ≤ 0

.

 at . p. Also, tracing (7.10) gives . M r − nλdv = 0, i.e. the average value of .r − nλ is .0. As .λ ≤ 0 for a steady or expending Ricci soliton, it turns out that .λ(r − nλ) ≥ 0 at . p. Consequently, minimum value of .r − nλ is zero which is also the average value of .r − nλ. So, .r = nλ on . M. substituting it in the trace of (7.10) and integrating shows that . f is constant, and hence .g is Einstein. For the shrinking case, .λ > 0. Again, let .r have a minimum at . p. Then Eq. (7.17) can be used to conclude that the minimum value of .r is .≥ 0, and hence .r ≥ 0 on . M. By the strong maximum principle, we find

7.4 Some Results on Ricci Solitons

95

that .r = 0 on . M, or .r > 0 on . M. But, if .r = 0, then . f = nλ which would imply λ = 0, contradicting the shrinking assumption. So, .r > 0, completing the proof. We can prove (i) also in the following way. Using Proposition 7.4 (part 4) and the trace of (7.10) gives 2 . f − |D f | + 2λ f − nλ = C, (7.18)

.

where .C is a constant. For .λ = 0, considering the maximum and minimum values of f from (7.18) we find that.C = 0, and then integrating (7.18) shows that. f is constant, and . Ric = 0. It remains to consider .λ < 0. The constant .C can be absorbed within . f , and so can be taken zero. Let . f have its maximum at . p. Then (7.18) provides n . f max ≤ and hence . f ≤ n2 on . M. Next, checking at a point of minimum of . f using 2 again (7.18) provides . f min = n2 . Thus . f max ≤ n2 = f min , and hence . f is constant on . M and, consequently, . g is Einstein. .

In dimensions .≤ 3, we have the following rigidity result. Proposition 7.3 In dimensions .≤ 3, there are no compact shrinking Ricci solitons other than those of positive constant curvature. It was proved by Hamilton [3] in dimension 2, and by Ivey [20] in dimension 3. An estimate of the potential function . f for a complete non-compact gradient shrinking Ricci soliton was given by Cao and Zhou [21] in the form of the following theorem. Theorem 7.4 (Cao–Zhou) Let (. M n , g, f ) be a complete non-compact gradient shrinking Ricci soliton. Then the potential function . f satisfies the estimates .

1 1 ((ρ(x) − c1 )2 ≤ f (x) ≤ ((ρ(x) + c2 )2 4 4

where .ρ(x) = d(x, x0 ) is the Riemannian distance function from some fixed point x ∈ M, and .c1 and .c2 are positive constants depending only on .n and the geometry of .g on the unit geodesic ball . Bx0 (1). In view of the Gaussian shrinker (. R n , g0 ) with its potential function .|x|2 /4, leading term . 41 (ρ(x))2 for the lower and upper bounds on√. f in the above theorem is optimal. Also, there is a proper distance-like function .2 f on . M n . We also state the following result based on the works of Perelman [5], Ni and Wallach [22] and Cao, Chen and Zhu [23]: “A .3-dimensional complete noncompact non-flat shrinking gradient soliton is necessarily the round cylinder . S 2 × R or one of its . Z 2 quotients. In addition, the following result was established in [21].

. 0

Theorem 7.5 (Cao–Zhou) Let (. M n , g, f ) be a complete non-compact gradient shrinking Ricci soliton. Then there exists some positive constant .C > 0 such that n . V ol(Bx 0 (ρ)) ≤ Cρ for sufficiently large .ρ > 0. The classification of gradient shrinking Ricci solitons has been a subject of interest to many researchers. Let us recall the following result of Hamilton [3]: Theorem 7.6 (Hamilton) The two-dimensional shrinking gradient Ricci solitons with bounded curvature are . S 2 , . R P 2 and .R2 with constant curvature.

96

7 Ricci Solitons

A work of Bohm and Wilking [24] states that Theorem 7.7 (Bohm–Wilking) Compact gradient shrinking Ricci solitons with positive curvature operator in any dimension are of constant curvature. For the non-compact case, Perelman [5] proved the following. Theorem 7.8 (Perelman) The 3-dimensional shrinking gradient Ricci solitons with bounded non-negative sectional curvature are . S 3 , . S 2 × R, .R3 , or their quotients. We also mention the following result of Brendle [25]. Theorem 7.9 (Brendle) Let (. M, g) be a 3-dimensional complete steady gradient Ricci soliton which is non-flat and .κ-noncollapsed. Then (. M, g) is rotationally symmetric and hence isometric to the Bryant soliton. Let us recall [26] that a Riemannian manifold is .κ-noncollapsed below the scale ≥ κ for any metric ball . B(x, ρ) with .|Rm| ≤ ρ −2 (where . Rm denotes s if . V ol B(x,ρ) ρn Riemann curvature tensor) in . B(x, ρ) and .ρ < s. Let us recall the Hamilton–Ivey estimate (stated and cited in [26]): “Let .in f x∈M 3 ν(x, 0) ≥ −1 on the initial metric, where .ν(x, t) is the smallest eigenvalue of the curvature operator. There exists a continuous positive nondecreasing function ψ(u) → 0 as .u → ∞, such that for .ψ : R → R with . decreasing for .u > 0 and . ψ(u) u u 3 any solution (. M , g(t)) of the Ricci flow on a closed.3-manifold, we have.ν ≥ −ψ(r ), where .r is the scalar curvature. That is, . Rm ≥ −ψ(r )id, where . Rm is the curvature operator and .id : 2 → 2 is the identity operator.” This estimate shows that 3-dimensional shrinking Ricci solitons with bounded curvature have non-negative sectional curvature (Theorem 6.44 of [8]). The aforementioned results imply that the the only 3-dimensional shrinking gradient Ricci solitons with bounded curvature are the finite quotients of .R3 , . S 2 × R and . S 3 . Subsequently, Ni and Wallach [22] studied complete gradient shrinking Ricci solitons with vanishing Weyl curvature tensor in any dimension, assuming non-negative Ricci curvature and at most exponential growth of the norm of curvature operator, and showed that the only such shrinkers are n n n−1 . S , .R , . S × R and their quotients. Cao, Wang and Zhang [27] relaxed the nonnegative Ricci curvature assumption to having the Ricci curvature bounded from below. Petersen and Wylie [28] obtained the same classification of complete locally conformally flat gradient shrinking solitons assuming only the integral bound .



|Ric|2 e− f < ∞.

.

(7.19)

M

The question whether an integral curvature estimate such as (7.19) are true for complete gradient shrinking Ricci solitons has been raised, for example, in [27–29]. In [30], Munteanu and Sesum did succeed in showing that (7.19) is true for any complete gradient shrinking Ricci soliton. They also proved the following nice classification result. Theorem 7.10 (Munteanu–Sesum) Any.n-dimensional complete gradient shrinking Ricci soliton with harmonic Weyl tensor is a finite quotient of .Rn , . S n , or . S n−1 × R.

7.4 Some Results on Ricci Solitons

97

The proof also uses  their result “Let (. M, g) be a gradient shrinking Ricci soliton. For some .μ < 1, if . M |∇ Rm|2 e−μ f < ∞, then . M |∇ Ric|2 e− f = M | div(Rm)|2 e− f < ∞.”, where . Rm denote the Riemannian curvature tensor. Next, we recall that a gradient shrinking Kaehler–Ricci soliton satisfies .

Rαβ¯ + f αβ¯ = λgαβ¯

for a smooth function . f having the property . f αβ = f α¯ β¯ = 0. This is equivalent to ∇∇ f + Ric = λg on a Kaehler manifold with its complex structure tensor . J such that .d f is real holomorphic, i.e. .L D f J = 0. The following Liouville type theorem was established in [30].

.

Theorem 7.11 (Munteanu–Sesum) Let (. M, g)  be a gradient steady or shrinking Ricci soliton. If .u is a harmonic function with . M |D f |2 < ∞, then .u is constant on . M. We now state the following results of Petersen and Wylie [28] on constant scalar curvature soliton. Theorem 7.12 (Petersen–Wylie) Let (. M n , g, f ) be a complete gradient Ricci soliton with .n ≥ 3, constant scalar curvature, and .W (D f, ., ., D f ) = O(|D f |2 ). Then . M is a flat bundle of rank 0, 1, or .n over an Einstein manifold. In particular, for .n = 3, the theorem yields the following nice corollary. “The only .3-dimensional complete expanding gradient Ricci solitons with constant scalar curvature are quotients of .R3 , . H 3 , . H 2 × R. In [28], the following result was also obtained. Proposition 7.4 If a shrinking (resp. expanding) gradient soliton has constant scalar curvature .r , then .0 ≤ r ≤ nλ (resp. .nλ ≤ r ≤ 0). Moreover, the metric is Ricci-flat (flat if moreover, . M is complete) when .r = 0 and Einstein when .r = nλ. In addition, . f is unbounded when . M is noncompact and .r = nλ. The following result was obtained by Fernández-López and García-Río in [31], which expresses the constant scalar curvature of a gradient Ricci soliton as finitely many discretized values. Theorem 7.13 (Fernández-López and García-Río) If a gradient Ricci soliton (M n , g) has constant scalar curvature .r then .r ∈ {0, λ, . . . , (n − 1)λ, nλ}.

.

They also showed the rigidity of gradient Ricci solitons with constant scalar curvature under the assumption that the Ricci operator has constant rank. According to Petersen and Wylie [32], a gradient Ricci soliton is said to be rigid if it is isometric to a quotient of . N × Rk , where . N is an Einstein manifold and . f = λ2 |x|2 on the Euclidean factor. Any rigid gradient Ricci soliton has constant scalar curvature. A special family of manifolds with constant scalar curvature are the homogeneous ones. In this context, it is known that any homogeneous gradient Ricci soliton is rigid (see [32]).

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7 Ricci Solitons

For a locally conformally flat gradient Ricci soliton (possibly incomplete) (. M, g), Fernández-López and García-Río [16] showed that. M is locally isometric to a warped product of an interval and a space form, and if . M is, in addition, complete and nonexpanding, then it is rotationally symmetric. A Lorentzian analogue of this result was obtained by Brozos-Vázquez, García-Río and Gavino-Fernández [33] as the following result “Let (. M, g, f ) be a locally conformally flat Lorentzian gradient Ricci soliton. (i) In a neighborhood of any point where .|D f | = 0, . M is locally isometric to a Robertson–Walker warped product . I ×ψ N with metric .dt 2 + ψ 2 g N (. 2 = 1), where . I is a real interval and (. N , g N ) is a space of constant curvature. (ii) If .|D f | = 0 on a non-empty open set, then (. M, g) is locally isometric to a plane wave, i.e. locally diffeomorphic to .R2 × Rn with metric .

g = 2dudv + H (u, x1 , . . . , xn )du 2 +

n

d xi2 ,

i=1

n

where . H (u, x1 , . . . , xn ) = a(u) i=1 xi2 + i = 1n bi (u)xi + c(u) for some functions .a(u), bi (u), c(u) and the potential function given by . f (u, x1 , . . . , xn ) = f 0 (u), with . f 0 (u) = −Ricuu = na(u).” For Bach-flat case, i.e. . Bi j = 0 (see Sect. 3 of Chap. 3 for definition of Bach tensor . Bi j ), we have the following results. Cao, Catino, Chen, Mantegazza and Mazzieri [34] proved the following result: Let (. M n , g, f ), .n ≥ 4, be a complete gradient steady Ricci soliton with . Ric > 0 such that scalar curvature attains its maximum at an interior point. If, in addition, . M is Bach-flat, then it is isometric to Bryant soliton up to a scaling factor. A complete gradient Ricci soliton (. M 3 , g, f ) with divergence free Bach tensor is either Einstein or locally conformally flat. Cao and Chen [35] proved the following results: (1) A complete Bach-flat gradient shrinking Ricci soliton (. M 4 , g, f ) is either Einstein or locally conformally flat, hence a finite quotient of either the Gaussian shrinking soliton .R4 or the round cylinder . S 3 × R. (2) A complete Bach-flat gradient shrinking Ricci soliton (. M n , g, f ), .n ≥ 5, is either (i) Einstein, (ii) a finite quotient of the Gaussian shrinking soliton .Rn , or (iii) a finite quotient of . N n−1 × R, where . N n−1 is positively Einstein. The proof uses a third order tensor . D introduced by Cao and Chen [36] and defined by .

1 1 (R jk ∇i f − Rik ∇ j f ) + (g jk ∇i r − gik ∇ j r ) n−2 2(n − 1)(n − 2) r (g jk ∇i f − gik ∇ j f ). − (n − 1)(n − 2)

Di jk =

The tensor . Di jk is tied up with the Cotton tensor and Weyl tensor through . Di jk = Ci jk − Wi jlk∇l f (in terms of our notation). The vanishing of . Bi j implies the vanishing of. Di jk for gradient Ricci solitons. Further, at any point. p ∈ M where.d f ( p) = 0,

7.4 Some Results on Ricci Solitons

99

we have |Di jk |2 =

.

H 1 2|D f |2 gab |2 + |∇a r |2 |h ab − 2 (n − 2) n−1 2(n − 1)(n − 2)

where .h ab and . H are the second fundamental form and the mean curvature for the level surface . = f = f ( p), and .gab is the induced metric on the level surface .. Having shown that . Bi j = 0 implies . Di jk = 0 and which implies, in turn, that the Cotton tensor vanishes at points where .d f = 0, Cao and Chen conclude that .W = 0 for .n = 4 and .div W = 0 for .n > 4, at points where . D f = 0, and complete the proof of their result by using the result of Munteanu–Sesum [30] and the fact that gradient Ricci solitons are analytic in harmonic coordinates. A classification of.4-dimensional gradient Ricci soliton with harmonic Weyl tensor was given by Kim [37], by proving the following result. Theorem 7.14 A .4-dimensional (not necessarily complete) gradient Ricci soliton (. M, g, f ) with harmonic Weyl tensor is locally isometric to one of the following: (i) Einstein with constant . f , (ii) .R2 × Nλ where .R2 is Euclidean and . Nλ is a 2dimensional manifold of constant curvature .λ, (iii) a domain in .R4 with coordinates (.s, t, x, y) and metric .ds 2 + s 2/3 dt 2 + s 4/3 g ∗ , s = 0, where .g ∗ is Euclidean, also .λ = 0 and . f = (2/3) log(s) modulo a constant and (iv) a locally conformally flat metric. We know that an .n-dimensional Riemannian manifold . M admitting a maximal, -parameter group of conformal transformations is conformally flat. i.e. . (n+1)(n+2) 2 Therefore, it would be interesting to study a manifold admitting a .1-parameter group of conformal transformations generated by a conformal vector field. In this direction, Jauregui and Wylie proved the following result in [38]. Theorem 7.15 (Jauregui–Wylie) A gradient Ricci soliton admitting a non-homothetic conformal vector field .V that preserves the gradient .1-form .d f (i.e. .∇ V f is constant) is Einstein. In a similar vein, Sharma [39] obtained the following result. Theorem 7.16 A gradient Ricci soliton with constant scalar curvature and admitting a non-homothetic conformal vector field leaving the potential vector field invariant is Einstein. A result of Diógenes, Ribeiro and Silva [40] is Theorem 7.17 (Diógenes–Ribeiro Jr.-Silva) A complete gradient Ricci soliton carrying a non-parallel closed conformal vector field must be either locally conformally flat for .n = 3 or .n = 4, or has harmonic Weyl tensor for .n ≥ 5. A recent result of Silva Filho and Sharma [41] is the following.  Theorem 7.18 Let . M n , g, f, λ , .n ≥ 3, be a complete gradient Ricci soliton with constant scalar curvature and carrying a closed conformal vector field. Then

100

7 Ricci Solitons

 .

M n , g is either (i) isometric to Euclidean space.Rn , or (ii) isometric to a Euclidean sphere . S n , or (iii) the product . N n−k × Rk , or (iv) negatively Einstein warped product of the real line with a non-positively complete Einstein manifold. In dimension 4, the class (iv) is a quotient of a hyperbolic space form . H n .

Intrigued by the following result: Theorem 7.19 (Munteanu–Wang [42]) Let (. M, g, f ) be a complete .4-dimensional shrinking gradient Ricci soliton with bounded scalar curvature .r . Then the curvature operator. Rm has bounded norm and.|Rm| ≤ cr , on. M where.c is a positive constant. Chan, Ma and Zhang proved the following theorem in the steady case. Theorem 7.20 (Chan–Ma–Zhang [43]) Let (. M, g, f ) be a complete .4-dimensional non-Ricci flat steady gradient Ricci soliton with bounded Riemann curvature. Then there exists a positive constant .c such that .|Rm| ≤ cr , on . M where .r denotes the scalar curvature. We conclude this chapter with a brief variational aspect  of compact Ricci solitons. Perelman [4] considered the functional .F(g, f ) = M (r + |D f |2 )e− f dv defined over the space of all Riemannian metrics .g and smooth functions . f (known as dilatons)on a compact Riemannian manifold . M. Let .λ(g) = in f [F(g, f ) : f ∈ C ∞ (M), M e− f dv = 1] be the associated energy. Using the first variation formula:  δF =

.

v [−vi j (Ri j + ∇i ∇ j f ) + ( − φ)(2 f − |D f |2 + r ]e− f dv 2 M

where .δgi j = vi j , .v = g i j vi j and .δ f = φ, Perelman showed for a solution .gi j to the Ricci flow on a compact manifold that the associated energy .λ(g) is nondecreasing in .t and the monotonicity is strict unless (. M, g) is a steady gradient soliton. Similarly, for a shrinking Ricci soliton, Perelman considered the entropy func tional .W(g, f, τ ) = M [τ (r + |D f |2 ) + f − n](4π τ )−n/2 e− f dv and the associated energy.μ(g) = in f [W(g, f, τ ) : f ∈ C ∞ (M), τ > 0, (4π τ )−n/2 M e− f dv = 1], and showed for a solution.g(t),.0 ≤ t < T of the Ricci flow on a compact manifold that .μ(g(t), T − t) is nondecreasing in .t; the monotonicity is strict unless (. M, g) is a shrinking gradient Ricci soliton. A similar result was  obtained by Feldman, Ilmanen, and Ni by considering the functional .W− = M [σ (r + |D f |2 ) − ( f − n)](4π σ )−3/2 e− f dv and the associated energy .μ− (g, σ ) = in f [W− (g, f, σ ) : f ∈ C ∞ (M), (4π σ )−n/2 M e− f dv = 1]. For details, we refer to the monograph [26] by Chow et al. and Cao’s survey article [44].

