Book description
This book, which focuses on the study of curvature, is an introduction to various aspects of pseudoRiemannian geometry. We shall use Walker manifolds (pseudoRiemannian manifolds which admit a nontrivial parallel null plane field) to exemplify some of the main differences between the geometry of Riemannian manifolds and the geometry of pseudoRiemannian manifolds and thereby illustrate phenomena in pseudoRiemannian 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 pseudoRiemannian 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 selfcontained as possible, we shall begin with two elementary chapters that provide an introduction to basic aspects of pseudoRiemannian geometry before beginning on our study of Walker geometry. An extensive bibliography is provided for further reading.
Math subject classifications : Primary: 53B20  (PACS: 02.40.Hw) Secondary: 32Q15, 51F25, 51P05, 53B30, 53C50, 53C80, 58A30, 83F05, 85A04
Table of Contents: Basic Algebraic Notions / Basic Geometrical Notions / Walker Structures / ThreeDimensional Lorentzian Walker Manifolds / FourDimensional Walker Manifolds / The Spectral Geometry of the Curvature Tensor / Hermitian Geometry / Special Walker Manifolds
Table of contents
 Preface

Basic Algebraic Notions
 Introduction
 A Historical Perspective in the Algebraic Context
 Algebraic Preliminaries

Spectral Geometry of the Curvature Operator
 Spectral Geometry of the Curvature Operator
 Basic Geometrical Notions (1/4)
 Basic Geometrical Notions (2/4)
 Basic Geometrical Notions (3/4)
 Basic Geometrical Notions (4/4)
 Walker Structures (1/4)
 Walker Structures (2/4)
 Walker Structures (3/4)

Walker Structures (4/4)
 Introduction
 Historical Development
 Walker Coordinates

Examples of Walker Manifolds
 Hypersurfaces with Nilpotent Shape Operators
 Locally Conformally Flat Metrics with Nilpotent Ricci Operator
 Locally Conformally Flat Metrics with Nilpotent Ricci Operator
 Degenerate PseudoRiemannian Homogeneous Structures
 Degenerate PseudoRiemannian Homogeneous Structures
 ParaKaehler Geometry
 Twostep Nilpotent Lie Groups with Degenerate Center
 Conformally Symmetric PseudoRiemannian Metrics

Riemannian Extensions
 The Affine Category
 Twisted Riemannian Extensions Defined by Flat Connections
 Twisted Riemannian Extensions Defined by Flat Connections
 Modified Riemannian Extensions Defined by Flat Connections
 Modified Riemannian Extensions Defined by Flat Connections
 Nilpotent Walker Manifolds
 Osserman Riemannian Extensions
 IvanovPetrova Riemannian Extensions
 ThreeDimensional Lorentzian Walker Manifolds (1/21)
 ThreeDimensional Lorentzian Walker Manifolds (2/21)
 ThreeDimensional Lorentzian Walker Manifolds (3/21)
 ThreeDimensional Lorentzian Walker Manifolds (4/21)
 ThreeDimensional Lorentzian Walker Manifolds (5/21)
 ThreeDimensional Lorentzian Walker Manifolds (6/21)
 ThreeDimensional Lorentzian Walker Manifolds (7/21)
 ThreeDimensional Lorentzian Walker Manifolds (8/21)
 ThreeDimensional Lorentzian Walker Manifolds (9/21)
 ThreeDimensional Lorentzian Walker Manifolds (10/21)
 ThreeDimensional Lorentzian Walker Manifolds (11/21)
 ThreeDimensional Lorentzian Walker Manifolds (12/21)
 ThreeDimensional Lorentzian Walker Manifolds (13/21)
 ThreeDimensional Lorentzian Walker Manifolds (14/21)
 ThreeDimensional Lorentzian Walker Manifolds (15/21)
 ThreeDimensional Lorentzian Walker Manifolds (16/21)
 ThreeDimensional Lorentzian Walker Manifolds (17/21)
 ThreeDimensional Lorentzian Walker Manifolds (18/21)
 ThreeDimensional Lorentzian Walker Manifolds (19/21)
 ThreeDimensional Lorentzian Walker Manifolds (20/21)

ThreeDimensional Lorentzian Walker Manifolds (21/21)

ThreeDimensional Lorentzian Walker Manifolds
 Introduction
 History
 Three Dimensional Walker Geometry
 Local geometry of Walker manifolds with =0
 Three dimensional homogeneous Lorentzian manifolds

Curvature Homogeneous Lorentzian Manifolds
 Curvature Homogeneous Lorentzian Manifolds

FourDimensional Walker Manifolds
 FourDimensional Walker Manifolds

The Spectral Geometry of the Curvature Tensor
 Introduction
 History
 FourDimensional Osserman Metrics
 Osserman and IvanovPetrova Metrics

Riemannian Extensions of Affine Surfaces
 Affine Surfaces with Skew Symmetric Ricci Tensor
 Affine Surfaces with Symmetric and Degenerate Ricci Tensor
 Affine Surfaces with Symmetric and Degenerate Ricci Tensor
 Riemannian Extensions with Commuting Curvature Operators
 Riemannian Extensions with Commuting Curvature Operators
 Other Examples with Commuting Curvature Operators
 Hermitian Geometry

ThreeDimensional Lorentzian Walker Manifolds
Product information
 Title: The Geometry of Walker Manifolds
 Author(s):
 Release date: July 2009
 Publisher(s): Morgan & Claypool Publishers
 ISBN: 9781598298208
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