Semi-Riemannian Geometry

Semi-Riemannian Geometry

by Stephen C. Newman
Released July 2019
Publisher(s): Wiley
ISBN: 9781119517535

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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

  1. Cover
  2. Preface
  3. Part I: Preliminaries
    1. Chapter 1: Vector Spaces
      1. 1.1 Vector Spaces
      2. 1.2 Dual Spaces
      3. 1.3 Pullback of Covectors
      4. 1.4 Annihilators
    2. Chapter 2: Matrices and Determinants
      1. 2.1 Matrices
      2. 2.2 Matrix Representations
      3. 2.3 Rank of Matrices
      4. 2.4 Determinant of Matrices
      5. 2.5 Trace and Determinant of Linear Maps
    3. Chapter 3: Bilinear Functions
      1. 3.1 Bilinear Functions
      2. 3.2 Symmetric Bilinear Functions
      3. 3.3 Flat Maps and Sharp Maps
    4. Chapter 4: Scalar Product Spaces
      1. 4.1 Scalar Product Spaces
      2. 4.2 Orthonormal Bases
      3. 4.3 Adjoints
      4. 4.4 Linear Isometries
      5. 4.5 Dual Scalar Product Spaces
      6. 4.6 Inner Product Spaces
      7. 4.7 Eigenvalues and Eigenvectors
      8. 4.8 Lorentz Vector Spaces
      9. 4.9 Time Cones
    5. Chapter 5: Tensors on Vector Spaces
      1. 5.1 Tensors
      2. 5.2 Pullback of Covariant Tensors
      3. 5.3 Representation of Tensors
      4. 5.4 Contraction of Tensors
    6. Chapter 6: Tensors on Scalar Product Spaces
      1. 6.1 Contraction of Tensors
      2. 6.2 Flat Maps
      3. 6.3 Sharp Maps
      4. 6.4 Representation of Tensors
      5. 6.5 Metric Contraction of Tensors
      6. 6.6 Symmetries of (0, 4)‐Tensors
    7. Chapter 7: Multicovectors
      1. 7.1 Multicovectors
      2. 7.2 Wedge Products
      3. 7.3 Pullback of Multicovectors
      4. 7.4 Interior Multiplication
      5. 7.5 Multicovector Scalar Product Spaces
    8. Chapter 8: Orientation
      1. 8.1 Orientation of ℝ m
      2. 8.2 Orientation of Vector Spaces
      3. 8.3 Orientation of Scalar Product Spaces
      4. 8.4 Vector Products
      5. 8.5 Hodge Star
    9. Chapter 9: Topology
      1. 9.1 Topology
      2. 9.2 Metric Spaces
      3. 9.3 Normed Vector Spaces
      4. 9.4 Euclidean Topology on ℝm
    10. Chapter 10: Analysis in ℝ m
      1. 10.1 Derivatives
      2. 10.2 Immersions and Diffeomorphisms
      3. 10.3 Euclidean Derivative and Vector Fields
      4. 10.4 Lie Bracket
      5. 10.5 Integrals
      6. 10.6 Vector Calculus
  4. Part II: Curves and Regular Surfaces
    1. Chapter 11: Curves and Regular Surfaces in ℝ3
      1. 11.1 Curves inℝ3
      2. 11.2 Regular Surfaces inℝ3
      3. 11.3 Tangent Planes inℝ3
      4. 11.4 Types of Regular Surfaces in ℝ3
      5. 11.5 Functions on Regular Surfaces in ℝ3
      6. 11.6 Maps on Regular Surfaces inℝ3
      7. 11.7 Vector Fields Along Regular Surfaces in ℝ3
    2. Chapter 12: Curves and Regular Surfaces in
      1. 12.1 Curves in
      2. 12.2 Regular Surfaces in
      3. 12.3 Induced Euclidean Derivative in
      4. 12.4 Covariant Derivative on Regular Surfaces in
      5. 12.5 Covariant Derivative on Curves in
      6. 12.6 Lie Bracket in
      7. 12.7 Orientation in
      8. 12.8 Gauss Curvature in
      9. 12.9 Riemann Curvature Tensor in
      10. 12.10 Computations for Regular Surfaces in
    3. Chapter 13: Examples of Regular Surfaces
      1. 13.1 Plane in
      2. 13.2 Cylinder in
      3. 13.3 Cone in
      4. 13.4 Sphere in
      5. 13.5 Tractoid in
      6. 13.6 Hyperboloid of One Sheet in
      7. 13.7 Hyperboloid of Two Sheets in
      8. 13.8 Torus in
      9. 13.9 Pseudosphere in
      10. 13.10 Hyperbolic Space in
  5. Part III: Smooth Manifolds and Semi‐Riemannian Manifolds
    1. Chapter 14: Smooth Manifolds
      1. 14.1 Smooth Manifolds