References 1. Scott Russell, J.: Report on Waves. Meet. Br. Assoc. 1845, 1842–1843 2. Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)

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3. Hamilton, R.S.: Ricci flow on surfaces. Contemp. Math. (AMS) 71, 237–261 (1988) 4. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). arXiv: math.DG0211159 5. Perelman, G.: Ricci flow with surgery on three manifolds (2003). arXiv: math.DG0303109 6. Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). arXiv: math.DG0307245 7. DeTurck, D.: Deforming metrics in the direction of their Ricci tensors. J. Diff. Geom. 18, 157–162 (1983) 8. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics, 3. Science Press and Graduate Studies in Mathematics, vol. 77. American Mathematical Society (co-publication) (2006) 9. Friedan, D.: Nonlinear models in .2 +  dimensions. Ann. Phys. 163, 318–419 (1985) 10. Baird, P., Danielo, L.: Three dimensional Ricci solitons which project to surfaces. J. Reine Angew. Math. 608, 65–71 (2007) 11. Lott, J.: On the long time behaviour of type III Ricci flow solutions (2007). arXiv:math.DG/0509639 12. Stepanov, S.E., Shandra, I.G.: Geometry of infinitesimal harmonic transformations. Ann. Glob. Anal. Geom. 24, 291–299 (2003) 13. Eminenti, M., La Nave, G., Mantegazza, C.: Ricci solitons: the equation point of view. Manuscripta Math. 127, 345–367 (2008) 14. Stepanov, S.E.: From infinitesimal harmonic transformation to Ricci solitons. arXiv: math.DG1101.1665v1 15. Wylie, W.: Complete shrinking Ricci solitonshave finite fundamental group. Proc. Amer. math. Soc. 136, 1803–1806 (2008) 16. Fernández-López, M., García-Río, E.: A remark on compact Ricci solitons. Math. Ann. 340, 893–896 (2008) 17. Naber, A.: Noncompact shrinking 4-solitons with nonnegative curvature. arXiv:math.DG/0710.5579 18. Sharma, R., Balasubramanian, S., Uday Kiran, N.: Some remarks on Ricci solitons. J. Geom. 108, 1031–1037 (2017) 19. Hamilton, R.S.: The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry (Cambridge, MA), vol. 2, pp. 7–136 (1993). International Press, Cambridge, MA (1995), MR 1375255 20. Ivey, T.: Ricci solitons on compact 3-manifolds. Diff. Geom. Appl. 3, 301–307 (1993) 21. Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85, 175– 185 (2011) 22. Ni, L., Wallach, N.: On a classification of the gradient shrinking solitons. Math. Res. Lett. 15, 941–955 (2008) 23. Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow. Surv. Diff. Geom. 12, 47–112 (2008) 24. Bohm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167, 1079–1097 (2008) 25. Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013) 26. Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: Techniques and Applications, Part I: Geometric Aspects, Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Rhode Island (2007) 27. Cao, X., Wang, B., Zhang, Z.: On locally conformally flat gradient shrinking Ricci solitons. Comm. Contemp. Math. 13, 269–282 (2011) 28. Petersen, P., Wylie, W.: On the classification of gradient Ricci solitons. Geom. Topol. 14, 2277–2300 (2010) 29. Cao, H.-D.: Geometry of complete gradient shrinkiong Ricci soliotons. Adv. Lect. Math. 17, 227–246 (2011)

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30. Munteanu, O., Sesum, N.: On gradient Ricci solitons. J. Geom. Anal. 23, 539–561 (2013) 31. Fernández-Lópes, M., García-Río, E.: On gradient Ricci solitons with constant scalar curvature. Proc. Amer. Math. Soc. 144, 369–378 (2016) 32. Petersen, P., Wylie, W.: On gradient Ricci solitons with symmetry. Proc. Amer. Math. Soc. 137, 2085–2092 (2009) 33. Brozos-Vázquez, M., García-Río, E., Gavino-Fernández, S.: Locally conformally flat Lorentzian gradient Ricci solitons. J. Geom. Anal. 23, 1196–1212 (2013) 34. Cao, H.-D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach flat gradient steady Ricci solitons. Calc. Var. Partial Diff. Eqn. 49, 125–138 (2014) 35. Cao, H.-D., Chen, Q.: On Bach flat gradient shrinking Ricci solitons. Duke Math. J. 162, 1149–1169 (2013) 36. Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady solitons. Trans. Amer. Math. Soc. 364, 2377–2391 (2012) 37. Kim, Jongsu: On a classification of .4 gradient Ricci soliton with harmonic Weyl curvature. J. Geom. Anal. 27, 986–1012 (2017) 38. Jauregui, J., Wylie, W.: Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds. J. Geom. Anal. 25, 668–708 (2015) 39. Sharma, R.: Gradient Ricci solitons with a conformal vector field. J. Geom. 109, 33 (2018). https://doi.org/10.1007/s00022-018-0439-x 40. Diógenes, R., Ribeiro, E., Jr., Silva, J.: Gradient Ricci solitons admitting a closed conformal vector field. J. Math. Anal. Appl. 455, 1975–1983 (2017) 41. Silva Filho, J.F., Sharma, R.: Gradient Ricci solitons carrying a conformal vector field. arXiv: math.DG2110.14103v1 42. Munteanu, O., Wang, J.: Geometry of shrinking Ricci solitons. Compos. Math. 151, 2273–2300 (2015) 43. Chan, P-Y., Ma, Z., Zhang, Y.: Hamilton-Ivey estimates for gradient Ricci solitons (2021). arXiv:2112.11025v1.math.DG. 44. Cao, H.-D.: Recent progress on Ricci solitons. Adv. Lect. Math. 11, 1–38 (2010)

Chapter 8

Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

8.1 Complex and Almost Complex Manifolds A complex manifold is a topological space . M with a family of pairs (.Ui , ϕi ) where Ui are open sets that cover . M and .ϕi are corresponding homeomorphisms from .Ui to open subsets of .Cn , such that, whenever .Ui ∩ U j = ∅, the transition maps

.

ϕ ◦ ϕ −1 j : ϕ j (Ui ∩ U j ) → ϕi (Ui ∩ U j )

. i

are holomorphic, i.e. complex analytic..n is the complex dimension of. M and hence the real dimension of . M is.2n. The pairs (.Ui , ϕi ) are called charts and assign complex coordinate systems, and the set .{(Ui , ϕi )} is an atlas on . M. Suppose that .z α = ϕi ( p) and .w β = ϕ j ( p) is the complex coordinates of a point . p ∈ Ui ∩ U j in the charts (.Ui , ϕi ) and (.U j , ϕ j ). Then the functions .w α = u α + iv α (.α run from 1 to .n) are holomorphic in .z β = x β + i y β , i.e. .

∂u α ∂v α , = ∂ yβ ∂xβ

∂v α ∂u α = − ∂xβ ∂ yβ

where .α, β run from 1 to .n. If the union of two atlases is again an atlas, then they are said to define the same complex structure. A connected orientable complex manifold of complex dimension 1 is known as a Riemann surface. Examples of complex manifolds are the .n-dimensional complex space .Cn , the unit 2-sphere . S 2 , the 2torus .T 2 and the complex projective space .CPn (the quotient of .Cn+1 \{0} by the equivalence relation.∼ defined by.z ∼ w if.w = az for some complex number.a = 0). The multiplicative group .C∗ of nonzero complex numbers acts freely on.Cn+1 \{0} by ∗ n+1 .(c, z) ∈ C × C \{0} → cz ∈ Cn+1 \{0}. Other examples of complex manifolds 2 p+1 1 × S (Hopf [1]) and . S 2 p+1 × S 2q+1 for integers . p, q ≥ 0 (Calabi and are . S Eckmann [2]).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_8

103

104

8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

At this point, let us consider a real.2n-dimensional smooth manifold. M. An almost complex structure on . M is defined by a tensor field . J of type (1, 1) satisfying the condition . J 2 = −I . The manifold . M with a fixed almost complex structure . J is called an almost complex manifold. Turning our attention back to complex manifolds, let (.z 1 , . . . , z n ) be a local complex coordinate system on a neighborhood .U of a point . p. Let us set .z α = x α + i y α (.α = 1, . . . , n) and define a tensor field . J on .U by  .

J

∂ ∂xα



∂ = α, J ∂y



∂ ∂ yα

 =−

∂ . ∂xα

(8.1)

Obviously, . J 2 = −I , and . J does not depend onthe choice of  the complex  coordinate  ∂ ∂ ∂ 1 ∂ 1 system. In view of the transformations. ∂z α = 2 ∂ x α − i ∂ y α , ∂ z¯ α = 2 ∂ ∂x α + i ∂ ∂y α , the structure Eq. (8.1) assumes the form  .

J

∂ ∂z α



∂ = i α, J ∂z



∂ ∂ z¯ α

 = −i

∂ . ∂ z¯ α

(8.2)

Let .T pC (M) denote the complexified tangent space of an almost complex (may be complex, in particular) manifold at a point. p. Then it can be decomposed as the direct sum of the eigenspaces .T p1,0 (M) and .T p0,1 (M) of . J , corresponding to the eigenvalues .i and .−i, respectively. A complex tangent vector is of type (1, 0) or (0, 1) accordingly as it is in .T p1,0 (M) or .T p0,1 (M), respectively. A complex tangent vector of type (1, 0) (resp. type (0, 1)) can be written as . X − i J X (resp. . X − i J X ) for some real tangent vector . X . An almost complex structure is said to be integrable if it is induced by a complex structure. This integrability condition can be expressed by either of the following criteria: (i) the Lie bracket of any two complex tangent vectors of type (1, 0) is again a complex tangent vector of type (1, 0); (ii) The torsion tensor (known as the Nijenhuis tensor) . N defined by .

N (X, Y ) = [J X, J Y ] − [X, Y ] − J [J X, Y ] − J [X, J Y ]

vanishes by Newlander–Nirenberg theorem [3]. Among all spheres, only . S 2 and 6 . S possess almost complex structures. Since the underlying differentiable manifold of .CP1 is . S 2 , . S 2 admits a complex structure. It is a famous open problem whether . S 6 admits a complex structure. . S 6 does have a non-integrable almost complex structure, as explained at the end of this section. A vector field .V on an almost complex manifold (. M, J ) is said to be an infinitesimal automorphism or almost analytic vector field of the almost complex structure . J if .L V J = 0 (see Yano [4]). A complex vector field . Z of type (1, 0) on a complex manifold is holomorphic if . Z f is holomorphic for every locally defined holomorphic function . f . . Z is holomorphic if . Z = f α ∂z∂ α and . f α are holomorphic. If .V is an infinitesimal automorphism of . J on a complex manifold . M with almost complex structure . J , then .V − i J V is a holomorphic vector field.

8.1 Complex and Almost Complex Manifolds

105

A Hermitian metric on an almost complex manifold (. M, J ) is a Riemannian metric .g on . M, satisfying the condition .

g(J X, J Y ) = g(X, Y )

for any . X, Y ∈ X(M). Then (. M, g, J ) is called an almost Hermitian manifold, and a complex manifold (. M, J ) with a Hermitian metric .g is called a Hermitian manifold. For an almost Hermitian manifold, .g(X, J Y ) = −g(J X, Y ), and one defines the fundamental 2-form .ω by .ω(X, Y ) = g(X, J Y ). .ω is . J -invariant, i.e. .ω(J X, J Y ) = ω(X, Y ). Also, the .2n-form .ω ∧ · · · ∧ ω (.n-times) on a real .2ndimensional almost Hermitian manifold . M is nonzero at each point, and hence . M is orientable. An almost Hermitian manifold satisfying .∇ J = 0 (.∇ denoting the Riemannian connection of.g) is called a Kähler manifold, and hence its almost complex structure is integrable. A Kähler manifold has the following properties: .

R(X, Y )J Z = J R(X, Y )Z ,

(8.3)

.

Ric(J X, J Y ) = Ric(X, Y ),

(8.4)

(∇ X Ric)(Y, Z ) = (∇Y Ric)(Z , X ) + (∇ J Z Ric)(J Y, X ).

(8.5)

.

A holomorphic sectional curvature of a Kähler manifold (. M, g, J ) at a point p is the sectional curvature of . M with respect to a plane section (holomorphic section) spanned by a unit vector . X at . p and . J X , and is given by .K( p) = g(R(X, J X )J X, X ). We have the following Kählerian analogue of Schur’s theorem “Let . M be a connected Kähler manifold of complex dimension .n > 1. If the holomorphic sectional curvature .K( p) does not depend on the choice of the holomorphic section at . p, and depends only on . p, then .K( p) is constant, say .c, on . M”. The Kähler manifold . M is of constant holomorphic sectional curvature .c if and only if

.

c [g(Y, Z )X − g(X, Z )Y + g(J Y, Z )J X 4 − g(J X, Z )J Y + 2g(X, J Y ) J Z ].

R(X, Y )Z =

(8.6)

A Kähler manifold of constant holomorphic sectional curvature is called a complex space form, and is Einstein. It is well known that a simply connected complete Kähler manifold of constant holomorphic sectional curvature .c can be identified with a complex projective space .C P n , the open unit ball . D n in .C n+1 or the complex Euclidean space .C n accordingly as .c > 0, c < 0 or .c = 0, respectively. It is also worth mentioning that a conformally flat Kähler manifold of real dimension .> 2 is flat. For details we refer to [4]. The Ricci form .ρ of a Kähler manifold is defined by ρ(X, Y ) = Ric(X, J Y ).

.

(8.7)

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

It is known [4] that .ρ is closed, and defines the first Chern class .c1 (M) = [ρ/2π ]. Complex manifolds with a Ricci-flat Kähler metric are called Calabi–Yau manifolds. In [5], Yau proved the existence of a Ricci-flat metric on any compact complex manifold . M with .c1 (M) = 0 and which admits a Kähler metric. The Calabi–Yau manifolds have their application in physics in superstring theory which is based on a 10-dimensional product manifold .V4 × M, where .V4 is the ordinary space-time and . M is a 6-dimensional Kähler manifold which is at least approximately Ricci-flat. Candelas et al. [6] have shown that for dimensional supersymmetry to be unbroken, . M must be Kähler. Also, Lebrun [7] constructed a 1-parameter family of complete non-flat Ricci-flat Kähler metrics, which are isometric (up to dilation and rescaling) to the Taub–NUT metric (Hawking [8]). An almost Hermitian manifold (. M, g, J ), whose fundamental 2-form .ω is closed, is called an almost Kähler manifold. For an almost Kähler manifold, the following formula holds: .2g((∇ X J )Y, Z ) = g(J X, N (Y, Z )). The almost Kähler manifold (. M, g, J ) is a symplectic manifold with symplectic form .ω (.dω = 0, ω ∧ · · · ∧ ω = 0 on . M). By a symplectic manifold, we mean an even (.2n)-dimensional smooth manifold. M with a global 2-form.ω (called symplectic form) which is closed and of maximal rank (.ωn = 0, where the power is defined by Wedge product). It may be noted that a symplectic manifold may not admit a Kähler metric, for example the Kodaira–Thurston manifold which is a compact symplectic manifold with no Kähler structure (Thurston [9]). Finally, we turn our attention to another important class of almost Hermitian manifolds. The Cayley multiplication can be used to define a vector cross product on the purely imaginary octonions .R7 by the formulas u×v =

.

1 (uv − vu), 2

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

.

The Cayley multiplication on the octonions .O is thus given by (r + u)(s + v) = r s − (u, v) + r v + su + (u × v)

.

for .r, s ∈ Re(O) and .u, v ∈ I m(O). An almost complex structure . J on the unit 6-sphere . S 6 = x ∈ R7 : ||x|| = 1 is defined by . J u = x × u where .x ∈ S 6 and 6 6 7 .u ∈ Tx S . The Riemannian metric . g on . S induced from .R is a Hermitian metric 6 with respect to . J , and its Riemannian connection .∇ of (. S , g) satisfies the equation .(∇ X J )Y + (∇Y J )X = 0. An almost Hermitian manifold satisfying this equation is known as a nearly Kähler manifold. Thus, . S 6 is nearly Kähler, but not Kähler. For a

8.2 Contact Manifolds

107

nearly Kähler manifold, .4J (∇ X J )Y + N (X, Y ) = 0. A 4-dimensional nearly Kähler manifold is Kähler. A result of Gray [10] says that a 6-dimensional non-Kähler nearly Kähler manifold is positively Einstein. For details, we refer to [4, 10], Yano and Kon [11] and Chen [12].

8.2 Contact Manifolds A (.2n + 1)-dimensional smooth manifold is said to be contact if it has a global 1-form .η such that .η ∧ (dη)n = 0 on . M. For a contact 1-form .η there exists a unique vector field .ξ called the Reeb vector field, such that .dη(ξ, X ) .= 0 and .η(ξ ) = 1. This implies that the contact distribution . D defined by .η = 0 is far from being integrable, and the maximum dimension of an integral submanifold of . D is .n. In [13], Martinet showed that every compact orientable 3-manifold carries a contact structure. About each point of a contact manifold, there exist local , x n , y 1 , . . . , y n , z) called Darboux coordinates, with respect coordinates (.x 1 , . . . n ∂ . Examples of a contact manifold are (i) y i d x i and .ξ = ∂z to which .η = dz − i=1 n 2n+1 1 n 1 .R with coordinates (.x , . . . , x , y , . . . , y n , z) and .η = dz − i=1 y i d x i , (ii) 3 3 3-dimensional torus .T = R /  (where . is the group of translations of the coordinates by .2π ) and .η = (sin y)d x + (cos y)dz, ξ = (sin y)∂x + (cos y)∂z , (iii) an odd dimensional sphere . S 2n+1 as a hypersurface of the complex Euclidean space .C n+1 , (iv) tangent sphere bundle of a Riemannian manifold and (v) Lie groups . SU (2) and . S L(2, R). The theory of contact structures has its roots in differential equations, optics, phase space of a dynamical system and thermodynamics (for details, we refer to Arnold [14], MacLane [15] and Nazaikinskii, Shatalov and Sternin et al. [16]). There is also the notion of a contact structure in the wider sense, often called simply a contact structure, and is defined as a hyperplane field . D defined locally by a contact form. A connected and simply connected contact manifold in the wider sense is a contact manifold in the restricted sense. Details can be found in the standard monograph of Blair [17]. In the present book, we will follow the definition of a contact structure in the restricted sense, as described in the previous paragraph. Given a (.2n + 1)-dimensional contact manifold (. M, η) with the Reeb vector field .ξ , a Riemannian metric .g on (. M, η) is said to be an associated metric if there exists a (1, 1)-tensor field .ϕ (called fundamental collineation) such that dη(X, Y ) = g(X, ϕY ), η(X ) = g(X, ξ ), ϕ 2 = −I + η ⊗ ξ.

.

(8.8)

The structure (.ϕ, η, ξ, g) is called a contact metric structure on . M. For a given contact form .η, the set of all associated contact metrics is infinite dimensional. Following [17], we recall two self-adjoint operators .h = 21 Lξ ϕ and . = R(., ξ )ξ . The tensors.h,.hϕ are trace-free and.hϕ = −ϕh. We also have the following formulas for a contact metric manifold:

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

∇ X ξ = −ϕ X − ϕh X,

(8.9)

− ϕ ϕ = −2(h 2 + ϕ 2 ),

(8.10)

∇ξ h = ϕ − ϕ − ϕh 2 ,

(8.11)

Tr = Ric(ξ, ξ ) = 2n − T r h 2 .

(8.12)

.

.

.

.

Let us also recall that a contact structure is regular if every point of . M has a neighborhood such that any integral curve of .ξ passing through the neighborhood passes through only once. For details we refer to [17]. A vector field .V on a contact metric manifold . M is said to be an infinitesimal contact transformation if.LV η = σ η for some smooth function.σ on. M. It is strictly so, if .σ = 0. A vector field .V is said to be an infinitesimal automorphism of the contact metric structure if it leaves all the structure tensors .η, ξ, g, ϕ invariant (see Tanno [18]). A contact metric structure is said to be . K -contact if .ξ is Killing with respect to .g, equivalently, .h = 0. A compact regular contact manifold . M 2n+1 carries a . K contact structure defined in terms of the almost Kähler structure of the base manifold (Example 4.5.4 in [17]). The contact metric structure on . M is said to be Sasakian if the almost Kähler structure on the cone manifold (. M × R + , r 2 g + dr 2 ) over . M is Kähler (Boyer and Galicki [19]). Sasakian manifolds are . K -contact and . K -contact 3-manifolds are Sasakian. For a Sasakian manifold, (∇ X ϕ)Y = g(X, Y )ξ − η(Y )X,

(8.13)

R(X, Y )ξ = η(Y )X − η(X )Y, Qξ = 2nξ.

(8.14)

.

.

