This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian...
This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest.
Author(s): O'Neill, Barrett
Series: PURE AND APPLIED MATHEMATICS: A series of Monographs and Textbooks
Edition: 1
Publisher: Academic Pr
Year: 1983
Language: English
Pages: 468
Tags: semi-riemannian; relativity; geometry; pseudo-riemannian
SEMI-RIEMANNIAN GEOMETRY
Copyright Page
CONTENTS
Preface
Notation and Terminology
CHAPTER 1. MANIFOLD THEORY
Smooth Manifolds
Smooth Mappings
Tangent Vectors
Differential Maps
Curves
Vector Fields
One-Forms
Submanifolds
Immersions and Submersions
Topology of Manifolds
Some Special Manifolds
Integral Curves
CHAPTER 2. TENSORS
Basic Algebra
Tensor Fields
Interpretations
Tensors at a Point
Tensor Components
Contraction
Covariant Tensors
Tensor Derivations
Symmetric Bilinear Forms
Scalar Products
CHAPTER 3. SEMI-RIEMANNIAN MANIFOLDS
Isometries
The Levi-Civita Connection
Parallel Translation
Geodesics
The Exponential Map
Curvature
Sectional Curvature
Semi-Riemannian Surfaces
Type-Changing and Metric Contraction
Frame Fields
Some Differential Operators
Ricci and Scalar Curvature
Semi-Riemannian Product Manifolds
Local Isometries
Levels of Structure
CHAPTER 4. SEMI-RIEMANNIAN SUBMANIFOLDS
Tangents and Normals
The Induced Connection
Geodesics in Submanifolds
Totally Geodesic Submanifolds
Semi-Riemannian Hypersurfaces
Hyperquadrics
The Codazzi Equation
Totally Umbilic Hypersurfaces
The Normal Connection
A Congruence Theorem
Isometric Immersions
Two-Parameter Maps
CHAPTER 5. RIEMANNIAN AND LORENTZ GEOMETRY
The Gauss Lemma
Convex Open Sets
Arc Length
Riemannian Distance
Riemannian Completeness
Lorentz Causal Character
Timecones
Local Lorentz Geometry
Geodesics in Hyperquadrics
Geodesics in Surfaces
Completeness and Extendibility
CHAPTER 6. SPECIAL RELATIVITY
Newtonian Space and Time
Newtonian Space–Time
Minkowski Spacetime
Minkowski Geometry
Particles Observed
Some Relativistic Effects
Lorentz–Fitzgerald Contraction
Energy–Momentum
Collisions
An Accelerating Observer
CHAPTER 7. CONSTRUCTIONS
Deck Transformations
Orbit Manifolds
Orientability
Semi-Riemannian Coverings
Lorentz Time-Orientability
Volume Elements
Vector Bundles
Local Isometries
Matched Coverings
Warped Products
Warped Product Geodesics
Curvature of Warped Products
Semi-Riemannian Submersions
CHAPTER 8. SYMMETRY AND CONSTANT CURVATURE
Jacobi Fields
Tidal Forces
Locally Symmetric Manifolds
Isometries of Normal Neighborhoods
Symmetric Spaces
Simply Connected Space Forms
Transvections
CHAPTER 9. ISOMETRIES
Semiorthogonal Groups
Some Isometry Groups
Time-Orientability and Space-Orientability
Linear Algebra
Space Forms
Killing Vector Fields
The Lie Algebra i(M)
I( M ) as Lie Group
Homogeneous Spaces
CHAPTER 10. CALCULUS OF VARIATIONS
First Variation
Second Variation
The Index Form
Conjugate Points
Local Minima and Maxima
Some Global Consequences
The Endmanifold Case
Focal Points
Applications
Variation of E
Focal Points along Null Geodesics
A Causality Theorem
CHAPTER 11. HOMOGENEOUS AND SYMMETRIC SPACES
More about Lie Groups
Bi-Invariant Metrics
Coset Manifolds
Reductive Homogeneous Spaces
Symmetric Spaces
Riemannian Symmetric Spaces
Duality
Some Complex Geometry
CHAPTER 12. GENERAL RELATIVITY; COSMOLOGY
Foundations
The Einstein Equation
Perfect Fluids
Robertson–Walker Spacetimes
The Robertson–Walker Flow
Robertson–Walker Cosmology
Friedmann Models
Geodesics and Redshift
Observer Fields
Static Spacetimes
CHAPTER 13. SCHWARZSCHILD GEOMETRY
Building the Model
Geometry of N and B
Schwarzschild Observers
Schwarzschild Geodesics
Free Fall Orbits
Perihelion Advance
Lightlike Orbits
Stellar Collapse
The Kruskal Plane
Kruskal Spacetime
Black Holes
Kruskal Geodesics
CHAPTER 14. CAUSALITY IN LORENTZ MANIFOLDS
Causality Relations
Quasi-Limits
Causality Conditions
Time Separation
Achronal Sets
Cauchy Hypersurfaces
Warped Products
Cauchy Developments
Spacelike Hypersurfaces
Cauchy Horizons
Hawking’s Singularity Theorem
Penrose’s Singularity Theorem
APPENDIX A. FUNDAMENTAL GROUPS AND COVERING MANIFOLDS
APPENDIX B. LIE GROUPS
Lie Algebras
Lie Exponential Map
The Classical Groups
APPENDIX C. NEWTONIAN GRAVITATION
References
Index
