Osserman Manifolds in Semi-Riemannian Geometry

by Garcia-Rio, Eduardo; Kupeli, Demir N.; Vazquez-Lorenzo, Ramon

• ISBN13: 9783540431442
• ISBN10: 3540431446
• Format: Paperback
• Copyright: 2002-04-01
• Publisher: Springer Verlag

Summary

The subject of this book is Osserman semi-Riemannian manifolds, and in particular, the Osserman conjecture in semi-Riemannian geometry. The treatment is pitched at the intermediate graduate level and requires some intermediate knowledge of differential geometry. The notation is mostly coordinate-free and the terminology is that of modern differential geometry. Known results toward the complete proof of Riemannian Osserman conjecture are given and the Osserman conjecture in Lorentzian geometry is proved completely. Counterexamples to the Osserman conjuncture in generic semi-Riemannian signature are provided and properties of semi-Riemannian Osserman manifolds are investigated.

Table of Contents

The Osserman Conditions in Semi-Riemannian Geometry	p. 1
The Jacobi Operator	p. 1
The Timelike and Spacelike Osserman Conditions at a Point	p. 3
Semi-Riemannian Pointwise and Global Osserman Manifolds	p. 9
The Osserman Conjecture in Riemannian Geometry	p. 21
Partial Solution to the Osserman Conjecture	p. 22
A Solution to the Osserman Conjecture with Additional Assumptions	p. 31
Lorentzian Osserman Manifolds	p. 39
The Complete Solution to the Osserman Problem	p. 39
The Null Osserman Condition	p. 44
Four-Dimensional Semi-Riemannian Osserman Manifolds with Metric Tensors of Signature (2,2)	p. 63
Nonsymmetric Semi-Riemannian Osserman Manifolds with Metric Tensors of Signature (2,2)	p. 64
Semi-Riemannian Osserman and Jordan-Osserman Manifolds with Metric Tensors of Signature (2,2)	p. 70
Semi-Riemannian Osserman Manifolds	p. 95
Nonsymmetric Semi-Riemannian Osserman Manifolds of Arbitrary Signature	p. 95
Special Semi-Riemannian Osserman Manifolds	p. 102
Classification of Semi-Riemannian Special Osserman Manifolds	p. 104
Generalizations and Osserman-Related Conditions	p. 137
Semi-Riemannian Generalized Osserman Manifolds	p. 137
The Osserman Condition in Affine Differential Geometry	p. 143
Riemannian Manifolds with Isoparametric Geodesic Spheres	p. 150
Riemannian [actual symbol not reproducible]-manifolds	p. 151
Skew-Symmetric Curvature Operators	p. 152
References	p. 157
Index	p. 165
Table of Contents provided by Blackwell. All Rights Reserved.