An example of a Sasakian manifold is .R2n+1 with contact form .η = 21 (dz − ∂ n and the Riemannian metric .g = η ⊗ η + y i d x i ), Reeb vector field .ξ = 2 ∂z i=1 1 n i 2 i 2  ((d x ) + (dy ) ). For this example, the sectional curvature of any plane 4 i=1 section containing .ξ is equal to 1. Also, the sectional curvature of a plane section spanned by a vector field . X ∈ D and .ϕ X is equal to .−3 and so this is an example of a Sasakian space form. Another example of a Sasakian manifold is the (.2n + 1)dimensional Heisenberg group described later in contact metrics as Ricci solitons. For a contact metric manifold (. M, η, ξ, g), we consider the connection .∇ D induced on the contact subbundle . D (.η = 0) from the Riemannian connection .∇ of .g, defined by .∇ XD Y = (∇ X Y )h , where . X is an arbitrary vector field on . M, .Y is an arbitrary vector field tangent to . D and .h denotes the projection onto . D. Propositions 3.2 and 3.5 of Boyer and Galicki [20] imply that the contact metric manifold is Sasakian if and only if .∇ D J = 0, where . J is the restriction of .ϕ to the contact subbundle . D. For Sasakian case, (. D, J, dη) defines a Kähler metric on . D, with the transverse Kähler metric .g T related to the Sasakian metric .g as .g = g T + η ⊗ η.

8.2 Contact Manifolds

109

One finds by a direct computation that the transverse Ricci tensor .RicT of .g T is given by T . Ric (X, Y ) = Ric(X, Y ) + 2g(X, Y ) for arbitrary vector fields . X, Y in . D. The Ricci form .ρ and transverse Ricci form .ρ T are defined by ρ(X, Y ) = Ric(X, ϕY ),

.

ρ T (X, Y ) = RicT (X, ϕY )

for . X, Y ∈ D. In case .ρ T = 0, the Sasakian structure is said to be null (transverse Calabi–Yau) in which case we can easily verify that .Ric = −2g + 2(n + 1)η ⊗ η. We refer to Boyer, Galicki and Matzeu [21] for details. At this point, we would like to point out the fact that a contact structure has an underlying structure called an almost contact structure. A (.2n + 1-dimensional smooth manifold . M is called an almost contact structure if it admits a 1-form .η, a vector field .ξ and a (1, 1)-tensor field .ϕ satisfying ϕ 2 = −I + η ⊗ ξ, η(ξ ) = 1.

.

Then .ϕξ = 0, .η ◦ ϕ = 0, .rank(ϕ) = 2n. Also, the structural group of the tangent bundle of . M reduces to .U(n) × 1. An almost contact manifold is called an almost contact metric manifold if it has a Riemannian metric .g satisfying .

g(ϕ X, ϕY ) = g(X, Y ) − η(X )η(Y ).

Another way of looking at a Sasakian manifold is the following. Consider the product M × R of an almost contact manifold . M with .R. Denoting a vector field on . M × R by (. X, f dtd ), where . X is tangent to . M, .t the coordinate on .R and . f a smooth function on . M × R, one defines an almost complex structure . J on . M × R by . J (X, f dtd ) = (ϕ X − f ξ, η(X ) dtd ). Then the almost contact structure on . M is said to be normal if . J is integrable. This integrability condition is equivalent to

.

[ϕ, ϕ] + 2(dη) ⊗ ξ = 0,

.

where .[ϕ, ϕ] denotes the Nijenhuis tensor defined by .[ϕ, ϕ](X, Y ) = ϕ 2 [X, Y ] + [ϕ X, ϕY ] − ϕ[ϕ X, Y ] − ϕ[X, ϕY ]. It is also worth mentioning that a contact metric structure is normal if and only if it is Sasakian. A Sasakian manifold is also defined as an almost contact metric manifold satisfying (∇ X ϕ)Y = g(X, Y ) − η(Y )X.

.

A contact metric manifold . M is said to be .η-Einstein, if the Ricci tensor can be written as . Ric(X, Y ) = αg(X, Y ) + βη(X )η(Y ) (8.15)

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

for some smooth functions .α and .β on . M. It is well-known (Yano and Kon [11]) that α and .β are constant if . M is . K -contact, and has dimension greater than 3. It may be noted that . K -contact, Sasakian and .η-Einstein manifolds belong to a wider class of contact metric manifolds called . H -contact manifolds studied by Perrone [22] and defined by the condition that a contact metric manifold whose Reeb vector field is an eigenvector of the Ricci tensor everywhere. . H -contact manifolds were also studied by Blair and Sharma in [23] in order to obtain a generalization of Myers’ theorem on a contact manifold. Given a contact metric structure (.η, ξ, g, ϕ), let .η¯ = aη, ξ¯ = a1 ξ, ϕ¯ = ϕ, g¯ = ¯ ξ¯ , ϕ, ¯ g) ¯ is again a contact ag + a(a − 1)η ⊗ η for a positive constant .a. Then (.η, metric structure. Such a change of structure is called a . D-homothetic deformation (see Tanno [24]), and preserves many basic properties like being . K -contact (in particular, Sasakian). It is straightforward to verify that, under a . D-homothetic deformation, a . K -contact .η-Einstein manifold transforms to a . K -contact .η-Einstein ¯ We remark here that the particuand .β¯ = 2n − α. manifold such that .α¯ = α+2−2a a lar value: .α = −2 remains fixed under a . D-homothetic deformation of a . K -contact structure (it is easy to verify this), and as .α + β = 2n [which follows from (8.15 with . X = Y = ξ and the second equation of (8.14))], .β also remains fixed. This fact prompted the following definition by Ghosh and Sharma in [25].

.

Definition 8.1 A . K -contact .η-Einstein manifold with .α = −2 is said to be . Dhomothetically fixed. At this point, we recall the following important result due to Boyer and Galicki [20] (that was proved earlier by a different method by Morimoto [26]): A compact . K -contact Einstein (more generally .η-Einstein with .α ≥ −2) manifold is Sasakian. As the Einstein constant of a . K -contact Einstein manifold is .2n, the compactness can be weakened to completeness in the case .α > −2 of the aforementioned result. It is known that the tangent sphere bundle . E n+1 × S n of a flat Riemannian manifold admits a contact metric structure satisfying the condition : . R(X, Y )ξ = 0 (this is supported by Theorem 7.5 of [17]). Applying a . D-homothetic deformation to this metric structure, Blair, Koufogiorgos and Papantoniou [27] obtained a generalization of a Sasakian manifold and . E n+1 × S n as a contact metric manifold satisfying the nullity condition: .

R(X, Y )ξ = k[η(Y )X − η(X )Y ] + μ[η(Y )h X − η(X )hY ],

(8.16)

for some real numbers .k and .μ. Such a manifold is called a (.k, μ)-contact manifold. This class of manifolds is preserved under a . D-homothetic deformation defined earlier, and includes Sasakian manifolds (for which .k = 1 and .h = 0) and the trivial sphere bundle . S n (4) × E n+1 (for which .k = μ = 0). A key factor in the introduction of (.k, μ)-contact manifolds is that the set of all contact metric manifolds whose Reeb vectors lie in the (.k)-nullity distribution (i.e. those satisfying . R(X, Y )ξ = k[η(Y )X − η(X )Y ], for a real constant .k) is not preserved under a . D-homothetic deformation. For .(k, μ)-contact manifolds, we know [27] that

8.2 Contact Manifolds

111

Ric(X, ξ ) = 2nkg(X, ξ )

(8.17)

h 2 = (1 − k)(I − η ⊗ ξ ), |h|2 = 2n(1 − k).

(8.18)

.

.

This shows that .k ≤ 1, and equality holds when . M is Sasakian. For the non-Sasakian case, i.e. .k < 1, the (.k, μ)-nullity condition determines the curvature of . M completely. Using this fact, Boeckx [28] proved that a non-Sasakian (.k, μ)-contact manifold is locally homogeneous and hence analytic. For non-Sasakian (.k, μ)-contact manifolds, the following formulas hold [27]: Q X = [2(n − 1) − nμ]X + [2(n − 1) + μ]h X +[2(1 − n) + n(2k + μ)]η(X )ξ.

(8.19)

(∇ X h)Y = ((1 − k)g(X, ϕY ) − g(X, ϕhY ))ξ − η(Y )((1 − k)ϕ X + ϕh X ) − μη(X )ϕhY.

(8.20)

(∇ X ϕ)Y = g(X + h X, Y )ξ − η(Y )(X + h X ),

(8.21)

where . X, Y denote arbitrary vector fields on . M. Moreover, the scalar curvature is constant and given by .r = 2n[2(n − 1) + k − nμ]. (8.22) The standard contact metric structure on the tangent sphere bundle of a Riemannian manifold . M is a (.k, μ)–manifold if and only if the base manifold . M is of constant curvature .c such that .k = c(2 − c), μ = −2c. We now state some important results on contact metric manifolds as follows (cited in [17]). An example of a (.k, μ)-contact manifold was constructed in [28] by defining a specific Lie algebra of an orthogonal basis. Now, we mention some important results on contact metric manifolds that are cited in [17]. Blair showed that there exist no flat Riemannian metrics associated with a contact structure in dimension .> 3; in contrast the 3-dimensional torus admits a flat contact metric. This result was further generalized by Olszak who showed that there are no contact metric manifolds of constant curvature in dimensions .> 3 unless the constant is 1 in which case the manifold is Sasakian. This raised an open question “Does there exist a 3-dimensional contact metric manifolds of constant curvature other than 0 or 1?” That these are the only possibilities was answered more generally by Blair and Sharma who proved that a 3-dimensional locally symmetric contact metric manifold is of constant curvature 0 or 1. This result generalized in dimension 3, an earlier result of Tanno (which in turn was a generalization of the corresponding result of Okumura on Sasakian manifolds) on . K -contact manifolds, who proved that a locally symmetric . K -contact manifold is of constant curvature 1.

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

Subsequently, Boeckx and Cho closed this topic by proving that a locally symmetric contact metric manifold . M 2n+1 is locally isometric to . S 2n+1 or . E n+1 × S n (4). On the other hand, Okumura [37] proved that a conformally flat Sasakian manifold of dimension .≥ 5 is of constant curvature 1; this result was extended by Tanno [53] to . K -contact case and for dimension .≥ 3. In 2004, Gouli-Andreou and Tsolakidou, and independently in 2005, Bang and Blair proved that a conformally flat contact metric manifold whose Reeb vector field is everywhere an eigenvector of the Ricci operator is of constant curvature. A plane section in the tangent space .T p M 2n+1 of a contact metric manifold . M 2n+1 is called a .ϕ-section if there exists a unit vector . X ∈ T p M 2n+1 orthogonal to .ξ such that the plane section is spanned by . X and .ϕ X . The sectional curvature with respect to a.ϕ-section is called a.ϕ-sectional curvature. It is known that if a Sasakian manifold has a .ϕ-sectional curvature .c which does not depend on the .ϕ-section at each point, then.c is constant on. M, and then. M is called a Sasakian space form and its curvature tensor is given by c−1 c+4 (g(Y, Z )X − g(X, Z )Y ) + [η(X )η(Z )Y 4 4 − η(Y )η(Z )X + g(X, Z )η(Y )ξ − g(Y, Z )η(X )ξ + g(Z , ϕY )ϕ X − g(Z , ϕ X )ϕY + 2g(X, ϕY )ϕ Z ].

R(X, Y )Z =

(8.23)

Contracting it at . X gives the Ricci tensor .

Ric =

n(c + 3) + c − 1 (n + 1)(c − 1) g− η⊗η 2 2

(8.24)

and hence it is .η-Einstein.

8.3 Conformal Vector Fields on Almost Hermitian Manifolds First, we mention some classical results of Ako, Goldberg, Lichnerowicz, Sawaki, Tachibana and Yano, cited in [4] as follows: • An analytic divergence free vector field on a compact Kähler manifold is Killing. A Killing vector field in a compact Kaehler manifold is analytic. • If .V is a Killing vector field on a compact Kähler manifold, then . J V is analytic and closed. • A Killing vector field on a compact almost Kähler manifold is almost analytic. • A conformal vector field .V on a compact Kähler manifold . M is Killing and hence analytic. If. M is almost Kähler, then.V is Killing provided.dim(M) > 2 and almost analytic if .dim(M) ≥ 2.

8.3 Conformal Vector Fields on Almost Hermitian Manifolds

113

• If, on a compact nearly Kähler manifold, a conformal vector field is almost analytic, then it is Killing. In the above-mentioned results, by an almost analytic vector field on an almost Hermitian manifold, we mean a vector field .V that satisfies .LV J = 0. For the noncompact case, we state the following result of Deshmukh [29]. Theorem 8.1 (Deshmukh) If .V is an analytic conformal vector field on a .2ndimensional (.n > 1) Kähler–Einstein manifold . M, then either .V is Killing, or . M is Ricci flat. For a closed conformal vector field on a Kähler (not necessarily compact) manifold, we have the following result (Goldberg [30]). Theorem G. A closed conformal vector field .V on a Kähler manifold. M is homothetic and analytic. Hence, if . M is locally non-flat and complete, then .V generates a 1-parameter group of automorphisms. In order to obtain a generalization of the aforementioned result, Sharma [31] considered a special conformal vector field on an almost Hermitian manifold. To do so, he considered the simplest complex manifold .C (i.e. . E 2 ) with the Euclidean metric . g and the complex structure . J defined by . J (∂/∂ x) = ∂/∂ y, J (∂/∂ y) = −∂/∂ x, and observed that, for a vector field .V on .C, the following three conditions are equivalent to one another: (i) .V is conformal (i.e. .LV g = 2σ g for a smooth function .σ on .C), (ii) . V is analytic, i.e. the components of . V satisfy the Cauchy–Riemann equations and (iii) .∇ X V = a X + b J X . Each of the three conditions can be expressed in terms of the coordinates .x, y as the following set of first order partial differential equations: .∂x V 1 = ∂ y V 2 (= σ ), ∂x V 2 = −∂ y V 1 (= b). However, the conditions .(i), (ii) and (iii) are not equivalent to one another, on a general almost Hermitian manifold. Condition (iii) motivated the following generalization of a closed conformal vector field in [31]. Definition 8.2 A vector field .V on an almost Hermitian manifold (. M, g, J ) is said to be a holomorphically planar conformal vector (abbreviated HPCV) field if ∇X V = σ X + b J X

.

(8.25)

where . X is an arbitrary vector field and .σ, b are smooth functions on . M. HPCV fields were also studied independently by Deshmukh [32]. Obviously, an HPCV field is conformal, and its integral curves are holomorphically planar (see [4], p. 258). It is easy to verify that, for an HPCV field .V on an almost Kähler manifold, .L V J = ∇ V J . Hence, . V is almost analytic (i.e. .L V J = 0) iff . J is parallel along . V . This fact implies that an HPCV field on a Kähler manifold is analytic. The following result is easy to verify “For an HPCV field .V on an almost Kähler manifold . M of dimension .> 2, the function .b is constant.” Noting that a closed conformal vector field is a special case (.b = 0) of an HPCV field, the following generalization of Theorem G was obtained in [31].

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Theorem 8.2 An HPCV field .V on a Kähler manifold . M of dimension .> 2 is homothetic and analytic. If, in addition, . M is complete and .V is not parallel, then . M is isometric to the complex Euclidean space. Its proof requires showing basically that .dω = −2σ  (where .ω is the 1-form metrically equivalent to . J V ) if . M is Kähler, hence by operating the last equation by .d we find that .σ is constant if .dim.M > 2, i.e. .V is homothetic. That .V is analytic was indicated earlier. Also, a routine calculation shows that .∇∇|V |2 = (σ 2 + b2 )g, and hence by a Tashiro’s theorem [33] and the non-parallelism of .V we obtain the second conclusion. We are now going to state and prove the following result that provides an example of an HPCV field on the unit 2-sphere (which is a Kähler manifold with its standard metric). Theorem 8.3 An HPCV field on the unit sphere . S 2 can be represented as .−(Dσ + J Db), where .σ, b are eigenfunctions of the Laplacian corresponding to the first eigenvalue .2. Proof Using Eq. (8.25), we have .

R(X, Y )V = (X σ )Y − (Y σ )X + (X b)J Y − (Y b)J X

where . R denotes the curvature tensor. Contracting the above equation with respect to . X , we get . Ric(Y, V ) = −Y σ − g(J Db, Y ). But .Ric = g on . S 2 . Therefore we obtain .

V = −(Dσ + J Db).

Use of this in (8.25) shows ∇ X Dσ − J ∇ X Db = −(σ X + b J X ).

.

Contracting it at . X gives .−σ = 2σ , i.e. .σ is an eigenfunction of the Laplacian .− corresponding to the first eigenvalue 2. Consequently, by a result of Tanno [34], we conclude that .σ, b satisfy Obata’s equation: ∇ X Dσ = −σ X, ∇ X Db = −bX.

.

One may note that .σ and .b are any two of the 3 linearly independent eigenfunctions corresponding to eigenvalue 2. Next we consider the flat.Cn with coordinates.(x i , y i ) where.i = 1, 2, 3 . . . , n and Kaehler structure. J defined by. J (∂/∂ x i ) = ∂/∂ y i , J (∂/∂ y i ) = −∂/∂ xi. Solving the HPCV Eq. (8.25) and noting that .σ, b are constants (as .Cn is Kaehler), we can show

8.3 Conformal Vector Fields on Almost Hermitian Manifolds

115

that .V = (σ I + b J )x + c where .x is the position vector of a point .(x i , y i ) and .c is a constant vector field. In the passing, we note that the Lie bracket of two HPCV fields on a Kähler manifold is a parallel vector field (easy to verify). Hence, the set of all HPCV fields on a Kähler manifold . M forms a sub-algebra of the conformal algebra on . M. We now state the following results (proofs are straightforward): • If a compact almost Kähler manifold. M admits an HPCV field.V , then.V is parallel. • Let an almost Kähler manifold . M (.dim.M > 2) admit a non-closed HPCV field . V . Then . V is homothetic and almost analytic. • Let .V be a nonzero non-closed non-Killing HPCV field on an almost Kähler manifold . M with harmonic Weyl conformal tensor, i.e. .div ·W = 0. Then . M is Ricci-flat. Now let us state and prove the result “An HPCV field on the nearly Kähler . S 6 is closed”. To prove it, we use (8.25), nearly Kaehler condition.(∇ X J )Y + (∇Y J )X = 0 and that . S 6 has constant curvature 1, so as to obtain < X, V + Dσ > Y − < Y, V + Dσ > X = (Y b)J X − (X b)J Y − 2b(∇ X J )Y.

(8.26)

Contracting this with respect to .Y yields .

V = −(1/5)(5Dσ + J Db).

(8.27)

Differentiating it and using (8.25) gives .

− (1/5)[5∇ X Dσ + (∇ X J )Db + J ∇ X Db] = σ X + b J X.

Taking its inner product with .Y , symmetrizing with respect to . X, Y and factoring out Y yields .∇ X Dσ = −σ X + (1/10)(∇ J X Db − J ∇ X Db).

.

We now contract this equation with respect to a local orthonormal . J -adapted frame (.ei , J ei ), .i = 1, 2, 3, and get .−σ = 6σ , i.e. .σ is the eigenfunction of the Laplacian 6 .− on . S with first eigenvalue 6. This implies (p. 175 of Tanno [34]) ∇ X Dσ = −σ X.

.

(8.28)

The use of (8.27) in (8.26) and taking inner product with .Y gives .

< Y, J Db >< X, Y > − < X, J Db > |Y |2 = 5(Y b) < J X, Y > .

Let us refer to the above-mentioned local . J -adapted frame and set .Y = ei , X = J ei in the above equation (leaving.i unsummed). This provides.ei b = 0. Similarly, setting

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

Y = J ei , X = ei in the above equation provides .(J ei )b = 0. Hence .b is constant, which reduces (8.27) to .V = −Dσ . Differentiating it and using (8.28) shows that .∇ X V = σ X , i.e. . V is closed. Actually, this result can be generalized on a 6-dimensional complete non-Kähler nearly Kähler manifold . M which is positively Einstein (as indicated earlier in this Sect. 8.1). To see this, we Lie-differentiate .Ric = nr g along an HPCV .V and use the equations .LV Ric = (2 − n)∇∇σ − 9σ )g and .LV r = −2r σ − 2(n − 1)σ (mentioned in Chap. 3), and thus get .