      2. 14.2 Functions and Maps
      3. 14.3 Tangent Spaces
      4. 14.4 Differential of Maps
      5. 14.5 Differential of Functions
      6. 14.6 Immersions and Diffeomorphisms
      7. 14.7 Curves
      8. 14.8 Submanifolds
      9. 14.9 Parametrized Surfaces
    2. Chapter 15: Fields on Smooth Manifolds
      1. 15.1 Vector Fields
      2. 15.2 Representation of Vector Fields
      3. 15.3 Lie Bracket
      4. 15.4 Covector Fields
      5. 15.5 Representation of Covector Fields
      6. 15.6 Tensor Fields
      7. 15.7 Representation of Tensor Fields
      8. 15.8 Differential Forms
      9. 15.9 Pushforward and Pullback of Functions
      10. 15.10 Pushforward and Pullback of Vector Fields
      11. 15.11 Pullback of Covector Fields
      12. 15.12 Pullback of Covariant Tensor Fields
      13. 15.13 Pullback of Differential Forms
      14. 15.14 Contraction of Tensor Fields
    3. Chapter 16: Differentiation and Integration on Smooth Manifolds
      1. 16.1 Exterior Derivatives
      2. 16.2 Tensor Derivations
      3. 16.3 Form Derivations
      4. 16.4 Lie Derivative
      5. 16.5 Interior Multiplication
      6. 16.6 orientation
      7. 16.7 Integration of Differential Forms
      8. 16.8 Line Integrals
      9. 16.9 Closed and Exact Covector Fields
      10. 16.10 Flows
    4. Chapter 17: Smooth Manifolds with Boundary
      1. 17.1 Smooth Manifolds with Boundary
      2. 17.2 Inward‐Pointing and Outward‐Pointing Vectors
      3. 17.3 orientation of Boundaries
      4. 17.4 Stokes's Theorem
    5. Chapter 18: Smooth Manifolds with a Connection
      1. 18.1 Covariant Derivatives
      2. 18.2 Christoffel Symbols
      3. 18.3 Covariant Derivative on Curves
      4. 18.4 Total Covariant Derivatives
      5. 18.5 Parallel Ranslation
      6. 18.6 Torsion Tensors
      7. 18.7 Curvature Tensors
      8. 18.8 Geodesics
      9. 18.9 Radial Geodesics and Exponential Maps
      10. 18.10 Normal Coordinates
      11. 18.11 Jacobi Fields
    6. Chapter 19: Semi‐Riemannian Manifolds
      1. 19.1 Semi‐Riemannian Manifolds
      2. 19.2 Curves
      3. 19.3 Fundamental Theorem of Semi‐Riemannian Manifolds
      4. 19.4 Flat Maps and Sharp Maps
      5. 19.5 Representation of Tensor Fields
      6. 19.6 Contraction of Tensor Fields
      7. 19.7 Isometries
      8. 19.8 Riemann Curvature Tensor
      9. 19.9 Geodesics
      10. 19.10 Volume Forms
      11. 19.11 orientation of Hypersurfaces
      12. 19.12 Induced Connections
    7. Chapter 20: Differential Operators on Semi‐Riemannian Manifolds
      1. 20.1 Hodge Star
      2. 20.2 Codifferential
      3. 20.3 Gradient
      4. 20.4 Divergence of Vector Fields
      5. 20.5 Curl
      6. 20.6 Hesse Operator
      7. 20.7 Laplace Operator
      8. 20.8 Laplace–de Rham Operator
      9. 20.9 Divergence of Symmetric 2‐Covariant Tensor Fields
    8. Chapter 21: Riemannian Manifolds
      1. 21.1 Geodesics and Curvature on Riemannian Manifolds
      2. 21.2 Classical Vector Calculus Theorems
    9. Chapter 22: Applications to Physics
      1. 22.1 Linear Isometries on Lorentz Vector Spaces
      2. 22.2 Maxwell's Equations
      3. 22.3 Einstein Tensor
  6. Part IV: Appendices
    1. Appendix A: Notation and Set Theory
    2. Appendix B: Abstract Algebra
      1. B.1. Groups
      2. B.2. Permutation Groups
      3. B.3. Rings
      4. B.4. Fields
      5. B.5. Modules
      6. B.6. Vector Spaces
      7. B.7. Lie Algebras
  7. Further Reading
  8. Index
  9. End User License Agreement

Product information

  • Title: Semi-Riemannian Geometry
  • Author(s): Stephen C. Newman
  • Release date: July 2019
  • Publisher(s): Wiley
  • ISBN: 9781119517535