Semi-Riemannian Geometry

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.

Inhalt

I Preliminaries 1

1 Vector Spaces 5

1.1 Vector Spaces 5

1.2 Dual Spaces 17

1.3 Pullback of Covectors 19

1.4 Annihilators 20

2 Matrices and Determinants 23

2.1 Matrices 23

2.2 Matrix Representations 27

2.3 Rank of Matrices 32

2.4 Determinant of Matrices 33

2.5 Trace and Determinant of Linear Maps 43

3 Bilinear Functions 45

3.1 Bilinear Functions 45

3.2 Symmetric Bilinear Functions 49

3.3 Flat Maps and Sharp Maps 51

4 Scalar Product Spaces 57

4.1 Scalar Product Spaces 57

4.2 Orthonormal Bases 62

4.3 Adjoints 65

4.4 Linear Isometries 68

4.5 Dual Scalar Product Spaces 72

4.6 Inner Product Spaces 75

4.7 Eigenvalues and Eigenvectors 81

4.8 Lorentz Vector Spaces 84

4.9 Time Cones 91

5 Tensors on Vector Spaces 97

5.1 Tensors 97

5.2 Pullback of Covariant Tensors 103

5.3 Representation of Tensors 104

5.4 Contraction of Tensors 106

6 Tensors on Scalar Product Spaces 113

6.1 Contraction of Tensors 113

6.2 Flat Maps 114

6.3 Sharp Maps 119

6.4 Representation of Tensors 123

6.5 Metric Contraction of Tensors 127

6.6 Symmetries of (0, 4)-Tensors 129

7 Multicovectors 133

7.1 Multicovectors 133

7.2 Wedge Products 137

7.3 Pullback of Multicovectors 144

7.4 Interior Multiplication 148

7.5 Multicovector Scalar Product Spaces 150

8 Orientation 155

8.1 Orientation of R*m *155

8.2 Orientation of Vector Spaces 158

8.3 Orientation of Scalar Product Spaces 163

8.4 Vector Products 166

8.5 Hodge Star 178

9 Topology 183

9.1 Topology 183

9.2 Metric Spaces 193

9.3 Normed Vector Spaces 195

9.4 Euclidean Topology on Rm 195

10 Analysis in R*m *199

10.1 Derivatives 199

10.2 Immersions and Diffeomorphisms 207

10.3 Euclidean Derivative and Vector Fields 209

10.4 Lie Bracket 213

10.5 Integrals 218

10.6 Vector Calculus 221

II Curves and Regular Surfaces 223

11 Curves and Regular Surfaces in R3 225

11.1 Curves in R3 225

11.2 Regular Surfaces in R3 226

11.3 Tangent Planes in R3 237

11.4 Types of Regular Surfaces in R3 240

11.5 Functions on Regular Surfaces in R3 246

11.6 Maps on Regular Surfaces in R3 248

11.7 Vector Fields along Regular Surfaces in R3 252

12 Curves and Regular Surfaces in R3*v *255

12.1 Curves in R3*v *256

12.2 Regular Surfaces in R3*v *257

12.3 Induced Euclidean Derivative in R3*v *266

12.4 Covariant Derivative on Regular Surfaces in R3*v *274

12.5 Covariant Derivative on Curves in R3*v *282

12.6 Lie Bracket in R3*v *285

12.7 Orientation in R3*v *288 12.8 Gauss Curvature in R3</...