Symmetries and Curvature Structure in General Relativity

The following sections are included:

• Preface

• Contents

The following sections are included:

• Geometry and Physics

• Preview of Future Chapters

The following sections are included:

• Introduction

• Groups

• Vector Spaces

• Dual Spaces

• Forms and Inner Products

• Similarity, Jordan Canonical Forms and Segre Types

• Lie Algebras

The following sections are included:

• Introduction

• Metric Spaces

• Topological Spaces

• Bases

• Subspace Topology

• Quotient Spaces

• Product Spaces

• Compactness and Paracompactness

• Connected Spaces

• Covering Spaces and the Fundamental Group

• The Rank Theorems

The following sections are included:

• Introduction

• Calculus on ℝⁿ

• Manifolds

• Functions on Manifolds

• The Manifold Topology

• The Tangent Space and Tangent Bundle

• Tensor Spaces and Tensor Bundles

• Vector and Tensor Fields

• Derived Maps and Pullbacks

• Integral Curves of Vector Fields

• Submanifolds

• Quotient Manifolds

• Distributions

• Curves and Coverings

• Metrics on Manifolds

• Linear Connections and Curvature

• Grassmann and Stiefel Manifolds

The following sections are included:

• Topological Groups

• Lie Groups

• Lie Subgroups

• Lie Algebras

• One Parameter Subgroups and the Exponential Map

• Transformation Groups

• Lie Transformation Groups

• Orbits and Isotropy Groups

• Complete Vector Fields

• Groups of Transformations

• Local Group Actions

• Lie Algebras of Vector Fields

• The Lie Derivative

The following sections are included:

• Minkowski Space

• The Lorentz Group

• The Lorentz Group as a Lie Group

• The Connected Lie Subgroups of the Lorentz Group

The following sections are included:

• Space-Times

• Bivectors and their Classification

• The Petrov Classification

• Alternative Approaches to the Petrov Classification

• The Classification of Second Order Symmetric Tensors

• The Anti–Self Dual Representation of Second Order Symmetric Tensors

• Examples and Applications

• The Local and Global Nature of Algebraic Classifications

The following sections are included:

• Introduction

• Holonomy Groups

• The Holonomy Group of a Space-Time

• Vacuum Space-Times

• Examples

The following sections are included:

• Introduction

• Metric and Connection

• Metric, Connection and Curvature

• Sectional Curvature

• Retrospect

The following sections are included:

• General Aspects of Symmetries

• Affine Vector Fields

• Subalgebras of the Affine Algebra; Isometries and Homotheties

• Fixed Point Structure

• Orbit Structure

• Space-Times admitting Proper Affine Vector Fields

• Examples and Summary

The following sections are included:

• Conformal Vector Fields

• Orbit Structure

• Fixed Point Structure

• Conformal Reduction of the Conformal Algebra

• Conformal Vector Fields in Vacuum Space-Times

• Other Examples

• Special Conformal Vector Fields

The following sections are included:

• Projective Vector Fields

• General Theorems on Projective Vector Fields

• Space-Times Admitting Projective Vector Fields

• Special Projective Vector Fields

• Projective Symmetry and Holonomy

The following sections are included:

• Introduction

• Curvature Collineations

• Some Techniques for Curvature Collineations

• Further Examples

The following sections are included:

• Bibliography

• Index