∇∇σ = −

.

r g n(n − 1)

and hence, by Obata’s theorem, . M is isometric to the round sphere . S 6 . Hence, by our previous result, .V is closed. As pointed out earlier, a strictly nearly Kähler 6-manifold. M is positively Einstein, the preceding theorem can be easily generalized for complete . M, first by taking the Lie derivative of .Ric = nr g along .V using the formulas for the Lie derivatives of .Ric and .r (as given in Chap. 3), and then using Obata’s theorem to show that . M is isometric to the round sphere.

8.4 Conformal Vector Fields on Contact Metric Manifolds The odd-dimensional sphere . S 2n+1 carries contact metric (specifically, Sasakian) structures and admits conformal vector fields. This motivates one to study and classify contact metric manifolds with conformal vector fields. In this direction, we recall the following result of [18]. Theorem 8.4 (Tanno) If a conformal vector field on a contact metric manifold. M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of . M. This result was generalized in [35] as the following result. Theorem 8.5 (Sharma-Vrancken) If a conformal vector field on a contact metric manifold . M is an infinitesimal contact transformation, then it is an infinitesimal automorphism of . M. At this point, we recall from [33] that a conformal vector field (.LV g = σ g) on a Riemannian manifold (. M, g) is said to be an infinitesimal special concircular transformation if the associated conformal scale function .σ satisfies the equation ∇∇σ = (−aσ + b)g

.

for real constants .a and .b. Let us also recall the following result [36].

8.4 Conformal Vector Fields on Contact Metric Manifolds

117

Theorem 8.6 (Sharma) If a conformal vector field .V on a contact metric manifold is pointwise either orthogonal to, or pointwise collinear with the Reeb vector field, then .V is necessarily Killing. We are now ready to recall the following important result from [37]. Theorem 8.7 (Okumura) Let. M be a Sasakian manifold of dimension.> 3, admitting a non-Killing conformal vector field .V . Then .V is special concircular. If, in addition, . M is complete and connected, then it is isometric to a unit sphere. This result shows that the existence of a non-Killing conformal vector field places a severe restriction on a Sasakian manifold of dimension.> 3, and its proof consists in taking the Lie derivative of the Sasakian property (8.14) along with the formulas for the Lie derivatives of curvature quantities along a conformal vector field (mentioned in Chap. 8), and also Obata’s theorem. It also shows that .V differs from . Dσ by a Killing vector field. For the 3-dimensional case, we note the following result [38]. Theorem 8.8 (Sharma-Blair) If a 3-dimensional Sasakian manifold . M of constant scalar curvature admits a non-Killing conformal vector field .V , then . M has constant curvature 1, and .V is special concircular. Hence, in addition, if . M is complete, then it is isometric to a unit sphere. This result was generalized by Sharma [39] replacing the constancy of the scalar curvature with the condition that .V leaves the scalar curvature invariant. The abovementioned results motivate one to examine the impact of the existence of a conformal vector field more generally on (.k, μ)-contact manifolds. The special case .μ = 0 was taken up in [38] where the following result was obtained. Theorem 8.9 (Sharma-Blair) Let . M 2n+1 be a (.k, 0)-contact manifold admitting a conformal vector field .V . For .n > 1, . M is either Sasakian or .V is Killing. In the second case, .V is an infinitesimal automorphism of the contact metric structure except when .k = 0. Further, for .k = 0, a Killing vector field orthogonal to .ξ cannot be an infinitesimal automorphism of the contact metric structure. For .n = 1, . M is either flat or Sasakian or .V is an infinitesimal automorphism of the contact metric structure. Generalizing this result on (.k, μ)-contact manifold, Sharma and Vrancken [35] established the following result. Theorem 8.10 (Sharma-Vrancken) Let a (.k, μ)-contact manifold . M 2n+1 admit a non-Killing conformal vector field .V . If . M is 3-dimensional and non-Sasakian, then it is locally flat. For .dim(M) > 3, (i) . M is Sasakian and .V is concircular, in which case if . M is complete then it is isometric to a unit sphere, or (ii) .μ = 1, k = −n − 1. In addition, if . M is compact, then it is necessarily isometric to the unit sphere. Corollary 8.1 Let .T1 M be the unit tangent bundle over a Riemannian manifold . M (.dim(M) > 2) of constant curvature .c and .g be the standard contact metric on .T1 M. Then a conformal vector field on (.T1 M, g) is Killing.

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As the class of (.k, μ)-contact manifolds does not include . K -contact manifolds, it would be interesting to examine . K -contact manifolds admitting a conformal vector field. However, it seems to be a formidable task and so it is natural to impose some extra assumption. Proceeding in this direction, Sharma [40] assumed the .η-Einstein condition and obtained the following classification result. Theorem 8.11 If a complete connected .η-Einstein . K -contact manifold . M of dimension .> 3 admits a conformal vector field .V , then either (i) . M is isometric to a unit sphere, or (ii) .V is an infinitesimal automorphism of the contact metric structure on . M. This provides the following characterization of a unit odd-dimensional sphere “Among all complete simply connected .η-Einstein . K -contact manifolds of dimension .> 3, only the unit sphere admits a non-isometric conformal vector field”. It is also known that a homothetic vector field on a . K -contact manifold is Killing [39].

8.5 Kähler–Ricci Solitons When the underlying manifold is a complex manifold, we have the following notion of a Kähler–Ricci soliton. Definition 8.3 A Kähler manifold (. M, g, J ) is said to be a Kähler–Ricci soliton if LV g + 2 Ric = 2λg,

.

(8.29)

where .λ is a real constant and .V is a real vector field which is an infinitesimal automorphism of the complex structure tensor . J , i.e. LV J = 0.

.

(8.30)

The reason behind the requirement (8.30) is that a solution of Ricci flow that starts with a Kähler metric on a complex manifold remains Kähler with respect to the same complex structure. The condition (8.30) means .LV (J X ) = J LV X which can be expressed, using .∇ J = 0, as ∇J X V = J ∇X V

.

(8.31)

for any . X ∈ X(M). Since .g(∇ X V, Y ) = 21 (LV g)(X, Y ) + (dv)(X, Y ) for arbitrary . X, Y ∈ X(M), using (8.29) and (8.31) we can show that (the reader may verify) the condition (8.30) is equivalent to (dv)(X, J Y ) = (dv)(Y, J X )

.

8.5 Kähler–Ricci Solitons

119

where .v is the metric dual of .V . Hence, it holds if .V is closed (in particular, .V is gradient). The Kähler–Ricci soliton is shrinking, steady or expanding accordingly as .λ > 0, = 0 or .< 0, respectively. Another way of looking at a gradient Kähler–Ricci soliton is to regard it as a complex manifold . M of complex dimension .n whose Ricci tensor . Rαβ¯ satisfies the equation .∇α Vβ¯ + ∇β¯ Vα + 2Rα β¯ = 2λgα β¯ where .V is a holomorphic vector field on . M and .λ is a real constant. It is called a gradient Kähler–Ricci soliton if .V is the gradient vector field of a real function . f such that .∇α ∇β¯ f + Rα β¯ = λgα β¯ , ∇α ∇β f = 0. We note that the second equation in the above pair means that . J (D f ) is Killing, and hence . f is a Killing potential. Examples: For real dimension 4, the first example of a compact shrinking soliton was constructed by Koiso [41] and Cao [42] on compact complex surface which is the connected sum of .C P 2 #(−C P 2 ), where .−C P 2 is the complex projective space with the opposite orientation. This is a gradient Kähler–Ricci soliton, has .U (2) symmetry and positive Ricci curvature. Another example of a gradient Kähler–Ricci soliton on.C P 2 #2(−C P 2 ) with.U (1) × U (1) symmetry was found by Wang and Zhu [43]. Feldman, Ilmanen and Knopf [44] found complete non-compact.U (n)-invariant shrinking gradient Kähler–Ricci solitons which are cone-like at infinity and satisfy quadratic decay for the curvature. Cao [42] found examples of non-compact gradient steady Kähler–Ricci solitons on .C n that are complete, rotationally symmetric and of positive curvature. Cao [45] also constructed a family of complete non-compact expanding Kähler–Ricci solitons on .C n that are expanding, have .U (n) symmetry and positive sectional curvature and are cone-like at infinity. All known examples of compact shrinking Kähler solitons are Kähler. Let .(M, g) be a compact Kähler manifold. Then we know that the only possible non-trivial Kähler–Ricci solitons on . M are shrinking gradient solitons and then it follows that .

Ric −λg + ∂ ∂¯ f = 0

for a real-valued function . f on . M and some positive real constant .λ. Thus, for . M to admit a gradient shrinking soliton, it is necessary that .c1 (M) > 0. So, let us assume .c1 (M) > 0 and the Kähler form .ω is cohomologous to the Ricci form .ρ. Then we have ¯β f . Rα β¯ − gα β¯ = ∂α ∂ for a real-valued function . f on . M. For a holomorphic vector field .V on . M, Futaki [46] defined  .

F(V ) =

(V f )ωn , M

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and showed that . F is independent of the choice of the Kähler metrics in the Kähler class .[ω]. . F is called the Futaki invariant. Thus, in order that . M admits a Kähler– shrinkEinstein metric, it is necessary that . F = 0. If . M admits a non-trivial gradient  ing Kähler–Ricci soliton with potential function . f , then . F(D f ) = M |D f |2 > 0. For details, we refer to the excellent survey article by Cao [47].

8.6 Contact Metrics as Ricci Solitons In [48] Sharma showed that if a. K -contact (in particular, Sasakian) metric is a gradient Ricci soliton, then it is Einstein. This was also shown independently by He and Zhu [49] for the Sasakian case. Subsequently, Sharma and Ghosh [50] proved that a 3dimensional Sasakian metric which is a non-trivial (i.e. non-Einstein) Ricci soliton is homothetic to the standard Sasakian metric on .nil 3 . Generalizing these results and also answering the following question of H.-D. Cao (cited in [49]):“Does there exist a shrinking Ricci soliton on a Sasakian manifold, which is not Einstein?”, Ghosh and Sharma [25] established the following result. Theorem 8.12 (Ghosh-Sharma) If the metric of a Sasakian manifold . M (.η, ξ, g, ϕ) is a non-trivial (non-Einstein) Ricci soliton, then (i) . M is null .η-Einstein (i.e. . Dhomothetically fixed and transverse Calabi–Yau), (ii) the Ricci soliton is expanding and (iii) the generating vector field .V leaves the structure tensor .ϕ invariant, and is an infinitesimal contact . D-homothetic transformation. Proof Using the Ricci soliton equation: .LV g + 2 Ric = 2λg in the commutation formula (Yano [51] p.23) (LV ∇ X g − ∇ X LV g − ∇[V ,X ] g)(Y, Z ) = − g((LV ∇)(X, Y ), Z ) − g((LV ∇)(X, Z ), Y ),

(8.32)

g((LV ∇)(X, Y ), Z ) = (∇ Z Ric)(X, Y ) − (∇ X Ric)(Y, Z ) − (∇Y Ric)(X, Z ).

(8.33)

we derive

As . M is Sasakian, .ξ is Killing, and hence .Lξ Ric = 0 which, in view of (8.9), the last equation of (8.14) and .h = 0, is equivalent to .∇ξ Q = Qϕ − ϕ Q. But for a Sasakian manifold, . Q commutes with .ϕ, and hence .Ric is parallel along .ξ . Moreover, differentiating the last equation of (8.14), we have .(∇ X Q)ξ = Qϕ X − 2nϕ X . Substituting .ξ for .Y in (8.33) and using these consequences, we obtain (LV ∇)(X, ξ ) = −2Qϕ X + 4nϕ X.

.

(8.34)

8.6 Contact Metrics as Ricci Solitons

121

Differentiating this along an arbitrary vector field.Y , using (8.13) and the last equation of (8.14), we find (∇Y LV ∇)(X, ξ ) − (LV ∇)(X, ϕY ) = −2(∇Y Q)ϕ X + 2η(X )QY − 4nη(X )Y. The use of the foregoing equation in the commutation formula [51]: (LV R)(X, Y )Z = (∇ X LV ∇)(Y, Z ) − (∇Y LV ∇)(X, Z )

.

(8.35)

for a Riemannian manifold shows that (LV R)(X, Y )ξ − (LV ∇)(Y, ϕ X ) + (LV ∇)(X, ϕY ) = −2(∇ X Q)ϕY +2(∇Y Q)ϕ X + 2η(Y )Q X − 2η(X )QY + 4nη(X )Y − 4nη(Y )X. Substituting.ξ for.Y in the foregoing equation, using (8.34) and the formula.∇ξ Q = 0 noted earlier, we find that (LV R)(X, ξ )ξ = 4(Q X − 2n X ).

.

(8.36)

The Ricci soliton equation, in conjunction with the last equation in (8.14), gives (LV g)(X, ξ ) + 2(2n − λ)η(X ) = 0, which in turn gives

.

(LV η)(X ) − g(LV ξ, X ) + 2(2n − λ)η(X ) = 0

(8.37)

η(LV ξ ) = (2n − λ)

(8.38)

.

.

where we used the Lie derivative of .g(ξ, ξ ) = 1 along .V . Next, Lie-differentiating the formula . R(X, ξ )ξ = X − η(X )ξ [a consequence of the first formula in (8.14)] along .V , and using Eqs. (8.36) and (8.38) provides 4(Q X − 2n X ) − g(LV ξ, X )ξ + 2(2n − λ)X = −((LV η)(X ))ξ.

.

By the direct application of (8.37) to the the above equation, we find     λ λ g(X, Y ) + n − η(X )η(Y ) . Ric(X, Y ) = n+ 2 2

(8.39)

which shows that . M is .η-Einstein with scalar curvature r = 2n(n + 1) + nλ.

.

At this point, we recall the following integrability formula [48]: LV r = r − 2λr + 2 |Q|2

.

(8.40)

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8 Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries

for a Ricci soliton, where .r = div Dr , and use it along with (8.39) to obtain the quadratic equation .(2n − λ)(2n + 4 + λ) = 0. As .λ = 2n corresponds to .g becoming Einstein, we must have.λ = −(2n + 4) and hence the soliton is expanding, which proves part (ii). Moreover, Eq. (8.40) reduces to .r = −2n. Thus Eq. (8.39) assumes the form . Ric(Y, Z ) = −2g(Y, Z ) + 2(n + 1)η(Y )η(Z ). (8.41) Hence, as defined in Sect. 8.2, . M is a . D-homothetically fixed null .η-Einstein manifold, proving part (i). Using (8.41) in (8.33) provides (LV ∇)(Y, Z ) = 4(n + 1){η(Y )ϕ Z + η(Z )ϕY }.

.

Differentiating this along an arbitrary . X , using Eqs. (8.9) and (8.13), incorporating the resulting equation in (8.35) and finally contracting at . X , we get (LV Ric)(Y, Z ) = 8(n + 1){g(Y, Z ) − (2n + 1)η(Y )η(Z )}.

.

(8.42)

Equation (8.41) reduces the Ricci soliton equation to the form (LV g)(Y, Z ) = −4(n + 1){g(Y, Z ) + η(Y )η(Z )}.

.

(8.43)

Next, Lie-differentiating (8.41) along .V , and using the above equation shows (LV Ric)(Y, Z ) = 8(n + 1){g(Y, Z ) + η(Y )η(Z )} + 2(n + 1){η(Z )(LV η)(Y ) + η(Y )(LV η)Z }. Comparing this with (8.42) and substituting .ξ for . Z leads to LV η = −4(n + 1)η.

.

(8.44)

Therefore, substituting .ξ for . Z in (8.43) and using the above equation, we immediately get .LV ξ = 4(n + 1)ξ . Operating (8.44) by .d, noting .d commutes with .LV and using the first equation of (8.8), we find (LV dη)(X, Y ) = −4(n + 1)g(X, ϕY ).

.

Its comparison with the Lie derivative of the first equation of (8.8) and the use of (8.43) yields .LV ϕ = 0, completing the proof. For a Kähler–Ricci soliton, the condition .LV J = 0 is added to the Ricci soliton equation. However, the foregoing result shows that .LV ϕ = 0 automatically holds for a non-trivial Sasakian Ricci soliton.

8.6 Contact Metrics as Ricci Solitons

123

A Supporting Example: An explicit example of non-trivial Ricci soliton as a Sasakian manifold is the (2n+1)-dimensional⎡Heisenberg⎤group .H 2n+1 (which arose in quantum mechanics) 1 Y z of matrices of type .⎣ O t In X t ⎦, where . X = (x1 , . . . , xn ), Y = (y1 , . . . , yn ), O = 0 O 1 (0, . . . , 0) ∈ Rn , z ∈ R. As a manifold, this is just .R2n+1 with coordinates (.x i , y i , z) where .i = 1, . . . , n, and has the left-invariant Sasakian structure   (.η, ξ, ϕ, g) defined

∂ 

∂ n 1 ∂ ∂ ∂ i i , .ϕ ∂z by .η = 2 (dz − i=1 y d x ), .ξ =2 ∂z , .ϕ ∂ x i = − ∂ y i , .ϕ ∂∂y i = ∂∂x i + y i ∂z n = 0, and the Riemannian metric .g = η ⊗ η + 41 i=1 ((d x i )2 + (dy i )2 ). Its .ϕsectional curvature (i.e. the sectional curvature of plane sections orthogonal to .ξ ) is equal to .−3, so its Ricci tensor satisfies Eq. (8.41), as shown by Okumura [37], 2n+1 fixed null .η-Einstein manifold. Setting and hence n .H i ∂ is ai . D-homothetically ∂ z ∂ ¯ .V = (V + V ) + V , using equations .LV ξ = 4(n + 1)ξ , .LV ϕ = 0 i i i=1 ∂z ∂x ∂y obtained in the proof of Theorem 8.1, and the aforementioned actions of .ϕ on the coordinate basis vectors shows that .V i and .V¯ i do not depend on .z and yields the PDEs: i ∂ V¯ i ∂V i ∂ V¯ i ∂V z ∂V i i ∂V = , = − , y = ∂x j ∂y j ∂y j ∂x j ∂y j ∂y j ∂V z ∂ V¯ i ∂V z − yi j , = −4(n + 1). V¯ j = y j ∂z ∂y ∂z

The last equation readily integrates as .V z = −4(n + 1)z + F(x i , y i ). For a special solution, assuming . F = 0, .V i = cx i , .V¯ i = cy i and substituting in the above PDEs, we get .c = −2(n + 1), and hence the Ricci soliton vector field .V = −2(n + ∂ ). For dimension 3, this reduces to .V = −4(x ∂∂x + y ∂∂y + 1)(x i ∂∂x i + y i ∂∂y i + 2z ∂z ∂ 2z ∂z ) which occurs on p. 37 of [52] without the factor 4, but gets adjusted with our .λ = 6 which is 4 times their .λ = 3/2. As a converse of the aforementioned result, the following question was also considered in [25]: “What can we say about an .η-Einstein contact metric manifold . M which admits a vector field .V that leaves .ϕ invariant?” and answered in the form of the following result “If an .η-Einstein contact metric manifold . M admits a vector field . V that leaves the structure tensor .ϕ and the scalar curvature invariant, then either . V is an infinitesimal automorphism, or . M is . D-homothetically fixed and . K -contact.” This result provides a generalization of the infinitesimal version of the following result of Tanno [53]: “The group of all diffeomorphisms . which leave the structure tensor .ϕ of a contact metric manifold . M invariant, is a Lie transformation group, and coincides with the automorphism group .A if . M is Einstein.” Gradient Ricci solitons were studied by Ghosh, Sharma and Cho [54] as a .(k, μ)contact metric. We now recall the following result from [54]: “If a non-Sasakian .(k, μ)-contact metric is a gradient Ricci soliton, then in dimension .3 it is flat and in

