This book, which focuses on the study of curvature, is an introduction to various aspects of pseudo-Riemannian geometry. We shall use Walker manifolds (pseudo-Riemannian manifolds which admit a non-trivial parallel null plane field) to exemplify some of the main differences between the geometry of Riemannian manifolds and the geometry of pseudo-Riemannian manifolds and thereby illustrate phenomena in pseudo-Riemannian geometry that are quite different from those which occur in Riemannian geometry, i.e. for indefinite as opposed to positive definite metrics. Indefinite metrics are important in many diverse physical contexts: classical cosmological models (general relativity) and string theory to name but two. Walker manifolds appear naturally in numerous physical settings and provide examples of extremal mathematical situations as will be discussed presently. To describe the geometry of a pseudo-Riemannian manifold, one must first understand the curvature of the manifold. We shall analyze a wide variety of curvature properties and we shall derive both geometrical and topological results. Special attention will be paid to manifolds of dimension 3 as these are quite tractable. We then pass to the 4 dimensional setting as a gateway to higher dimensions. Since the book is aimed at a very general audience (and in particular to an advanced undergraduate or to a beginning graduate student), no more than a basic course in differential geometry is required in the way of background. To keep our treatment as self-contained as possible, we shall begin with two elementary chapters that provide an introduction to basic aspects of pseudo-Riemannian geometry before beginning on our study of Walker geometry. An extensive bibliography is provided for further reading. Table of Contents: Basic Algebraic Notions / Basic Geometrical Notions / Walker Structures / Three-Dimensional Lorentzian Walker Manifolds / Four-Dimensional Walker Manifolds / The Spectral Geometry of the Curvature Tensor / Hermitian Geometry / Special Walker Manifolds
Author(s): Eduardo García-Río, Miguel Brozos-Vázquez, Rámon Vázquez-Lorenzo, Stana Nikcevic
Series: Synthesis Lectures on Mathematics and Statistics
Publisher: Morgan and Claypool Publishers
Year: 2009
Language: English
Pages: 177
Preface......Page 15
Introduction......Page 19
A Historical Perspective in the Algebraic Context......Page 20
Jordan Normal Form......Page 21
Indefinite Geometry......Page 22
Algebraic Curvature Tensors......Page 23
Hermitian and Para-Hermitian Geometry......Page 24
Sectional, Ricci, Scalar, and Weyl Curvature......Page 26
Curvature Decompositions......Page 27
Self-Duality and Anti-Self-Duality Conditions......Page 29
Spectral Geometry of the Curvature Operator......Page 30
Osserman and Conformally Osserman Models......Page 31
Osserman Curvature Models in Signature (2,2)......Page 32
Ivanov--Petrova Curvature Models......Page 35
Osserman Ivanov--Petrova Curvature Models......Page 36
Commuting Curvature Models......Page 37
Basic Manifold Theory......Page 39
The Tangent Bundle, Lie Bracket, and Lie Groups......Page 40
The Cotangent Bundle and Symplectic Geometry......Page 41
Connections, Curvature, Geodesics, and Holonomy......Page 43
Pseudo-Riemannian Geometry......Page 45
The Levi-Civita Connection......Page 46
Associated Natural Operators......Page 47
Weyl Scalar Invariants......Page 48
Null Distributions......Page 49
Pseudo-Riemannian Holonomy......Page 50
Pseudo-Hermitian and Para-Hermitian Structures......Page 51
Hyper-Para-Hermitian Structures......Page 52
Geometric Realizations......Page 53
Homogeneous Spaces, and Curvature Homogeneity......Page 54
Technical Results in Differential Equations......Page 55
Historical Development......Page 57
Walker Coordinates......Page 58
Examples of Walker Manifolds......Page 61
Locally Conformally Flat Metrics with Nilpotent Ricci Operator......Page 62
Degenerate Pseudo-Riemannian Homogeneous Structures......Page 63
Para-Kaehler Geometry......Page 64
Two-step Nilpotent Lie Groups with Degenerate Center......Page 65
Conformally Symmetric Pseudo-Riemannian Metrics......Page 66
The Affine Category......Page 67
Twisted Riemannian Extensions Defined by Flat Connections......Page 69
Nilpotent Walker Manifolds......Page 71
Osserman Riemannian Extensions......Page 72
Ivanov--Petrova Riemannian Extensions......Page 73
History......Page 75
Adapted Coordinates......Page 76
The Jordan Normal Form of the Ricci Operator......Page 77
Christoffel Symbols, Curvature, and the Ricci Tensor......Page 78
Locally Symmetric Walker Manifolds......Page 79
Einstein-Like Manifolds......Page 80
The Spectral Geometry of the Curvature Tensor......Page 82
Curvature Commutativity Properties......Page 83
Foliated Walker Manifolds......Page 84
Contact Walker Manifolds......Page 87
Strict Walker Manifolds......Page 88
Three dimensional homogeneous Lorentzian manifolds......Page 89
Three dimensional Lie groups and Lie algebras......Page 90
Diagonalizable Ricci Operator......Page 92
Type II Ricci Operator......Page 93
History......Page 95
Four-Dimensional Walker Manifolds......Page 96
Almost Para-Hermitian Geometry......Page 101
Isotropic Almost Para-Hermitian Structures......Page 102
Characteristic Classes......Page 103
Self-Dual Walker Manifolds......Page 104
History......Page 107
Osserman Metrics with Diagonalizable Jacobi Operator......Page 108
Osserman Walker Type II Metrics......Page 109
Osserman and Ivanov--Petrova Metrics......Page 113
Affine Surfaces with Skew Symmetric Ricci Tensor......Page 115
Affine Surfaces with Symmetric and Degenerate Ricci Tensor......Page 117
Riemannian Extensions with Commuting Curvature Operators......Page 118
Other Examples with Commuting Curvature Operators......Page 119
History......Page 121
The Proper Almost Hermitian Structure of a Walker Manifold......Page 122
Proper Almost Hyper-Para-Hermitian Structures......Page 124
Hermitian Walker Manifolds of Dimension Four......Page 125
Proper Hermitian Walker Structures......Page 126
Locally Conformally Kaehler Structures......Page 129
Almost Kaehler Walker Four-Dimensional Manifolds......Page 132
History......Page 137
Curvature Commuting Conditions......Page 138
Curvature Homogeneous Strict Walker Manifolds......Page 143
Bibliography......Page 147
Glossary......Page 167
Biography......Page 169
Index......Page 171
