Book description
An introduction to semi-Riemannian geometry as a foundation for general relativity
Semi-Riemannian Geometry: The Mathematical Language of General Relativity is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell’s equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.
STEPHEN C. NEWMAN is Professor Emeritus at the University of Alberta, Edmonton, Alberta, Canada. He is the author of Biostatistical Methods in Epidemiology and A Classical Introduction to Galois Theory, both published by Wiley.
Table of contents
- Cover
- Preface
-
Part I: Preliminaries
- Chapter 1: Vector Spaces
- Chapter 2: Matrices and Determinants
- Chapter 3: Bilinear Functions
- Chapter 4: Scalar Product Spaces
- Chapter 5: Tensors on Vector Spaces
- Chapter 6: Tensors on Scalar Product Spaces
- Chapter 7: Multicovectors
- Chapter 8: Orientation
- Chapter 9: Topology
- Chapter 10: Analysis in ℝ m
-
Part II: Curves and Regular Surfaces
- Chapter 11: Curves and Regular Surfaces in ℝ3
-
Chapter 12: Curves and Regular Surfaces in
- 12.1 Curves in
- 12.2 Regular Surfaces in
- 12.3 Induced Euclidean Derivative in
- 12.4 Covariant Derivative on Regular Surfaces in
- 12.5 Covariant Derivative on Curves in
- 12.6 Lie Bracket in
- 12.7 Orientation in
- 12.8 Gauss Curvature in
- 12.9 Riemann Curvature Tensor in
- 12.10 Computations for Regular Surfaces in
- Chapter 13: Examples of Regular Surfaces
-
Part III: Smooth Manifolds and Semi‐Riemannian Manifolds
- Chapter 14: Smooth Manifolds
-
Chapter 15: Fields on Smooth Manifolds
- 15.1 Vector Fields
- 15.2 Representation of Vector Fields
- 15.3 Lie Bracket
- 15.4 Covector Fields
- 15.5 Representation of Covector Fields
- 15.6 Tensor Fields
- 15.7 Representation of Tensor Fields
- 15.8 Differential Forms
- 15.9 Pushforward and Pullback of Functions
- 15.10 Pushforward and Pullback of Vector Fields
- 15.11 Pullback of Covector Fields
- 15.12 Pullback of Covariant Tensor Fields
- 15.13 Pullback of Differential Forms
- 15.14 Contraction of Tensor Fields
- Chapter 16: Differentiation and Integration on Smooth Manifolds
- Chapter 17: Smooth Manifolds with Boundary
- Chapter 18: Smooth Manifolds with a Connection
-
Chapter 19: Semi‐Riemannian Manifolds
- 19.1 Semi‐Riemannian Manifolds
- 19.2 Curves
- 19.3 Fundamental Theorem of Semi‐Riemannian Manifolds
- 19.4 Flat Maps and Sharp Maps
- 19.5 Representation of Tensor Fields
- 19.6 Contraction of Tensor Fields
- 19.7 Isometries
- 19.8 Riemann Curvature Tensor
- 19.9 Geodesics
- 19.10 Volume Forms
- 19.11 orientation of Hypersurfaces
- 19.12 Induced Connections
- Chapter 20: Differential Operators on Semi‐Riemannian Manifolds
- Chapter 21: Riemannian Manifolds
- Chapter 22: Applications to Physics
- Part IV: Appendices
- Further Reading
- Index
- End User License Agreement
Product information
- Title: Semi-Riemannian Geometry
- Author(s):
- Release date: July 2019
- Publisher(s): Wiley
- ISBN: 9781119517535
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