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higher dimensions it is locally isometric to the trivial sphere bundle . S n (4) × E n+1 ”. This result was generalized by Ghosh and Sharma in [55] by waiving the gradient condition and obtaining the following classification result. Theorem 8.13 If a non-Sasakian (.k, μ)-contact metric .g is a non-trivial Ricci soliton on a (.2n + 1)-dimensional smooth manifold . M, then (. M, g) is locally a 3dimensional Gaussian soliton, or a gradient shrinking rigid Ricci soliton on the trivial sphere bundle . S n (4) × E n+1 , or a non-gradient expanding Ricci soliton with .k = 0, μ = 4. The last case occurs on a Lie group with a left-invariant metric, especially for dimension 3, on . Sol 3 also regarded as the group . E(1, 1) of rigid motions of the Minkowski 2-space. As the standard contact metric .g on the unit tangent bundle .T1 M(c) over a space of constant curvature .c is a (.k, μ)-contact metric with .k = c(2 − c), μ = −2c, with .c = 1 corresponding to the Sasakian case, the above-mentioned result shows that (.T1 M(c), g) would be a non-Sasakian Ricci soliton if and only if .c = 0. Hence, we have the following corollary. Corollary 8.2 The standard contact metric .g on the unit tangent bundle .T1 M(c) over a space of constant curvature .c is a non-Sasakian Ricci soliton if and only if (.T1 M(c), g) is the trivial sphere bundle . S n (4) × E n+1 . An Example Supporting The Last Theorem: Following Boeckx [28], we begin with a (.2n + 1)-dimensional Lie algebra with basis (.ξ, X 1 , . . . , X n , Y1 , . . . , Yn ) and Lie bracket defined by [ξ, X i ] = 0, [ξ, Yi ] = 2X i , [X i , X j ] = 0, [Y2 , Yi ] = 2Yi (i = 2), [Yi , Y j ] = 0(i, j = 2), [X 1 , Y1 ] = 2(ξ − X 2 ), [X 1 , Yi ] = 0(i ≥ 2), .[X 2 , Y1 ] = 2X 1 , [X 2 , Y2 ] = 2ξ, .[X 2 , Yi ] = 2X i , [X i , Y1 ] = [X i , Y2 ] = 0(i ≥ 3), .[X i , Y j ] = 2δi j (ξ − X 2 )(i, j ≥ 3). Let us define a left-invariant contact metric structure with metric .g on the associated Lie group .G as follows: (.ξ, X 1 , . . . , X n , Y1 , . . . , Yn ) is .g-orthogonal, .ξ is the characteristic vector field, the contact 1-form .η is the metric dual of .ξ and the (1, 1)tensor field .ϕ is determined by .ϕ(ξ ) = 0, ϕ(X i ) = Yi , ϕ(Yi ) = −X i . Upon a lengthy computation, it turns out that (.G, ξ, η, ϕ, g) is a (.k, μ)-contact metric manifold with .k = 0, μ = 4. For dimension 3 (.n = 1), we set . X = e, Y = ϕe and thus obtain the Lie algebra .[ξ, e] = 0, [ξ, ϕe] = 2e, [e, ϕe] = 2ξ which corresponds to the unimodular Lie group . E(1, 1) of rigid motions of the Minkowski 2-space, and which is also regarded as the 3-dimensional solvable Lie group . Sol 3 = R  R2 such that the action of .u ∈ R sends .(v, w) ∈ R2 to (.eu v, e−u w). As .sol 3 is diffeomorphic to .R3 , following the construction procedure given on p. 37 of [52], we define the frame field . F1 = 2∂1 , F2 = 2(e−x1 ∂2 + e x1 ∂3 ), F3 = 2(e−x1 ∂2 − e x1 ∂3 ), where ∂ .∂i = and .x i are standard coordinates on .R3 . A direct calculation shows that ∂ xi . .

8.6 Contact Metrics as Ricci Solitons

125

[F1 , F2 ] = −2F3 , [F2 , F3 ] = 0, [F3 , F1 ] = 2F2 which are identical to the Lie algebra equations for (.ϕe, e, ξ ). So, we can take.ϕe = F1 , e = √12 F2 , ξ = √12 F3 . The dual

.

frame of (.ϕe, e, ξ ) is (.θ 1 = 21 d x1 , θ 2 = 2√1 2 (e x1 d x2 + e−x1 d x3 ), θ 3 = 2√1 2 (e x1 d x2 − e−x1 d x3 ). Using this, we define the left-invariant metric .g = θ 1 ⊗ θ 1 + θ 2 ⊗ θ 2 + θ 3 ⊗ θ 3 . Its Ricci tensor turns out to be .Ric = −8θ 1 ⊗ θ 1 . For .t ∈ R, we consider the vector field .V = 4t[−F1 − e−x1 x3 F2 + e−x1 x3 F3 ] + 4(1 − t)[F1 − e x1 x2 F2 − e x1 x2 F3 ], and compute the Lie derivative of .g along .V . This provides .LV g = −16(θ 2 ⊗ θ 2 + θ 3 ⊗ θ 3 ). Hence, we obtain the Ricci soliton equation .LV g + 2 Ric +16g = 0 with .λ = 8. Thus, it is expanding and is non-gradient. Also, as indicated before, .g is a (.k, μ)-contact metric with .k = 0, μ = 4.

Remark: The above example represents the known non-gradient expanding Ricci soliton on . Sol 3 (given on p. 37 of [52]) as a contact metric manifold. Regarding Ricci soliton as a . K -contact manifold, Ghosh and Sharma [56] provided the following characterization. Theorem 8.14 (Ghosh-Sharma) If a. K -contact.eta-Einstein manifold. M (.η, ϕ, ξ, g) is a non-trivial (non-Einstein) Ricci soliton with potential vector field .V , then (i) .V is Jacobi along the geodesics determined by .ξ , (ii) .V is a non-strict infinitesimal contact transformation equal to .− 21 ϕ D f + f ξ for a smooth function . f on . M, (ii) . V preserves the fundamental collineation .ϕ and (iv) Ricci soliton is expanding. They also proved the following result in [57]: “If a . K -contact manifold of dimension 2n + 1 is a Ricci soliton, then its scalar curvature .r is constant and satisfies either (i) .r ≥ 2n(2n + 1) for shrinking case, or (ii) .r ≤ −2n for expanding case. The inequalities are sharp, because the equalities hold on Sasakian Ricci solitons.” Recently, in an unpublished article, they have proved the following generalization of the Sasakian Ricci soliton case (mentioned earlier): “Let (. M, g) be a . K -contact manifold whose Ricci operator commutes with its fundamental collineation .ϕ. If .(M, g) is a Ricci soliton, then it is either Einstein, or . D-homothetically fixed .η-Einstein. In the second case, (i) it is expanding, (ii) its Ricci tensor is Killing, (iii) the potential vector field .V leaves the structure tensor .ϕ invariant, and (iv) .V is an infinitesimal contact transformation”. Finally, we mention the following result of Cho and Sharma [58]. .

Theorem 8.15 (Cho-Sharma) A compact contact Ricci soliton with a potential vector field collinear with the Reeb vector field is Einstein. A homogeneous . H -contact gradient Ricci soliton is locally isometric to . E n+1 × S n (4). They also obtained conditions in order that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle of a Riemannian manifold. Concluding Remark: The problem of characterizing and classifying a general contact metric manifold as a Ricci soliton is open.

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56. Ghosh, A., Sharma, R.: . K -contact metrics as Ricci solitons. Beitr. Algebra Geom. 53, 25–30 (2012) 57. Ghosh, A., Sharma, R.: . K -contact and Sasakian metrics as Ricci almost solitons. Int. J. Geom. Methods Math. Phy. 18, 2150047 (12 pages) (2021) 58. Cho, J.T., Sharma, R.: Contact geometry and Ricci solitons. Int. J. Geom. Methods Math. Phy. 7, 951–960 (2010)

Chapter 9

Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

9.1 Ricci Almost Solitons Modifying the Ricci soliton equation by allowing the dilation constant .λ to become a variable function, Pigola et al. [1] defined a Ricci almost soliton as a Riemannian manifold (. M, g) satisfying the condition: LV gi j + 2Ri j = 2λgi j ,

.

(9.1)

where .V is a vector field on . M, .gi j and . Ri j are the components of the metric tensor .g and its Ricci tensor in local coordinates (.x i ), .LV is the Lie derivative operator along . V and .λ is a smooth function on . M. A simple example is the canonical metric . g on a Euclidean sphere with .V a non-homothetic conformal vector field. For .λ constant, (9.1) becomes the Ricci soliton. The Ricci almost soliton is said to be shrinking, steady and expanding according as .λ is positive, zero and negative, respectively; otherwise it is indefinite. If the vector field .V is the gradient of a smooth function . f , up to the addition of a Killing vector field, (9.1) defines a gradient Ricci almost soliton, and assumes the form: ∇i ∇ j f + Ri j = λgi j .

.

(9.2)

For a Ricci almost soliton of dimension .> 2 with .V homothetic, .g is Einstein and hence .λ becomes constant and it becomes the trivial Ricci soliton. For .V nonhomothetic, .g is a non-trivial almost Ricci soliton. We also note for a Ricci almost soliton that .V is conformal if and only if .g is Einstein. We can view Ricci almost soliton as a special solution of Ricci flow: .∂t gi j = −2Ri j , by considering the ansatz: g (t) = σ (t, x k )ψt∗ gi j ,

. ij

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_9

(9.3)

129

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9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

where .ψt are diffeomorphisms of . M generated by the family of vector fields .Y (t), and .σ (t, x k ) can be viewed as a pointwise scaling function that depends not only on time .t, but also on the coordinates .x k of points. The initial conditions: .gi j (0) = gi j , k .ψ0 = identity, imply .σ (0, x ) = 1. Differentiating (9.3) with respect to .t, using the Ricci flow equation and substituting .t = 0 shows .

− 2Ri j = (

∂ σ (t, x k ))|t=0 gi j + LY (0) gi j . ∂t

Labeling .Y (0) as .V and the time-independent function .( ∂t∂ σ (t, x k ))|t=0 as .−2λ, we obtain the Ricci almost soliton equation (9.1). We present two examples of a Ricci almost solitons as follows. Example 9.1 The standard sphere. S n with the conformal vector field.V = a T , where n+1 .a is a fixed vector in .R and .a T is its orthogonal projection over .T S n and is the gradient of the height function .h a . For details, we refer to Barros and Ribeiro, Jr. [2] and Theorem 2.3 in Pigola et al. [1] Example 9.2 In [1], Pigola et al. constructed some examples of non-compact gradient Ricci almost solitons on warped product manifolds . M n+1 = R ×cosh t H n with metric .g = dt 2 + (cosh2 t)g0 , where .g0 is the standard metric on the hyperbolic space . H n . Then (. M n+1 , g, D f, λ) is a Ricci almost soliton with . f (x, t) = sinh t and .λ(x, t) = sinh t − n. The following results were obtained in [2]. Theorem 9.1 (Barros–Ribeiro, Jr.) If a compact Ricci almost soliton is also a gradient Ricci almost soliton with potential function . f , then, up to a constant, . f agrees with the Hodge–de Rham potential. Theorem 9.2 (Barros–Ribeiro, Jr.) Let (. M n , g, V, λ) be a compact Ricci almost soliton with .n ≥ 3. (i) If .V is a non-trivial conformal vector field, then . M is isometric to a Euclidean sphere{. S n . (ii) If . M (Ric(V, V ) + (n − 2)g(Dλ, V )dvg ≤ 0, then .V is Killing and hence . M n is trivial Ricci soliton. Theorem 9.3 (Barros–Ribeiro, Jr.) Let (. M n , g, D f, λ) be a compact gradient Ricci almost { soliton. Then { g(Dr, D f )dvg , and (i) . M | Hess f − Δnf g|2 dvg = n−2 2n {M { Δf n−2 2 (ii) . M | Hess f − n g| dvg = 2n M (Ric(D f, D f ) + (n − 1)g(Dλ, D f ))dvg . For the locally conformally flat case, Catino [3] proved the following result. Theorem 9.4 (Catino) Around any regular point of the potential function. f , a locally conformally flat gradient Ricci almost soliton is locally a warped product with (.n − 1)-dimensional fibers of constant sectional curvature.

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131

Perelman’s classical result that a compact Ricci soliton is gradient need not be true for almost Ricci solitons. Intrigued by this fact and that a compact Ricci soliton with constant scalar curvature is trivial, Barros, Batista and Ribeiro Jr. [4] obtained the following result. Theorem 9.5 (Barros–Batista–Ribeiro, Jr.) Let (. M n , g, V, λ) be a compact oriented Ricci almost soliton. Then { { n−2 r g(Dr, V )dvg (9.4) . |Ric − g|2 dvg = 2n n M M where .r is the scalar curvature of .g. If, in addition, .n > 2, the Ricci almost soliton is non-trivial and the scalar curvature is constant, then (. M, g) is isometric to a Euclidean sphere and the Ricci almost soliton is gradient. Its proof was based on the Hodge–de Rham decomposition of .V . In [5], the above result was proved by a Lie derivative theoretic method, based on equations of evolution of Christoffel symbols and curvature quantities along .V . We provide this proof as follows. Proof Let us denote the inverse of .gi j by .g i j . Taking the Lie derivative of the relation .gi j g jk = δik along .V , using Eq. (9.1) and subsequently operating the resulting equation by .g il , we immediately get LV g kl = 2R kl − 2λg kl .

(9.5)

.

Next, the use of equation (9.1) in the formula (page 23, Yano [6]): LV Γihj =

.

1 ht g [∇ j (LV git ) + ∇i (LV g jt ) − ∇t (LV gi j )] 2

yields the evolution equation LV Γihj = ∇ h Ri j − ∇ j Rih − ∇i R hj − (∇ h λ)gi j + (∇ j λ)δih + (∇i λ)δ hj . Using this equation the commutation formula (page 23, [6]): h ∇k (LV Γihj ) − ∇ j (LV Γik ) = LV R h

.

ik j ,

subsequently contracting with .g hk and using the twice contracted Bianchi identity: 1 i .∇i R j = ∇ j r , we have 2 LV R ji = ∇ j ∇i r − ∇h ∇ j Rih − ∇h ∇i R hj + ΔRi j − (Δλ)gi j − (n − 2)∇i ∇ j λ.

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9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

Lie-differentiating .r = Ri j g i j along .V , and using the above equation and Eq. (9.5) provides the evolution equation for the scalar curvature: LV r = 2Ri j R i j + Δr − 2λr − 2(n − 1)Δλ.

.

(9.6)

Writing .LV r as .g(∇r, V ), integrating the above equation over the compact . M and using the Gauss divergence theorem, we get { .

1 [Ri j R i j − λr − g(Dr, V )]dvg = 0. 2 M

(9.7)

At this point, we note .

div(r V ) = ∇i (r V i ) = g(Dr, V ) + r div V,

and integrate it over . M in order to get { [g(Dr, V ) + r div V ]dvg = 0.

.

(9.8)

M

Now we contract equation (9.1) with .g i j in order to get .div V = nλ − r , and use it in (9.8) to obtain { (nλr − r 2 + g(Dr, V ))dvg = 0.

.

M

{ Eliminating . M (λr )dvg between the above equation and (9.7) and noting that r2 r 2 ij we obtain Eq. (9.4), proving the first part of the theo.| Ric − g| = Ri j R − n n rem. For the second part, we use the hypothesis that.r is constant in Eq. (9.4) to conclude that .g is Einstein. Thus, Eq. (9.1) reduces to .LV gi j = 2(λ − nr )gi j , i.e. .V is a nonhomothetic conformal vector field on . M. With the setting .λ − nr = ρ, the foregoing conformal equation assumes the form LV gi j = 2ρgi j .

.

(9.9)

Using the conformal integrability condition (p. 26, [6]) LV Ri j = (2 − n)∇i ∇ j ρ − (Δρ)gi j

.

and the Einstein condition . Ri j = nr gi j , we get (Δρ +

.

2r ρ)gi j = (2 − n)∇i ∇ j ρ. n

(9.10)

9.2 Generalized Quasi-Einstein Manifolds

133

r ρ. Using this in the identity .Δρ 2 = Contracting it with .g i j gives .Δρ = − n−1 { r ∇ i ∇ (ρ 2 ) = 2[|Dρ|2 + ρΔρ], and integrating over . M gives . M |Dρ|2 dvg = n−1 { i2 M ρ dvg . This shows that .r > 0. Consequently, Eq. (9.10) becomes

∇i ∇ j ρ = −

.

r ρgi j . n(n − 1)

(9.11)

This implies, by virtue of Obata’s theorem [7]: “A complete Riemannian manifold (. M, g) of dimension .n ≥ 2 admits a non-trivial solution .ρ of the system of partial differential equations .∇i ∇ j ρ = −c2 ρgi j (.c a positive constant) if and only if . M is isometric to a Euclidean sphere of radius .1/c” that (. M, g) is isometric to a Euclidean / sphere of radius .

n(n−1) . r

Equation (9.11) can also be expressed as .L Dρ gi j = with (9.9), we obtain .L V − n(n−1) Dρ gi j = 0.

2r ρgi j . n(1−n)

Combining this

r

Hence, .V = D( n(1−n) ρ) + a Killing vector field, i.e. the almost Ricci soliton is r gradient, completing the proof. We also mention the following result of Barros and Evangelista [8] “A compact Ricci almost soliton with harmonic Weyl tensor (.div ·W = 0) is isometric to the round sphere provided that the second symmetric function of the Schouten tensor is constant and positive”.

9.2 Generalized Quasi-Einstein Manifolds In [3], Catino defined a generalized quasi-Einstein manifold as an n-dimensional smooth Riemannian manifold (. M, g) satisfying the condition .

Ric + Hess f − μd f ⊗ d f = λg

(9.12)

where . f , .μ and .λ denote smooth functions on . M, .Ric is the Ricci tensor and .Hess f is the Hessian of . f with respect to .g. It is said to be trivial when . f is constant, in which case it becomes Einstein. However, if . M is Einstein, it need not be trivial. In [3], Catino proved the following characterization result. Theorem 9.6 Let (. M n , g) (.n ≥ 3) be a generalized quasi-Einstein manifold with harmonic Weyl tensor and satisfying .W (D f, ., ., .) = 0. Then, around any regular point of . f , (. M n , g) is locally a warped product of a line with (.n − 1)-dimensional Einstein fibers.

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9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

It may be noted that, for .n = 3, the Weyl tensor vanishes and the hypotheses in the above result are trivially satisfied. So, for .n = 3, the hypotheses should be replaced n−3 Cabc . by the vanishing of the Cotton tensor, in view of the formula .∇ d Wabcd = − n−2 The following corollary was also stated in [3]. Corollary 9.1 Let (. M 4 , g) be a 4-dimensional generalized quasi-Einstein manifold with harmonic Weyl tensor and .W (D f, ., ., .) = 0. Then, around any regular point of 4 . f , (. M , g) is locally a warped product of a line with 3-dimensional fibers of constant curvature. In particular, if it is non-trivial, then it is locally conformally flat. The last part of this corollary also follows from the fact that .W (D f, ., ., .) = 0 in dimension 4 implies that .W = 0 at any regular point of . f [this follows by a straightforward combinatorial computations using the complete tracelessness and symmetry properties of the Weyl tensor]. We now give an outline of the proof of the above-mentioned theorem. By a straightn−3 Cabc , where .C is the Cotton tensor forward computation, we have .∇ d Wabcd = − n−2 Cabc = ∇a Rbc − ∇b Rac −

.

1 [(∇a r )gbc − (∇b r )gac ]. 2(n − 1) 2f

Consider the conformally deformed metric .g¯ = e− n−2 g. Under this deformation, the Cotton tensor transforms as (n − 2)C¯ abc = (n − 2)Cabc +

.

