Geometric Realizations of Curvature

by Vazquez, Miguel Brozos; Gilkey, Peter B.; Nikcevic, Stana

• ISBN13: 9781848167414
• ISBN10: 1848167415
• Format: Hardcover
• Copyright: 2012-03-16
• Publisher: World Scientific Pub Co Inc

Summary

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 SingerThorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the TricerriVanhecke decomposition, the GrayHervella 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.

Table of Contents

Preface	p. v
Introduction and Statement of Results	p. 1
Notational Conventions	p. 4
Representation Theory	p. 8
Affine Structures	p. 10
Mixed Structures	p. 13
Affine Kähler Structures	p. 16
Riemannian Structures	p. 19
Weyl Geometry I	p. 21
Almost Pseudo-Hermitian Geometry	p. 23
The Gray Identity	p. 25
Kähler Geometry in the Riemannian Setting I	p. 27
Curvature Kähler-Weyl Geometry	p. 28
The Covariant Derivative of the Kähler Form I	p. 31
Hyper-Hermitian Geometry	p. 34
Representation Theory	p. 37
Modules for a Group G	p. 37
Quadratic Invariants	p. 44
Weyl's Theory of Invariants	p. 47
Some Orthogonal Modules	p. 53
Some Unitary Modules	p. 58
Compact Lie Groups	p. 63
Connections, Curvature, and Differential Geometry	p. 69
Affine Connections	p. 69
Equiaffine Connections	p. 72
The Levi-Civita Connection	p. 73
Complex Geometry	p. 77
The Gray Identity	p. 81
Kähler Geometry in the Riemannian Setting II	p. 84
Real Affine Geometry	p. 89
Decomposition of 21 and R as Orthogonal Modules	p. 91
The Modules R, S2o, and ¿2 in U	p. 99
The Modules Wo6, Wo7, and Wo8 in U	p. 104
Decomposition of U as a General Linear Module	p. 106
Geometric Readability of Affine Curvature Operators	p. 111
Decomposition of U as an Orthogonal Module	p. 124
Affine Kähler Geometry	p. 125
Affine Kähler Curvature Tensor Quadratic Invariants	p. 125
The Ricci Tensor for a Kähler Affine Connection	p. 134
Constructing Affine (Para)-Kähler Manifolds	p. 136
Affine Kahler Curvature Operators	p. 140
Affine Para-Kähler Curvature Operators	p. 149
Structure of RU± as a GI± Module	p. 155
Riemannian Geometry	p. 173
The Riemann Curvature Tensor	p. 174
The Weyl Conformal Curvature Tensor	p. 178
The Cauchy-Kovalevskaya Theorem	p. 180
Geometric Realizations of Riemann Curvature Tensors	p. 181
Weyl Geometry II	p. 183
Complex Riemannian Geometry	p. 189
The Decomposition of R as Modules over U±	p. 190
The Submodules of R Arising from the Ricci Tensors	p. 204
Para-Hermitian and Pseudo-Hermitian Geometry	p. 210
Almost Para-Hermitian and Almost Pseudo-Hermitian Geometry	p. 212
Kähler Geometry in the Riemannian Setting III	p. 213
Complex Weyl Geometry
The Covariant Derivative of the Kähler Form II	p. 221
Notational Conventions	p. 235
Bibliography	p. 239
Index	p. 249
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