Geometric Realizations Of Curvature

by Miguel Brozos-vazquez, Peter B Gilkey, Stana Z Nikcevic

Overview

A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer-Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri-Vanhecke decomposition, the Gray-Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions.The book organizes, in one coherent volume, the results of research completed by many different investigators over the past 30 years. Complete proofs are given of results that are often only outlined in the original publications. Whereas the original results are usually in the positive definite (Riemannian setting), here the authors extend the results to the pseudo-Riemannian setting and then further, in a complex framework, to para-Hermitian geometry as well. In addition to that, new results are obtained as well, making this an ideal text for anyone wishing to further their knowledge of the science of curvature.

Product Details

ISBN-13:	9781848167414
Publisher:	Imperial College Press
Publication date:	03/19/2012
Series:	Icp Advanced Texts In Mathematics , #6
Pages:	264
Product dimensions:	6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface v

1 Introduction and Statement of Results 1

1.1 Notational Conventions 4

1.2 Representation Theory 8

1.3 Affine Structures 10

1.4 Mixed Structures 13

1.5 Affine Kähler Structures 16

1.6 Riemannian Structures 19

1.7 Weyl Geometry I 21

1.8 Almost Pseudo-Hermitian Geometry 23

1.9 The Gray Identity 25

1.10 Kähler Geometry in the Riemannian Setting I 27

1.11 Curvature Kähler-Weyl Geometry 28

1.12 The Covariant Derivative of the Kähler Form I 31

1.13 Hyper-Hermitian Geometry 34

2 Representation Theory 37

2.1 Modules for a Group G 37

2.2 Quadratic Invariants 44

2.3 Weyl's Theory of Invariants 47

2.4 Some Orthogonal Modules 53

2.5 Some Unitary Modules 58

2.6 Compact Lie Groups 63

3 Connections, Curvature, and Differential Geometry 69

3.1 Affine Connections 69

3.2 Equiaffine Connections 72

3.3 The Levi-Civita Connection 73

3.4 Complex Geometry 77

3.5 The Gray Identity 81

3.6 Kähler Geometry in the Riemannian Setting II 84

4 Real Affine Geometry 89

4.1 Decomposition of 21 and R as Orthogonal Modules 91

4.2 The Modules R, S²_o, and Λ² in U 99

4.3 The Modules W^o₆, W^o₇, and W^o₈ in U 104

4.4 Decomposition of U as a General Linear Module 106

4.5 Geometric Readability of Affine Curvature Operators 111

4.6 Decomposition of U as an Orthogonal Module 124

5 Affine Kähler Geometry 125

5.1 Affine Kähler Curvature Tensor Quadratic Invariants 125

5.2 The Ricci Tensor for a Kähler Affine Connection 134

5.3 Constructing Affine (Para)-Kähler Manifolds 136

5.4 Affine Kahler Curvature Operators 140

5.5 Affine Para-Kähler Curvature Operators 149

5.6 Structure of RU± as a GI± Module 155

6 Riemannian Geometry 173

6.1 The Riemann Curvature Tensor 174

6.2 The Weyl Conformal Curvature Tensor 178

6.3 The Cauchy-Kovalevskaya Theorem 180

6.4 Geometric Realizations of Riemann Curvature Tensors 181

6.5 Weyl Geometry II 183

7 Complex Riemannian Geometry 189

7.1 The Decomposition of R as Modules over U± 190

7.2 The Submodules of R Arising from the Ricci Tensors 204

7.3 Para-Hermitian and Pseudo-Hermitian Geometry 210

7.4 Almost Para-Hermitian and Almost Pseudo-Hermitian Geometry 212

7.5 Kähler Geometry in the Riemannian Setting III 213

7.6 Complex Weyl Geometry

7.7 The Covariant Derivative of the Kähler Form II 221

Notational Conventions 235

Bibliography 239

Index 249