1 Wabcd ∇ d f. n−2

Hence, for .n > 3, .div W = 0 is equivalent to .C = 0. By hypothesis, .Wabcd ∇ d f = 0, 1 ¯ − ¯ given by . S¯ = n−2 (Ric and hence .C¯ = 0. Therefore, the Schouten tensor of .g, r¯ g), ¯ is Codazzi. Furthermore, the conformal transformation law for Ricci tensor, 2(n−1) in conjunction with (9.12), gives ¯ = (μ + Ric +

1 )d f ⊗ d f n−2

2f 1 ¯ (Δ f − |D f |2 + (n − 2)λ)e n−2 g. n−2

This shows that, at every regular point. p of. f , the Ricci tensor and hence the Schouten tensor of .g¯ have a unique eigenvalue (in which case .g is conformally Einstein) or two distinct eigenvalues .σ1 (multiplicity 1) and .σ2 (multiplicity (.n − 1)). Applying a result of Derdzinski [9], we see that the tangent bundle of a neighborhood of . p splits as the orthogonal direct sum of two integrable eigendistributions, a line field.Vσ1 and a codimension 1 distribution .Vσ2 with totally umbilical leaves. By the standard Gauss 1 and Codazzi equations, we find that .g¯ i j (x 1 , ..., x n ) = eψ(x ) G i j (x 2 , ...x n ), where (.x 1 , ..., x n ) is a local chart on . M such that . ∂∂x 1 ∈ Vσ1 and . ∂∂x i ∈ Vσ2 . So, around any ¯ is locally a warped product with (.n − 1)-dimensional regular point of . f , (. M n , g) fibers. By the conformal structure of conformal deformation, this also holds for

9.2 Generalized Quasi-Einstein Manifolds

135

(. M n , g). As .g has harmonic Weyl tensor, (.n − 1)-dimensional fibers are Einstein, by a result of Gebarowski [10]. In particular, if.μ = m1 for a real m such that.0 < m ≤ ∞, then Eq. (9.12) becomes .

Hess f + Ric −

1 d f ⊗ d f = λg m

(9.13)

where the left side is the so-called .m-Bakry–Émery tensor denoted by .Ricmf and introduced by Bakry and Émery in [11], motivated by diffusion processes. The Riemannian manifold (. M, g) satisfying Eq. (9.13) is called a generalized .m-quasiEinstein manifold (in particular, .m-quasi-Einstein if .λ is constant). The case .m = 1 includes the static metrics which have been studied in depth for their connections to the positive mass theorem and general relativity (Anderson and Khuri [12]). The .m-quasi-Einstein manifold is said to be expanding, steady or shrinking accordingly as .λ < 0, .λ = 0, or .λ > 0. For .m = ∞, and .λ constant in (9.13), we get a gradient Ricci soliton. For .m = ∞, but .λ non-constant, (9.13) defines a gradient Ricci almost soliton discussed earlier. For .λ constant and .m a positive integer, an .m-quasi-Einstein manifold is the base of an (.n + m)-dimensional Einstein warped product (Case et al., [13], He et al. [14] and Barros and Ribeiro [15]). For an .m-quasi-Einstein manifold with finite .m, let .u = e− f m. Then . Du = − m1 e− f /m D f and . mu Hess u = − Hess f + m1 d f ⊗ d f . Hence (9.13) takes the form .

Ric −

m Hess u = λg. u

(9.14)

Its trace gives.Δu = mu (r − nλ). As.u > 0, it follows that a compact.m-quasi-Einstein metric with constant scalar curvature is trivial [13]. It was shown in [13] that a quasiEinstein metric with .1 ≤ m < ∞ and .λ > 0 has positive scalar curvature. We also mention the following results established in [13]. Theorem 9.7 (Case-Shu-Wei) All 2-dimensional .m-quasi-Einstein metrics on compact manifolds are trivial. Theorem 9.8 (Case-Shu-Wei) (. M n , g) satisfies the quasi-Einstein equation (9.13) if and only if the warped product metric . M ×e− f /m F m is Einstein, where . F m is an .m-dimensional Einstein manifold with Einstein constant .c satisfying ce2 f /m = λ −

.

1 (Δ f − |D f |2 ). m

For .1-quasi-Einstein metric, the constant .c = 0. Combining this with the above equation and the trace .r + Δ f − |D f |2 = nλ of (9.13) we find that the scalar curvature is constant. For a finite .m, an .m-quasi-Einstein manifold is compact (as shown by Qian [16]). The following generalizations of the corresponding formulas of a Ricci soliton were obtained in [13]:

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9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

.

1 2 Δ|D f |2 = | Hess f |2 − Ric(D f, D F) + |D f |2 Δ f, 2 m .

1 m−1 1 Dr = Q(D f ) + (r − (n − 1)λ)D f, 2 m m

1 m−1 m+2 r Δr − ∇D f r = − | Ric − g|2 2 2m m n m+n−1 n(n − 1) − (r − nλ)(r − λ). mn m+n−1 The use of the above equations gives bounds on the scalar curvature, as obtained in [13]. More precisely, we have the following result. Proposition 9.1 (Case-Shu-Wei) Let an .m-quasi-Einstein manifold . M have .m ≥ 1. n(n−1) λ, equality holds (a) If .λ > 0 and . M is compact then the scalar curvature .r ≥ m+n−1 if and only if .m = 1. (b) .λ = 0, .r is constant and .m > 1, then . M is Ricci-flat. (c) .λ < 0, .r is constant, then n(n − 1) .nλ ≤ r ≤ λ m+n−1 and when.m > 1,.r equals either one of the extreme values if and only if. M is Einstein. In [13], the following splitting result was obtained for the Kaehler .m-quasiEinstein manifolds. Theorem 9.9 (Case-Shu-Wei) If (. M, g) is an .n-dimensional complete simply connected Riemannian manifold with a Kaehler .m-quasi-Einstein metric for finite .m, then . M = M1 × M2 is a Riemannian product, and . f can be considered as a function of . M2 , where . M1 is an (.n − 2)-dimensional Einstein manifold manifold with Einstein constant .λ, and . M2 is a 2-dimensional quasi-Einstein manifold. Turning our attention to generalized .m-quasi-Einstein manifolds, we recall that Barros and Ribeiro, Jr. [15] provided examples of generalized .m-quasi-Einstein manifolds on the model spaces . S n , . E n and . H n , and proved the following rigidity and triviality results. Theorem 9.10 (Barros–Ribeiro, Jr.) Let (. M n , g, D f, λ) be a non-trivial generalized −f .m-quasi-Einstein manifold with .n ≥ 3, and be either Einstein or . Du (.u = e m) be n n n n conformal. Then . M is isometric to . S , or . E or . H . In each case . f is determined by height and distance functions. . Theorem 9.11 (Barros–Ribeiro, Jr.) Let (. M n , g, D f, λ) be a compact generalized n .m-quasi-Einstein manifold. Then . M is trivial provided: { { { 2 . Ric(D f, D f )dv ≤ |D f |2 Δ f dvg − (n − 2) M g(Dλ, D f )dvg . (1) g M m M (2) .r ≥ nλ or .r ≤ nλ.

9.2 Generalized Quasi-Einstein Manifolds

137

Theorem 9.12 (Barros–Ribeiro, Jr.) Let (. M n , g, D f, λ) be a complete generalized .m-quasi-Einstein manifold with finite .m. Then . D f = 0, if one of the following conditions holds: (1) . M n is non-compact, .nλ ≥ r and .|D f | ∈ L 1 (M n ). In particular, . M n is Einstein. (2) (. M n , g) is Einstein and . D f is conformal. The following formulas were derived in [15] for a generalized .m-quasi-Einstein manifold: .

1 2 Δ|D f |2 = | Hess f |2 − Ric(D f, D f ) + |D f |2 Δ f − (n − 2) < Dλ, D f >, 2 m .

.

1 m−1 1 Dr = Ric(D f ) + (r − (n − 1)λ)D f + (n − 1)Dλ, 2 m m

D(r + |D f |2 − 2(n − 1)λ) = 2λD f +

2 [∇ D f D f + (|D f |2 − Δ f )D f ], m

1 n Δf 2 m + n Δr = −| Hess f − g| − (Δ f )2 − < D f, Dλ > + < D f, Dr > 2 n mn 2 1 m−2 < D f, DΔ f ) + div(∇ D f D f ) + (n − 1)Δλ + λD f. + 2m m The preceding two theorems generalize the corresponding results for Ricci solitons, Ricci almost solitons and .m-quasi-Einstein manifolds. Next, we state the following result of Barros and Gomes [17] that generalize a corresponding result for Ricci almost soliton established in [4]. Theorem 9.13 (Barros–Gomes) Let (. M n , g, D f, λ) be a non-trivial compact generalized .m-quasi-Einstein manifold with .n ≥ 3. In addition, { if .L Du r ≥ 0 (where −f .u = e m), then . M n is isometric to a round sphere. (1) . M Ric(D f, D f )dvg ≤ { { 2 |D f |2 Δ f dvg − (n − 2) M g(Dλ, D f )dvg . (2) .r ≥ nλ or .r ≤ nλ. m M Before proceeding further, let us consider the generalization of Bakry–Émery tensor from a function . f to a vector field .V as .

1 1 RicmV = Ric + LV g − V b ⊗ V b , 2 m

where .m is defined as before, and .V b is the 1-form metrically equivalent to .V . A Riemannian manifold (. M, g) satisfying .RicmV = λg is called an .m-quasi-Einstein manifold (not necessarily gradient). Barros and Ribeiro, Jr. considered such manifolds in [18] and proved the following results. Theorem 9.14 (Barros–Ribeiro, Jr.) Let (. M n , g, V ), .n ≥ 3, be a compact Riemannian manifold satisfying . RVm = λg. Then . M n is Einstein provided:

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9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds

{ { (1) . M Ric(V, V )dvg ≤ m2 M |V |2 div{ V dvg . (2) .V is a conformal vector { field and . M Ric(V, V )dvg ≤ 0. (3) .|V | is constant and . M Ric(V, V )dvg ≤ 0. Theorem 9.15 (Barros–Ribeiro, Jr.) Let (. M n , g, V ) be a complete non-compact Riemannian manifold satisfying .RicmV = λg. If .nλ ≥ r and .|V | ∈ L 1 (M n ), then . M n is Einstein. The proofs of the above two results use the formula: .

div(LV g)(V ) =

1 Δ|V |2 − |∇V |2 + Ric(V, V ) + ∇V div V, 2

for a vector field .V on a Riemannian manifold, and the following result of Yau [19] (which is a generalization of Hopf’s theorem): A subharmonic function. f : M n → R on a complete non-compact Riemannian manifold is constant, if its gradient belongs to . L 1 (M n ); this was generalized for a vector field by Camargo et al. cited in [18]. Generalized .m-quasi-Einstein manifolds with parallel Ricci tensor and constant scalar curvature were classified by Hu, Li and Xu in [20] as follows. Theorem 9.16 (Hu–Li–Xu) Let (. M n , g, f, λ) be a complete .n-dimensional (.n ≥ 3) non-trivial generalized .m-quasi-Einstein manifold with parallel Ricci tensor. Then n . M is isometric to one of the following manifolds: a, (5) (1) a space form, (2) .Dcn , (3) .R × N n−1 (b), (4) .H p (a) × N n− p (b), b = m+n−1 p−1 p n− p 2 k .Dc × N (b), b = (1 − m − p)c , where .a, b are negative constants, .N (b) a .kdimensional Einstein manifold with scalar curvature .kb; .H p (a) the . p-dimensional hyperbolic space with Einstein constant .a; .Dck a .k-dimensional Einstein warped product .R ×c−1 ecr F k−1 endowed with the metric .dr 2 + (c−1 ecr )2 gF , .c is a positive constant, .F k−1 with metric .gF is a Ricci flat manifold. Theorem 9.17 (Hu–Li–Xu) Let (. M n , g, f, λ) be a complete .n-dimensional nontrivial .m-quasi-Einstein manifold with constant scalar curvature .r , then .λ ≤ 0. Moreover, (1) if .m = 1, then .r = (n − 1)λ, (2) if .m > 1, then .λ < 0, and if moreover − f /m .e attains its maximum or minimum at some point on M, then the scalar curvature gets discretized as { r∈

| } mn − (m − n) p − n || λ | p = 1, 2, ..., n . m+ p−1

Conversely, for each. p ∈ {1, 2, ..., n}, there exists a nontrivial.m-quasi-Einstein manp−n λ. ifold with .λ < 0 and constant scalar curvature, which satisfies .r = mn−(m−n) m+ p−1 Subsequently, Hu, Li and Zhai [21] considered an .n-dimensional (.n ≥ 3) proper generalized .m-quasi-Einstein manifold (. M n , g) with eigenvalues of the Ricci tensor

9.3 Generalized m-Quasi-Einstein Manifolds with Conformal Vector Fields

139

constants, and showed that (i) if .m /= 1, then (. M n , g) is Einstein and (ii) if .m = 1 and .Ric is Codazzi or . Du (.u defined as before) is an eigenvector of .Ric, then (. M n , g) is Einstein.

9.3 Generalized m-Quasi-Einstein Manifolds with Conformal Vector Fields Let us recall the following result of Yano and Nagano [22]: “If a complete Einstein manifold admits a complete non-homothetic conformal vector field, then it is isometric to a round sphere”. A question now arises whether this result can be generalized on a generalized quasi-Einstein manifold. Jauregui and Wylie [23] achieved this goal by assuming that the conformal vector field .V preserves the generalized quasi-Einstein structure, and obtaining the following generalization. Theorem 9.18 (Jauregui–Wylie) Let (. M, g, f, μ, λ) be a complete generalized 1 that admits a non-homothetic conformal quasi-Einstein manifold, with .μ /= n−2 vector field .V : .LV g = 2σ g such that .V preserves .d f and .μ, i.e. .LV d f = 0 and .L V μ = 0. If the conformal scale function .σ has a critical point (e.g. if . M is compact), then . f is constant and (. M, g) is isometric to a simply connected space form. In addition, if .V is complete, then (. M, g) is isometric to a round sphere. The last part generalizes Yano-Nagano’s result mentioned earlier. For the case 1 , we note that .g is conformal to an Einstein metric. μ = n−2

.

In particular, when the conformal vector field is closed, we recall the following ¨ [24], which gives a nice expression of the Ricci tensor. result of Demirba˘g and G.uler Theorem 9.19 (Demirba˘g–G.uler) ¨ Let (. M n , g, f, λ = ρr + λ0 ) (for constants .ρ and .λ0 and scalar curvature .r ) be an (.m, ρ)-quasi-Einstein manifold admitting a closed conformal vector field .V with conformal factor .ψ. Then its Ricci tensor is given by 1−n . Ric = (ρr + λ0 )g + [ V ψ − (ρr + λ0 )]u ⊗ u |V |2 where .u is a 1-form metrically equivalent to a unit vector field along .V . This result has been recently generalized over a generalized .m-quasi-Einstein manifold by Poddar, Balasubramanian and Sharma in [25] as follows. Theorem 9.20 (Poddar–Balasubramanian–Sharma) Let (. M, g, f, λ) be an n-dimensional complete generalized .m-quasi-Einstein manifold. If it admits a nonhomothetic, non-parallel complete closed conformal vector field .V with conformal factor .ψ, then either it is isometric to a round sphere, or its Ricci tensor is expressed in terms of conformal data as

.

140

9 Ricci Almost Solitons and Generalized Quasi-Einstein Manifolds .

Ric = (

r r + nα)u ⊗ u + α)g − ( n−1 n−1

Vψ , and where .u is a 1-form metrically over an open dense subset of . M, and .α = |V |2 equivalent to a unit vector field along .V . In this second case, . M is locally a warped product of an open real interval with an Einstein manifold and has vanishing Cotton and Bach tensors.

9.4 Generalized m-Quasi-Einstein Manifolds in Contact Geometry We exhibit an interplay between the generalized .m quasi-Einstein structure and contact Riemannian geometry. For this, we list the following results obtained by Ghosh in [26, 27]. Theorem 9.21 (Ghosh) Let (. M 2n+1 , g) be a Sasakian manifold with an associated .m-quasi-Einstein structure. Then . f is constant and . M is Einstein. Theorem 9.22 (Ghosh) Let (. M 2n+1 , g) be a complete . K -contact manifold with an associated .m-quasi-Einstein structure. Then . f is constant, . M is compact, Einstein and Sasakian. Theorem 9.23 (Ghosh) Let (. M 2n+1 , g) be a (.k, μ)-contact manifold with an associated .m-quasi-Einstein structure. If .n = 1, then it has constant curvature 0 or 1, and in higher dimensions, it is either Einstein Sasakian or locally isometric to the trivial sphere bundle . E n+1 × S n (4). Theorem 9.24 (Ghosh) Let (. M 2n+1 , g) be a Sasakian manifold with an associated generalized .m-quasi-Einstein structure. Then it is Einstein. In addition, if . M is complete, then it is compact and isometric to . S 2n+1 . Theorem 9.25 (Ghosh) Let (. M 2n+1 , g) be a complete . K -contact manifold with an associated generalized .m-quasi-Einstein structure with .m /= 1. Then it is compact and Einstein. In addition, if . M is complete, then is compact Sasakian and isometric to . S 2n+1 . Remark It would be desirable to examine generalized quasi-Einstein manifolds (including generalized .m-quasi-Einstein manifold as special case) further, in more general contact geometry.

