Semi-Riemannian Geometry

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.

• I Preliminaries 11 Vector Spaces 51.1 Vector Spaces 51.2 Dual Spaces 171.3 Pullback of Covectors 191.4 Annihilators 202 Matrices and Determinants 232.1 Matrices 232.2 Matrix Representations 272.3 Rank of Matrices 322.4 Determinant of Matrices 332.5 Trace and Determinant of Linear Maps 433 Bilinear Functions 453.1 Bilinear Functions 453.2 Symmetric Bilinear Functions 493.3 Flat Maps and Sharp Maps 514 Scalar Product Spaces 574.1 Scalar Product Spaces 574.2 Orthonormal Bases 624.3 Adjoints 654.4 Linear Isometries 684.5 Dual Scalar Product Spaces 724.6 Inner Product Spaces 754.7 Eigenvalues and Eigenvectors 814.8 Lorentz Vector Spaces 844.9 Time Cones 915 Tensors on Vector Spaces 975.1 Tensors 975.2 Pullback of Covariant Tensors 1035.3 Representation of Tensors 1045.4 Contraction of Tensors 1066 Tensors on Scalar Product Spaces 1136.1 Contraction of Tensors 1136.2 Flat Maps 1146.3 Sharp Maps 1196.4 Representation of Tensors 1236.5 Metric Contraction of Tensors 1276.6 Symmetries of (0, 4)-Tensors 1297 Multicovectors 1337.1 Multicovectors 1337.2 Wedge Products 1377.3 Pullback of Multicovectors 1447.4 Interior Multiplication 1487.5 Multicovector Scalar Product Spaces 1508 Orientation 1558.1 Orientation of Rm 1558.2 Orientation of Vector Spaces 1588.3 Orientation of Scalar Product Spaces 1638.4 Vector Products 1668.5 Hodge Star 1789 Topology 1839.1 Topology 1839.2 Metric Spaces 1939.3 Normed Vector Spaces 1959.4 Euclidean Topology on Rm 19510 Analysis in Rm 19910.1 Derivatives 19910.2 Immersions and Diffeomorphisms 20710.3 Euclidean Derivative and Vector Fields 20910.4 Lie Bracket 21310.5 Integrals 21810.6 Vector Calculus 221II Curves and Regular Surfaces 22311 Curves and Regular Surfaces in R3 22511.1 Curves in R3 22511.2 Regular Surfaces in R3 22611.3 Tangent Planes in R3 23711.4 Types of Regular Surfaces in R3 24011.5 Functions on Regular Surfaces in R3 24611.6 Maps on Regular Surfaces in R3 24811.7 Vector Fields along Regular Surfaces in R3 25212 Curves and Regular Surfaces in R3v 25512.1 Curves in R3v 25612.2 Regular Surfaces in R3v 25712.3 Induced Euclidean Derivative in R3v 26612.4 Covariant Derivative on Regular Surfaces in R3v 27412.5 Covariant Derivative on Curves in R3v 28212.6 Lie Bracket in R3v 28512.7 Orientation in R3v 28812.8 Gauss Curvature in R3v 29212.9 Riemann Curvature Tensor in R3v 29912.10 Computations for Regular Surfaces in R3v 31013 Examples of Regular Surfaces 32113.1 Plane in R30 32113.2 Cylinder in R30 32213.3 Cone in R30 32313.4 Sphere in R30 32413.5 Tractoid in R30 32513.6 Hyperboloid of One Sheet in R30 32613.7 Hyperboloid of Two Sheets in R30 32713.8 Torus in R30 32913.9 Pseudosphere in R31 33013.10 Hyperbolic Space in R31 331III Smooth Manifolds and Semi-Riemannian Manifolds 33314 Smooth Manifolds 33714.1 Smooth Manifolds 33714.2 Functions and Maps 34014.3 Tangent Spaces 34414.4 Differential of Maps 35114.5 Differential of Functions 35314.6 Immersions and Diffeomorphisms 35714.7 Curves 35814.8 Submanifolds 36014.9 Parametrized Surfaces 36415 Fields on Smooth Manifolds 36715.1 Vector Fields 36715.2 Representation of Vector Fields 37215.3 Lie Bracket 37415.4 Covector Fields 37615.5 Representation of Covector Fields 37915.6 Tensor Fields 38215.7 Representation of Tensor Fields 38515.8 Differential Forms 38715.9 Pushforward and Pullback of Functions 38915.10 Pushforward and Pullback of Vector Fields 39115.11 Pullback of Covector Fields 39315.12 Pullback of Covariant Tensor Fields 39815.13 Pullback of Differential Forms 40115.14 Contraction of Tensor Fields 40516 Differentiation and Integration on Smooth Manifolds 40716.1 Exterior Derivatives 40716.2 Tensor Derivations 41316.3 Form Derivations 41716.4 Lie Derivative 41916.5 Interior Multiplication 42316.6 Orientation 42516.7 Integration of Differential Forms 43216.8 Line Integrals 43516.9 Closed and Exact Covector Fields 43716.10 Flows 44317 Smooth Manifolds with Boundary 44917.1 Smooth Manifolds with Boundary 44917.2 Inward-Pointing and Outward-Pointing Vectors 45217.3 Orientation of Boundaries 45617.4 Stokes's Theorem 45918 Smooth Manifolds with a Connection 46318.1 Covariant Derivatives 46318.2 Christoffel Symbols 46618.3 Covariant Derivative on Curves 47218.4 Total Covariant Derivatives 47618.5 Parallel Translation 47918.6 Torsion Tensors 48518.7 Curvature Tensors 48818.8 Geodesics 49718.9 Radial Geodesics and Exponential Maps 50218.10 Normal Coordinates 50718.11 Jacobi Fields 50919 Semi-Riemannian Manifolds 51519.1 Semi-Riemannian Manifolds 51519.2 Curves 51919.3 Fundamental Theorem of Semi-Riemannian Manifolds 51919.4 Flat Maps and Sharp Maps 52619.5 Representation of Tensor Fields 52919.6 Contraction of Tensor Fields 53219.7 Isometries 53519.8 Riemann Curvature Tensor 53919.9 Geodesics 54619.10 Volume Forms 55019.11 Orientation of Hypersurfaces 55119.12 Induced Connections 55820 Differential Operators on Semi-Riemannian Manifolds 56120.1 Hodge Star 56120.2 Codifferential 56220.3 Gradient 56620.4 Divergence of Vector Fields 56820.5 Curl 57220.6 Hesse Operator 57320.7 Laplace Operator 57520.8 Laplace-de Rham Operator 57620.9 Divergence of Symmetric 2-Covariant Tensor Fields 57721 Riemannian Manifolds 57921.1 Geodesics and Curvature on Riemannian Manifolds 57921.2 Classical Vector Calculus Theorems 58222 Applications to Physics 58722.1 Linear Isometries on Lorentz Vector Spaces 58722.2 Maxwell's Equations 59822.3 Einstein Tensor 603IV Appendices 609A Notation and Set Theory 611B Abstract Algebra 617B.1 Groups 617B.2 Permutation Groups 618B.3 Rings 623B.4 Fields 623B.5 Modules 624B.6 Vector Spaces 625B.7 Lie Algebras 626Further Reading 627Index 629