References

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References 1. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.: Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. X(5), 757–799 (2011) 2. Some characterizations for compact almost Ricci solitons: Barros, A., Ribeiro, Jr. Proc. Am. Math. Soc. 140, 1033–1040 (2012) 3. Catino, G.: Generalized quasi-Einstein manifolds with harmonic Weyl tensor. Math. Z. 271, 751–756 (2012) 4. Barros, A., Batista, R., Ribeiro, Jr.: Compact almost Ricci solitons with constant scalar curvature are gradient. Monatsh. Math. (2014). https://doi.org/10.1007/s00605-013-0581-3 5. Sharma, R.: Almost Ricci solitons and . K -contact geometry. Monatsh. Math. (2014). https:// doi.org/10.1007/s00605-014-0657-8 6. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) 7. Obata, M.: Certain conditions for a Riemannian manifold to be isometric to a sphere. J. Math. Soc. Japan 14, 333–340 (1962) 8. Barros, A., Evangelista, I.: Some results on compact almost Ricci solitons. Illinois J. Math. 60, 529–540 (2016) 9. Derdzinski, A.: Some remarks on the local structure of Codazzi tensors. In: Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 838. Springer, Berlin 1981, 243–299 (1979) 10. Gebarowski, A.: Doubly warped products with harmonic Weyl conformal curvature tensor. Colloq. Math. 67, 73–89 (1994) 11. Bakry, D., Émery, M.: Diffusions Hypercontractives. In: Séminaire de probabilités XIX, 1983–84, 177–206, Lecture Notes in Mathematics, vol. 1123. Springer, Berlin (1985) 12. Anderson, M., Khuri, M.: The static extension problem in general relativity. ArXiv: 0909.4550v1 [math.DG] (2009) 13. Case, J., Shu, Y., Wei, G.: Rigidity of quasi-Einstein metrics. Differ. Geom. Appl. 29, 93–100 (2011) 14. He, C., Petersen, P., Wylie, W.: On the classification of warped product Einstein metrics. Commun. Anal. Geom. 20, 271–312 (2012) 15. Barros, A., Ribeiro, E., Jr.: Characterizations and integral formulae for generalized quasiEinstein metrics. Bull. Braz. Math. Soc. 45, 325–341 (2014) 16. Qian, Z.: Estimates for weighted volumes and applications. Quart. J. Math. Oxford Ser. 48(2), 235–242 (1997) 17. Barros, A., Gomes, J.N.: A compact gradient generalized quasi-Einstein metric with constant scalar curvature. J. Math. Anal. Appl. 401, 702–705 (2013) 18. Barros, A., Ribeiro, E., Jr.: Integral formulae on quasi-Einstein manifolds and applications. Glasgow Math. J. 54, 213–223 (2012) 19. Yau, S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976) 20. Hu, Z., Li, D., Xu, J.: On generalized .m-quasi-Einstein manifolds with constant scalar curvature. J. Math. Anal. Appl. 432, 733–743 (2015) 21. Hu, Z., Li, D., Zhai, S.: On generalized .m-quasi-Einstein manifolds with constant Ricci curvatures. J. Math. Anal. Appl. 446, 843–851 (2017) 22. Yano, K., Nagano, T.: Einstein spaces admitting a one-parameter group of conformal transformations. Ann. Math. 69, 451–461 (1959) 23. Jauregui, J., Wylie, W.: Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds. J. Geom. Anal. 25, 668–708 (2015) ¨ S.: Rigidity of (.m, ρ)-quasi Einstein manifolds. Math. Nachr. 290, 24. Demirba˘g, A.A., G.uler, 2100–2110 (2017) 25. Poddar, R., Balasubramanian, S., Sharma, R.: Generalized.m-quasi Einstein manifolds admitting a closed conformal vector field. Ann. Mat. Pura Ed Appl. (2023). https://doi.org/10.1007/ s10231-023-01335-w

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26. Ghosh, A.: Quasi-Einstein contact metric manifolds. Glasgow Math. J. 57, 569–577 (2015) 27. Ghosh, A.: Generalized .m-Quasi-Einstein metric within the framework of Sasakian and . K contact manifolds. Ann. Polon. Math. 115, 33–41 (2015)

Chapter 10

Yamabe Solitons

10.1 Yamabe Problem Let us recall that a Riemannian metric .g on a smooth manifold . M is conformal to a Riemannian metric .g0 on . M if .g = e2 f g0 for a smooth function . f : M → R. The set of all metrics conformal to a Riemannian metric .g0 is denoted by .[g0 ] and is called the conformal class of .g0 . Let us now recall the well-known uniformization theorem “If.g0 is a Riemannian metric on a compact surface . M, then there exists a smooth function . f : M → R such that the metric .e2 f g0 has constant Gaussian curvature”. This motivated Yamabe [1] to seek a generalization of the uniformization theorem to a compact Riemannian manifold of dimension .≥ 3, and he proposed the following problem/conjecture. Yamabe Conjecture: If (. M, g0 ) is a compact Riemannian manifold of dimension .n ≥ 3, then there exists a metric . g on . M which is conformal to . g0 and has constant scalar curvature. 4 In order to reformulate this conjecture, let .g = u n−2 g0 for a positive smooth function .u : M → R. Then the scalar curvatures .r g and .r g0 of .g and .g0 are related by r = u − n−2 [− n+2

. g

4(n − 1) /\g0 u + r g0 u], n−2

(10.1)

where ./\g0 denotes the Laplace operator of .g0 . Thus, .g has constant scalar curvature c if and only if .u satisfies the semi-linear partial differential equation:

.

.

4(n − 1) n+2 /\g0 u − r g0 u + cu n−2 = 0, n−2

(10.2)

called the Yamabe equation. Restricting the Einstein–Hilbert action: {

r g dvg

.

M

[vol(M, g)]

n−2 n

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4_10

143

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10 Yamabe Solitons 4

whose critical points are Einstein metrics, to the conformal class .{g = u n−2 g0 }, and using (10.1), we have the Yamabe functional } .

E g0 (u) =

4(n−1) 2 M [ n−2 |∇u|g0

[

}

2n

+ r g0 u 2 ]dvg

n−2 dv ] g0 Mu

n−2 n

.

It turns out that .u is a critical point of . E g0 (u) if and only if .u satisfies (10.2) for some constant .c, and therefore the Yamabe problem is equivalent to finding critical points of . E g0 (u). Next, one defines the Yamabe constant .Y (M, g0 ) of a compact Riemannian manifold (. M, g0 ) as .

Y (M, g0 ) = in f 0 0 can be shown by the product . S n−1 × S 1 (l) where . S 1 (l) has length .l. For sufficiently large .l, Eq. (10.2) has an arbitrarily large number of solutions. For details, we refer to the survey article [6] of Brendle and Marques.

10.2 Yamabe Flow and Yamabe Solitons Let (. M n , g0 ) be an .n-dimensional closed (compact without boundary) Riemannian manifold. In order to solve the Yamabe problem, Hamilton [7] introduced the Yamabe flow as the evolution equation

10.2 Yamabe Flow and Yamabe Solitons

.

145

∂g = −r g g ∂t

(10.3)

from the initial metric .g0 to a metric .g(t) over time .t, where .r g denotes the scalar curvature of .g(t). The normalized Yamabe flow is given by .

}

∂g = (¯r − r g )g ∂t

(10.4)

r dv

g g where .r¯ = }M dev is average scalar curvature of .r g , and preserves the volume of the g M initial Riemannian manifold. Equations (10.3) and (10.4) differ only by a change of scale in space and a change of parametrization in time. Although the Yamabe problem was solved using a different approach (as described in Sect. 10.1), the Yamabe flow is a natural geometric deformation to metrics of constant scalar curvature and was used by Brendle to solve the Yamabe problem for .3 ≤ n ≤ 5 or if .n ≥ 6 (in the latter case, Brendle assumes that the metric is either locally conformally flat or satisfies a certain condition on the rate of vanishing of Weyl tensor at points where it vanishes), then showing that normalized Yamabe flow exists for a long time and converges to a metric of constant scalar curvature. We may note that the Yamabe flow corresponds to the fast diffusion case of the porous medium equation (the plasma equation) in mathematical physics (Burchard, McCann and Smith [8]). Just as a Ricci soliton is a special solution of the Ricci flow, a Yamabe soliton is a special solution of the Yamabe flow that moves by one parameter family of diffeomorphisms .φt generated by a time-dependent vector field .Wt on . M, and homotheties, i.e. .g(t) = σ (t)φt∗ g where .σ is a positive real-valued function of the parameter .t. Substituting the foregoing equation in the Yamabe flow equation (10.3) and setting .−σ ˙ (0) = ρ gives the equation

LV g = 2(r − ρ)g

.

(10.5)

where .g is the initial metric .g(0) of the Yamabe flow, .V is a time-independent vector field such that .Wt = − 2σ1(t) V , .r is the scalar curvature of .g and .ρ is a real constant defined earlier. The Riemannian manifold (. M, g) with a vector field .V and a constant .ρ, satisfying the Eq. (10.5), is called a Yamabe soliton. The Yamabe soliton is said to be shrinking, steady or expanding when .ρ > 0, = 0, < 0, respectively. In particular, if .V = D f (up to the addition of a Killing vector field) for a smooth function . f , where . D denotes the gradient operator of .g, then a Yamabe soliton is called a gradient Yamabe soliton in which case equation (10.5) assumes the form ∇∇ f = (r − ρ)g.

.

(10.6)

It is evident from Eq. (10.5) that the vector field.V is conformal. The gradient Yamabe soliton is trivial when . f is constant and .r is constant.

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10 Yamabe Solitons

Let us first characterize a closed (compact without boundary) Yamabe soliton. In this direction, we first recall the following result of Cerbo and Disconzi [9]. Theorem 10.3 (Cerbo–Disconzi) A Yamabe soliton on a closed.n-dimensional manifold with .n ≥ 3 has constant scalar curvature. Its proof depends on the following calculation of the derivative of the total scalar functional along the normalized Yamabe flow: d dt

{

{ dr g d r g dvg dvg + dt M dt M { n = (1 − ) (¯r − r g )2 dvg , 2 M

{

r g dvg = M

where we have used the formulas: ∂ r = −/\(tr.vi j ) + div(div(vi j )− < (vi j ), Ri j >,

. t g

vi j = ∂t gi j

and √ ∂t (dvg ) = ∂t ( gd x 1 ∧ ... ∧ d x n ) 1 n = g −1/2 gg i j (∂t gi j )d x 1 ∧ ... ∧ d x n = (r − r g )dvg . 2 2 Thus, the total scalar curvature functional is scale-invariant and decreases along the non-normalized Yamabe flow. Hence, the normalized total scalar curvature functional decreases monotonically along the Yamabe flow unless the initial scalar curvature is constant, completing the proof. For dimension .2, we have the following result proved in [9]. Theorem 10.4 (Cerbo–Disconzi) There are no non-trivial compact .2-dimensional Yamabe solitons. In [10], Daskalopoulos and Sesum gave another non-variational proof of the above-mentioned theorem of Cerbo–Disconzi for a compact gradient Yamabe soliton, as follows. Tracing gradient Yamabe equation (10.6) gives ./\ f = n(r − ρ). Therefore, { { (/\ f )2 dvg n 2 (r − ρ)2 dvg = M {M { = (∇ i ∇i f )(∇ j ∇ j f )dvg = − (∇i f )(∇i ∇ j ∇ j f )dvg . M

M

The use of the curvature identity .∇i ∇ j ∇ j f = ∇ j ∇i ∇ j f − Rik ∇k f in the above, and integrating by parts again, gives {

{

{

(/\ f ) dvg =

|∇ f | dvg +

2

.

M

2

M

Ric(D f, D f )dvg .

2

M

10.2 Yamabe Flow and Yamabe Solitons

147

At this point, we note from (10.6) f, .) = 21 Dr . Using this in the preceding } that .Ric(D 2 equations provides .2n(n − 1) M (r − ρ) dvg = 0. This implies .r = ρ, and hence . f is constant, completing the proof. A simpler proof was given by Hsu [11] for .n ≥ 3 using the formula: (n − 1)/\r +

.

1 < Dr, D f > +r (r − ρ) = 0, 2

given on page 20 of [10] and is essentially the formula for the Lie derivative of the scalar curvature .r along the soliton vector field . D f that is a conformal vector field. Another proof of this result occurs in the monograph [12] of Chow, Lu and Ni that } uses the formula (Bourguignon and Ezin [13]): . M (L V r )dvg = 0 for a conformal vector field on a closed Riemannian manifold of dimension .≥ 3. Yet another proof (including the non-gradient case) has been recently found in an unpublished result of Balasubramanian, Poddar and Sharma and is based on the following formula: (n − 2)(r − ρ)2 = div[(r −

.

n−2 ρ)V + 2(n − 1)∇r ] n

which can be derived from the integrability equations for a conformal vector field .V (as given in chapter 4). The use of the above formula gives the following result for the compact Yamabe soliton with boundary. Proposition 10.1 Let .(M, g) be a compact Yamabe soliton with boundary .∂ M and ρ)V + 2(n − 1)∇r is associated vector field .V . If .dim M = n > 2 and .(r − n−2 n tangential to .∂ M, then .r is constant and .V is Killing. In the non-compact case, the following result holds (can be proved by the conformal integrability equations). Proposition 10.2 Let.(M, g) be a Yamabe soliton whose soliton vector field.V leaves the Ricci tensor invariant. Then its scalar curvature is constant. Unlike the compact Yamabe solitons, the complete Yamabe solitons are not well understood. In [10], Daskalopoulus and Sesum established the rotational symmetry of locally conformally flat Yamabe solitons by proving the following nice result. Theorem 10.5 (Daskalopoulus–Sesum) All locally conformally flat complete Yamabe gradient solitons with positive sectional curvature are rotationally symmetric. An .n-dimensional rotationally symmetric metric is of the form: .ψ 2 (r )dr 2 + ϕ (r )g S n−1 , where .g S n−1 is the canonical metric of the unit .(n − 1)-dimensional unit sphere. In addition, the following classification result for gradient Yamabe solitons was obtained in [10]. 2

Theorem 10.6 (Daskalopoulus–Sesum) The metric .g is a complete locally conformally flat Yamabe gradient soliton with positive sectional curvature if and only if 4 2 . g = u n+2 d x , where .u satisfies the elliptic equation

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10 Yamabe Solitons

.

n−1 m /\u + βx.Du + γ u = 0 m

> 0. This elliptic equation admits non-trivial on .Rn , for some .β ≥ 0 and .γ = 2β+ρ 1−m radially symmetric smooth solutions. Inspired by the first result mentioned above, Cao et al. [14] noticed from the special nature of gradient Yamabe soliton that it admits a special global warped product structure with a 1-dimensional base and the warping function given by .|D f |, removed the locally conformally flat condition and proved the following result. Theorem 10.7 (Cao–Sun–Zhang) Let (. M n , g, f ) be a non-trivial complete gradient Yamabe soliton. Then .|D f |2 is constant on regular level hypersurfaces of . f , and either (i) . f has a unique critical point at some point .x0 ∈ M n , and (. M n , g, f ) is ∗ ), rotationally symmetric and is the warped product (.[0, ∞), dr 2 ) ×|D f | (S n−1 , gcan ∗ n−1 n where .gcan is the round metric on . S , or (ii) . f has no critical point and (. M , g, f ) is the warped product. (.R, dr 2 ) ×|D f | (N n−1 , g ∗ ), where (. N , g ∗ ) is a Riemannian manifold of constant scalar curvature .r ∗ . Moreover, if (. M n , g, f ) has non-negative Ricci curvature, then (. M n , g) is isometric to the Riemannian product (. R, dr 2 ) × (N n−1 , g ∗ ); if the scalar curvature .r ≥ 0 on . M n , then either .r ∗ > 0, or .r = r ∗ = 0 and (. M n , g) is isometric to the Riemannian product (. R, dr 2 ) × (N n−1 , g ∗ ). In particular, for a locally conformally flat case, they proved the following result. Theorem 10.8 (Cao–Sun–Zhang) Let (. M n , g, f ) be a non-trivial complete gradient locally conformally flat Yamabe soliton such that . f has no critical point. Then (. M n , g, f ) is the warped product (. R, dr 2 ) ×|D f | (N n−1 , g ∗ ) where (. N n−1 , g ∗ ) is of constant sectional curvature. The proofs of the above two results make use of the following result derived in [14]. Proposition 10.3 Let (. M n , g, f ) be a complete gradient Yamabe soliton, and .Σc − f −1 (c) be a regular level surface. Then (a) .|D f |2 is constant on .Σc , (b) the scalar curvature .r is constant on .Σc , (c) the second fundamental form of .Σc is given by r −ρ r −ρ . Aab = is constant on .Σc and (e) in g , (d) the mean curvature . A = (n − 1) |D f| |D f | ab any open neighborhood .Uαβ = f −1 ((α, β)) of .Σc in which . f has no critical points, and the solution metric .g can be expressed as ds 2 = dt 2 + ( f , (t))2 gt∗0 ,

.

where (.θ 2 , ..., θ n ) is any local coordinate system on .Σc and .gt∗0 = gab (t0 , θ )dθ a dθ b is the induced metric on .Σc = t −1 (t0 ). The part (a) was observed in [10], and parts (b–d) were proved in [10] assuming that (. M n , g, f ) is locally conformally flat. Subsequently, Catino, Mantegazza and Mazzieri [15] established the following classification result for non-negative Ricci tensor.

10.2 Yamabe Flow and Yamabe Solitons

149

Theorem 10.9 (Catino–Mantegazza–Mazzieri) Let (. M n , g) be a complete noncompact gradient Yamabe soliton with non-negative Ricci tensor and non-constant potential function . f . Then, we have the following two cases: (1) either (. M n , g) is a direct product .R × N n−1 where .(N n−1 , g N ) is an (.n − 1)-dimensional complete Riemannian manifold with non-negative Ricci tensor. If, in addition, (. M, g) is locally conformally flat, then either it is flat or the manifold (. N n−1 , g N ) is a quotient of the round sphere . S n−1 . (2) Or (. M n , g) is rotationally symmetric and globally conformally equivalent to n n n .R . More precisely, there exists a point . O ∈ M such that on . M \{O}, the metric has the form 2 2 2 S n−1 . g = v (t)(dt + t g ), where .v : R+ → R is some positive smooth function. The following result follows as a corollary. Corollary 10.1 Let (. M n , g) be a complete non-compact gradient Yamabe soliton with non-negative Ricci tensor and non-constant potential function . f . If the Ricci tensor is positive definite at some point, then (. M n , g) is rotationally symmetric and globally equivalent to .Rn , in particular, it is locally conformally flat. This result is an improvement over the corresponding result in [10] because it removes the locally conformally flat assumption and relaxes the assumption on the sectional secure. Next, we mention a result obtained by Ma and Miquel [16] that imposes some condition on the scalar curvature to show that it is constant. Theorem 10.10 (Ma–Miquel) Let (. M, }g) be a complete non-compact gradient Yamabe soliton such that .|r − ρ| ∈ L 1 (M), . M Ric(D f, D f ) ≤ 0 and the potential function satisfy .| f (x)| ≤ Cd(x, x0 )2 , .|D f | ≤ C(1 + d(x, x0 )2 ) near infinity, where .C is some uniform positive constant and .d(x, x0 ) is the distance function from the point . x to a fixed point . x 0 . Then .r = ρ on (. M, g). Argument of this result follows from the following proposition g) be } “Let (. M, 2 .n(n − 1) (r − ρ) dv a} gradient Yamabe soliton with smooth boundary. Then g − M } Ric(D f, D f )dv = (n − 1) (r − ρ)(ν f )dσ , where . ν is the unit outward norg M ∂M mal to .∂ M and .σ is the surface element of .∂ M”. In particular, for the non-expanding case, we recall the following results of Maeta [17]. Theorem 10.11 (Maeta, [17]) Let (. M, g) be a steady or shrinking complete gradient Yamabe soliton with non-positive Ricci curvature. If the scalar curvature .r ∈ L p (M) for some .0 < p < ∞, then . M is Ricci-flat. Theorem 10.12 (Maeta, [17]) Let (. M, g) be a steady or shrinking complete gradient Yamabe soliton with non-positive scalar curvature and .Ric ≥ φrg for some nonnegative function .φ on . M. If the scalar curvature .r ∈ L p (M) for some .0 < p < ∞, then the scalar curvature is zero.

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10 Yamabe Solitons

In [18], Maeta classified 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor and obtained the following result. Theorem 10.13 (Maeta, [18]) Let (. M 3 , g, f ) be a non-trivial non-flat 3-dimensional complete gradient Yamabe soliton with divergence-free Cotton tensor. I. If . M is steady, then it is rotationally symmetric and equal to the warped product ([0, ∞), dr 2 ) ×|D f | (S 2 , g S∗ ), where .g S∗ is the round metric on . S 2 . II. If . M is shrinking, then either (1) . M is rotationally symmetric and equal to the warped product (.[0, ∞), dr 2 ) ×|D f | (S 2 , g S∗ ), where .g S∗ is the round metric on . S 2 , or (2) .|D f | is constant and . M is isometric to the Riemannian product .(R, dr 2 ) × (S 2 ( 21 ρ|D f |2 ), g ∗ ) where . S 2 ( 21 ρ|D f |2 ) is the sphere of constant Gaussian curvature 1 2 . ρ|D f | . 2 III. If . M is expanding, then either (1) it is rotationally symmetric and equal to the warped product .([0, ∞), dr 2 ) ×|D f | (S 2 , g S∗ ), where .g S∗ is the round metric on . S 2 , or (2) .|D f | is constant and . M is isometric to the Riemannian product 2 ∗ 2 1 2 2 2 1 .(R, dr ) × (H ( ρ|D f | ), g ), where .H ( ρ|D f | ) is the hyperbolic space of con2 2 1 2 stant Gaussian curvature . 2 ρ|D f | .

.

Its proof is based on tensorial equations derived from Yamabe equation, the expression for Cotton tensor and the result of Cao, Sun and Zhang, stated earlier. We also mention the following result of Shaikh et al. [19]. Theorem 10.14 (Shaikh–Cunha–Mandal) Let (. M, g, f, λ) be a complete noncompact gradient Yamabe soliton of} dimension .n with non-negative scalar curvature f and its potential function . f satisfy . M−B(q,a) d(x,q) 2 < ∞, where . B(q, a) is a ball of radius .a > 0 and centered at .q and .d(x, q) is the distance function from some fixed point .q ∈ M. (i) If . M is expanding, then it is isometric to .Rn . (ii) If . M is steady, then . f is harmonic. Even though the authors did not include completeness and non-compactness in their original statements, these two assumptions are relevant. Its proof uses the cut-off function.ψa ∈ C0∞ (B(q, 2a)) for.a > 0 such that.0 ≤ ψa ≤ 1 in. B(q, 2a),.ψa = 1 in C C 2 . B(q, a), .|∇ψa | ≤ 2 in . B(q, 2a) and ./\ψa ≤ 2 in . B(q, 2a), where .C is a constant. a a In case (ii), it is noteworthy that .r = 0 and .Hess f = 0, and hence . M is isometric to the product of a real line and a scalar-flat Riemannian manifold. It is also interesting to study Yamabe solitons on submanifolds of a Riemannian manifold. For an isometric immersion .i : (M, g) → Em of a Riemannian manifold (. M, g) into a Euclidean .m-space .Em , we denote by .xT and .x N the tangential and normal components of the position vector field .x of . M in .Em , respectively. In this direction, Chen and Deshmukh [20] have proved the following result. Theorem 10.15 (Chen–Deshmukh) Let (. M, g) be a Riemannian manifold. Then an isometric immersion .i : (M, g) → S m−1 (a) ⊂ Em of . M into the hypersphere m−1 .S (a) of radius .a with center at the origin is a Yamabe soliton with .xT as its soliton field if and only if (. M, g) has constant scalar curvature.

10.3 Conformal Solutions of Yamabe Flow

151

The proof uses the standard equations of submanifold theory. Their next result provides the following classification of Yamabe soliton on Euclidean hypersurfaces with .xT as the soliton vector field: “Let (. M, g) be a Euclidean hypersurface of .En+1 . Then (. M, g, xT , λ) is a Yamabe soliton if and only if either (1) .λ = −1 and . M is an open part of a hyperplane of .En+1 , or (2) .λ = r > 0 and . M is an open part of a hypersphere of .En+1 centered at the origin.”

10.3 Conformal Solutions of Yamabe Flow Just as Ricci soliton was generalized to Ricci almost soliton (Chapter 9), in the same vein, Barbosa and Ribeiro Jr. [21] generalized Yamabe soliton to almost Yamabe soliton defined by Eq. (10.5) where the constant .ρ is taken as a smooth function on . M. The almost Yamabe soliton will be shrinking, steady or expanding accordingly as .ρ > 0, = 0, < 0 respectively, otherwise it will be indefinite. It is gradient when . V = D f for some potential function . f . It will be called trivial when either . V is trivial or . f is constant on . M. Examples of an almost Yamabe soliton are (i) . S n , (ii) Gaussian soliton on .Rn with the standard Euclidean metric and the potential function . f (x) = ρ2 |x|2 where n=1 .ρ is constant and (iii) . M = R ×cosh t S n with metric .g = dt 2 + cosh2 tg0 where n . g0 is the standard metric of . S , . f (x, t) = sinh t and .λ(x, t) = sinh t + n. The following results were obtained in [21]. Theorem 10.16 (Barbosa–Ribeiro Jr.) The soliton vector field .V of an almost Yamabe soliton (. M n , g, V,} ρ) is Killing if one of the following holds: 1. . M is compact and .} M [ 21 Ric(V, V ) + (n − 2) < Dρ, V >]dvg ≤ 0. 2. . M is compact and . M < Dρ, V > dvg ≤ 0. 3. .|V | ∈ L1 (M) and either .r ≥ ρ or .r ≤ ρ. Corollary 10.2 A compact Riemannian manifold of negative Ricci curvature cannot be a non-trivial almost Yamabe soliton. Theorem 10.17 (Barbosa–Ribeiro Jr) Let (. M n , g, D f, ρ) be a non-trivial gradient almost Yamabe soliton. The following statements hold: 1.. M is compact with constant scalar curvature.r , then. M n is isometric to a Euclidean sphere . S n . 2. Suppose that .r − ρ is a nonzero constant function and . M n is complete and noncompact. Then . M n is isometric to a Euclidean space .Rn . Corollary 10.3 Let (. M n , g, V, ρ) be a compact non-trivial almost Yamabe soliton with closed conformal vector field .V vanishing at some point of . M n . If . M n has constant scalar curvature, then (. M n , g) admits a gradient structure. Theorem 10.18 (Barbosa–Ribeiro Jr.) Let (. M n , g, D f, ρ) be a homogeneous gradient almost Yamabe soliton. Then one of the following statements hold: 1. . M n is the round sphere . S n . 2. . M n is the Gaussian soliton. 3. . M n is the direct product . N n−1 × I of an Einstein manifold . N n−1 with a line . I .

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Proofs of the above results use the following lemma. Lemma 10.1 For an almost Yamabe soliton (. M n , g, V, ρ), .

1 /\|V |2 + (n − 2) < D(r − ρ), V >= |∇V |2 − Ric(V, V ). 2

For a gradient almost Yamabe soliton (. M n , g, D f, ρ), n(r − ρ) = /\ f, (n − 1)Dr + Ric(D f ) = (n − 1)Dρ

.

and (n − 1)/\r +

.

1 < Dr, D f > +r (r − ρ) = (n − 1)/\ρ. 2

10.4 Yamabe Solitons in Contact Geometry In [22], Sharma studied Yamabe soliton in contact Riemannian geometry, and proved the following result. Theorem 10.19 (Sharma) A 3-dimensional Yamabe soliton whose metric is Sasakian has constant scalar curvature and the soliton vector field .V is Killing. The proof dwells on the following expression for Ricci tensor for a 3-dimensional Sasakian manifold (Blair, Koufogiorgos and Sharma [23]): .

r r Ric = ( − 1)g + (3 − )η ⊗ η, 2 2

evolution of Ricci tensor along the soliton vector field (which is conformal) as given in Chap. 4, and the properties of a contact metric structure. Recently, this result has been extended to higher dimensions by Poddar, Balasubramanian and Sharma [24], in the form of the following result. Theorem 10.20 Let .(M, g) be a Yamabe soliton with soliton vector field .V . If .g is a Sasakian metric, then it has constant scalar curvature and .V is Killing. The crucial idea of the proof is based on the following result of Okumura [25]. Theorem 10.21 (Okumura) A conformal vector field .V (defined by .LV g = 2σ g) on a Sasakian manifold of dimension .2n + 1 > 3 differs from .−Dσ by a Killing vector field, i.e. .V = W − Dσ where .W is a Killing vector field on . M, and the conformal scale function .σ satisfies the concircular equation ∇∇σ = −σ g,

.

and that .σ = r − λ for a Yamabe soliton.

10.5 Concluding Remarks

153

Furthermore, it has been shown in [24] that the same conclusion holds (as in the Sasakian case) on a (.2n + 1)-dimensional . K -contact manifold . M, under some additional conditions. More precisely, the following result was established. Theorem 10.22 (Poddar–Balasubramanian–Sharma) Let .(M, g) be a (.2n + 1)dimensional Yamabe soliton with soliton vector field .V , and .g be a . K -contact metric. Then its scalar curvature is constant and .V is Killing, if any one of the following conditions hold: (i) The scalar curvature is constant along .V . (ii) .V is an eigenvector of the Ricci operator with eigenvalue .2n. (iii) .V is gradient. In case (iii), the Yamabe soliton becomes trivial (i.e. . f is constant). The proof of this result uses Eq. (8.14) of Chap. 8, Lie derivative of curvature tensor, Ricci tensor and scalar curvature along a conformal vector field, given in [26]. The condition (ii) in the above-mentioned theorem is motivated by the fact that, if we take .V as the Reeb vector field .ξ , then the Yamabe soliton equation (10.5) implies .r = ρ because .ξ is Killing for a . K -contact metric. Remarks: 1. In the proof of the above theorem, it was shown for a Yamabe soliton as a . K -contact manifold that the conformal scale function .r − ρ is an eigenfunction of the Laplacian (–.∇ i ∇i ) with eigenvalue .4n. This implies that, if the eigenvalues of the Laplacian are different from .4n, then .r = ρ, i.e. the metric of the . K -contact Yamabe soliton in dimension .> 3 is a Yamabe metric. This assumption holds for the unit sphere . S 2n+1 (which is Sasakian and hence . K -contact) because the spectral values of the Laplacian acting on functions on . S 2n+1 are well known (Tanno, p. 91 in [27]) to be .k(k + 2n) for .k = 0, 1, 2, ..., ∞ and hence do not include .4n. 2. It was also pointed out in [24] that, under the hypothesis of the above-mentioned theorem, the Yamabe soliton vector field .V commutes with the Reeb vector field .ξ , i.e. .[V, ξ ] = 0.

10.5 Concluding Remarks Thus, the classification of a . K -contact manifold as a Yamabe soliton seems to be a formidable problem and is an open question. In addition, it would be interesting to study Yamabe solitons in Lorentzian geometry (in particular, space-time manifolds of general relativity).

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References 1. Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960) 2. Trudinger, N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa 22, 265–274 (1968) 3. Aubin, T.: Equations differerentielles non lineares et probleme de Yamabe concernant la coubure scalaire. J. Math. Pures Appl. 55, 269–296 (1976) 4. Schoen, R.M.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984) 5. Schoen, R.M., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979) 6. Brendle, S., Marques, F.C.: Recent progress on the Yamabe problem, arXiv: 1010.4960v1 [math.DG] (2010) 7. Hamilton, R.S.: Ricci flow on surfaces. Contemp. Math. (AMS) 71, 237–261 (1988) 8. Burchard, A., McCann, R.J., Smith, A.: Explicit Yamabe flow of an asymptotic cigar. Methods Appl. Anal. 15, 065–080 (2008) 9. Cerbo, L.F.C., Disconzi, M.M.: Yamabe solitons, determinant of the Laplacian and the uniformization theorem for Riemannian surfaces. Lett. Math. Phys. 83, 13–18 (2008) 10. Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346–369 (2013) 11. Hsu, S.Y.: A note on compact gradient Yamabe solitons. J. Math. Anal. Appl. 388, 725–726 (2012) 12. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, Lectures in Contemporary Mathematics. Science Press and Graduate Studies in Mathematics, vols. 3, 77. American Mathematical Society (Copublication) (2006) 13. Bourgignon, J.-P., Ezin, J.-P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. A.M.S. 301, 723–736 (1987) 14. Cao, H.-D., Sun, X., Zhang, Y.: On the structure of gradient Yamabe solitons. Math. Res. Lett. 19 (2011). arXiv: 1108.6316v2 [math.DG] 22 Sept 2011 15. Catino, G., Mantegazza, C., Mazzieri, L.: On the global structure of conformal gradient solitons with nonnegative Ricci tensor. Commun. Contemp. Math. 14, 12 (2012) 16. Ma, L., Miquel, V.: Remarks on scalar curvature of Yamabe solitons. Ann. Glob. Anal. Geom. 42, 195–205 (2012) 17. Maeta, S.: Complete Yamabe solitons with finite total scalar curvature. Differ. Geom. Appl. 66, 75–81 (2019) 18. Maeta, S.: 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. Ann. Glob. Anal. Geom. 58, 227–237 (2020) 19. Shaikh, A.A., Cunha, A.W., Mandal, P.: Some characterizations of gradient Yamabe solitons. J. Geom. Phys. 167, 104293 (1–6) (2021) 20. Chen, B.Y., Deshmukh, S.: Yamabe and quasi-Yamabe solitons on Euclidean submanifolds. Mediterr. J. Math. 15, 9 (2018) 21. Barbosa, E., Ribeiro, Jr., E.: On conformal solutions of the Yamabe flow. Arch. Math. 101, 79–89 (2013) 22. Sharma, R.: A 3-dimensional Sasakian metric as a Yamabe soliton. Internat. J. Geom. Methods Mod. Phys. 9 1220003 (5 pages) (2012) 23. Blair, D.E., Koufogiorgos, T., Sharma, R.: A class of 3-dimensional contact metric manifolds with . Qϕ = ϕ Q. Kodai Math. J. 13, 391–401 (1990) 24. Poddar, R., Balasubramanian, R., Sharma, R.: Yamabe solitons in contact geometry. To appear in New Zealand J. Math. 25. Okumura, M.: On infinitesimal conformal and projective transformations of normal contact spaces. Tohoku Math. J. 14, 398–412 (1962) 26. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970) 27. Tanno, S.: Promenades on Spheres. Department of Mathematics, Tokyo Institute of Technology (1996)

Index

A Almost analytic vector field, 104, 113 Almost complex manifold, 104 Almost complex structure, 104 Almost contact structure, 109 Almost Hermitian manifold, 105 Almost Kähler manifold, 106 Almost Yamabe soliton, 151 Automorphism group, 123

B Bach-flat, 33, 98 Bach tensor, 32 Bakry–Émery tensor, 135 Bianchi’s identities, 4 Bochner’s lemma, 49 Bochner–Weitzenbock formula, 10 Bryant soliton, 91

C Calabi–Yau manifolds, 106 Calibration, 44 Cartan’s magic formula, 22 Cayley multiplication, 106 Chart, 1 Christoffel symbols of second kind, 7 Cigar soliton, 91 Closed form, 6 Co-differential operator, 51 Complete vector field, 21 Complex analytic, 103 Complex Euclidean space, 105

Complex manifold, 103 Complex projective space, 103, 105 Complex space form, 105 Conformal class, 143 Conformal diffeomorphism, 28 Conformal Einstein equation, 32 Conformal Killing vector field, 39 Conformally Einstein, 31 Conformally equivalent, 27 Conformally flat, 30 Conformal map, 27 Conformal scale function, 43 Conformal symmetry, 39 Conformal transformation, 28 Conformal vector field, 39 Connection, 3 Connection coefficients, 4 Constant curvature, 11 Contact 1-form, 107 Contact distribution, 107 Contact manifold, 107 Contact metric structure, 107 Contact structure, 107 Coordinate system, 1 Cotangent space, 3 Cotton tensor, 30 Covariant derivative, 3 .C R sphere, 45 CR structure, 44 Curvature tensor, 4 Cut-off function, 150

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 R. Sharma and S. Deshmukh, Conformal Vector Fields, Ricci Solitons and Related Topics, Infosys Science Foundation Series, https://doi.org/10.1007/978-981-99-9258-4

155

156 D Darboux coordinates, 107 . D-homothetically fixed, 110 . D-homothetic deformation, 110 Differential . p-form, 5 Dilatons, 100 Distribution, 14 Divergence of a vector field, 10 Dual vector space, 3

E Effectively, 19 Einstein manifold, 9 Einstein metric, 9 Einstein’s field equations, 11 Einstein’s summation convention, 3 Embedded submanifold, 12 Embedding, 12 Entropy functional, 100 Essential conformal vector field, 41 Exact form, 6 Exterior algebra, 5 Exterior derivative, 5 Extrinsic curvature, 13

F First Chern class, 106 Foliation, 14 Freely, 19 Fundamental 2-form, 105 Fundamental collineation, 107 Futaki invariant, 120

Index Harmonic coordinates, 89 Hawking–Ellis construction, 7 Heat flow, 89 Heisenberg group, 45 Hermitian metric, 105 Hessian, 10 Hodge Laplacian, 51 Holomorphic, 103 Holomorphically planar conformal vector, 113 Holomorphic section, 105 Holomorphic sectional curvature, 105 Holomorphic vector field, 104 Homogeneous gradient Ricci soliton, 97 Homogeneous space, 19, 20 Homothetic vector field, 39 Hopf’s lemma, 48 Hypersurface, 12

I Immersed submanifold, 12 Immersion, 12 Index, 7 Infinitesimal automorphism, 104, 108 Infinitesimal contact transformation, 108 Infinitesimal harmonic transformation, 92 Infinitesimal isometry, 33 Integral curve, 5 Integral manifolds, 14 Inversion, 27 Involutive, 14 Isotropy group, 20

J Jacobi identity, 18

G Gauss and Codazzi–Mainardi equations, 13 Gauss’ divergence Theorem, 47 Gauss equation, 13 Gauss formula, 12 Gaussian soliton, 91, 151 Gauss–Kronecker curvature, 13 General complex linear group, 18 Generalized .m-quasi-Einstein, 135 General linear group, 18 Geodesic, 5 Gradient of a smooth function, 10 Gradient Ricci almost soliton, 129 Gradient Ricci soliton, 90

K Kaehler–Ricci soliton, 97 Kähler manifold, 105 Kähler–Ricci soliton, 118 . K -contact, 108 Killing equation, 34 Killing vector field, 34 Kodaira–Thurston manifold, 106 Koszul’s formula, 8 (.k, μ)-contact manifold, 110 Kulkarni–Nomizu product, 31

H Hamilton–Ivey estimate, 96

L Laplacian, 10

Index Left and right multiplications, 18 Left-invariant metric, 19 Levi-Civita connection, 8 Levi form, 44 Lichnerowicz conjecture, 44 Lie algebra, 3, 18 Lie bracket, 2 Lie derivative, 21 Lie group, 17 Lie transformation group, 19 Lightlike, 7 Lightlike submanifold, 12 Liouville map, 28 Liouville theorem, 27 Local orthonormal frame, 8

M Manifold, 1 Manifold with boundary, 2 Maximal atlas, 1 Mean curvature, 13 Mean curvature vector, 13 Metric tensor, 6 .m-quasi-Einstein, 135

N Nearly Kähler manifold, 106 Newlander–Nirenberg theorem, 104 Nijenhuis tensor, 104 Normalized Ricci flow, 88 Normalized Yamabe flow, 145 Normal space, 12 Null cone, 7 .η-Einstein, 109 Null .η-Einstein, 120 Null vectors, 7

O Obata’s theorem, 133 Orthogonal group, 18

P Parallel, 5 Poincaré Conjecture, 87 Poincare Lemma, 6 Positive mass theorem, 144 Potential function, 90 Principal curvatures, 13 Principal directions, 13 Pull-back differential map, 6

157 R Reaction–diffusion equation, 89 Reeb vector field, 107 Ricci almost soliton, 129 Ricci–DeTurck flow, 88 Ricci flow, 87 Ricci form, 105 Ricci soliton, 90 Ricci tensor, 8 Riemann surface, 103 Robertson–Walker space-time, 14 Rotationally symmetric, 91 Rough Laplacian, 49

S Sasakian, 108 Sasakian manifolds, 108 Sasakian space form, 108, 112 Scalar curvature, 9 Schouten tensor, 30 Schur’s lemma, 92 Schur’s theorem, 11 Schwarzschild exterior space-time, 14 Second fundamental form, 13 Sectional curvature, 11 Semi-Riemannian manifold, 6 Shape operator, 13 Signature, 7 Simply connected, 19 Singular solutions of Ricci flow, 90 Smooth manifold, 1 Solitary wave, 87 Solvable Lie group, 92, 124 Space form, 11 Spacelike, 7 Spacelike vectors, 7 Special complex linear group, 18 Special linear group, 18 Special orthogonal group, 18 Special unitary group, 18 Stokes’ theorem, 47 Strictly pseudoconvex, 44 Structure constants, 19 Structure equations, 19 Symplectic form, 106 Symplectic manifold, 106

T Tangent space, 2 Tangent vector, 2 Taub–NUT metric, 106

158 Tensor field, 3 Thurston’s geometrization conjecture, 87 Timelike, 7 Timelike vectors, 7 Torsion of a connection, 4 Totally geodesic, 13 Totally umbilical, 13 Total umbilicity, 13 Transitively, 19 Transverse Calabi–Yau, 109 Transverse Kähler metric, 108 U Uniformization theorem, 143 Unitary group, 18 V Vector field, 2

Index W Warped product, 14 Webster metric, 44 Weyl conformal tensor, 9 Weyl–Schouten theorem, 31 Weyl’s connection, 28

Y Yamabe conjecture, 143 Yamabe constant, 144 Yamabe equation, 143 Yamabe flow, 144 Yamabe functional, 144 Yamabe soliton, 145 Yano operator, 51, 92