book

Semi-Riemannian Geometry

Name: Semi-Riemannian Geometry
Author: Stephen C. Newman
ISBN: 9781119517535

by Stephen C. Newman

July 2019

Intermediate to advanced

625 pages

20h 7m

English

Wiley

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Cover
Preface
Part I: Preliminaries
Chapter 1: Vector Spaces
1.1 Vector Spaces1.2 Dual Spaces1.3 Pullback of Covectors1.4 Annihilators
Chapter 2: Matrices and Determinants
2.1 Matrices2.2 Matrix Representations2.3 Rank of Matrices2.4 Determinant of Matrices2.5 Trace and Determinant of Linear Maps
Chapter 3: Bilinear Functions
3.1 Bilinear Functions3.2 Symmetric Bilinear Functions3.3 Flat Maps and Sharp Maps
Chapter 4: Scalar Product Spaces
4.1 Scalar Product Spaces4.2 Orthonormal Bases4.3 Adjoints4.4 Linear Isometries4.5 Dual Scalar Product Spaces4.6 Inner Product Spaces4.7 Eigenvalues and Eigenvectors4.8 Lorentz Vector Spaces4.9 Time Cones
Chapter 5: Tensors on Vector Spaces
5.1 Tensors5.2 Pullback of Covariant Tensors5.3 Representation of Tensors5.4 Contraction of Tensors
Chapter 6: Tensors on Scalar Product Spaces
6.1 Contraction of Tensors6.2 Flat Maps6.3 Sharp Maps6.4 Representation of Tensors6.5 Metric Contraction of Tensors6.6 Symmetries of (0, 4)‐Tensors
Chapter 7: Multicovectors
7.1 Multicovectors7.2 Wedge Products7.3 Pullback of Multicovectors7.4 Interior Multiplication7.5 Multicovector Scalar Product Spaces

Chapter 8: Orientation
8.1 Orientation of ℝ m 8.2 Orientation of Vector Spaces8.3 Orientation of Scalar Product Spaces8.4 Vector Products8.5 Hodge Star
Chapter 9: Topology
9.1 Topology9.2 Metric Spaces9.3 Normed Vector Spaces9.4 Euclidean Topology on ℝm
Chapter 10: Analysis in ℝ m
10.1 Derivatives10.2 Immersions and Diffeomorphisms10.3 Euclidean Derivative and Vector Fields10.4 Lie Bracket10.5 Integrals10.6 Vector Calculus
Part II: Curves and Regular Surfaces
Chapter 11: Curves and Regular Surfaces in ℝ3
11.1 Curves inℝ3 11.2 Regular Surfaces inℝ3 11.3 Tangent Planes inℝ3 11.4 Types of Regular Surfaces in ℝ3 11.5 Functions on Regular Surfaces in ℝ3 11.6 Maps on Regular Surfaces inℝ3 11.7 Vector Fields Along 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
13.1 Plane in 13.2 Cylinder in 13.3 Cone in 13.4 Sphere in 13.5 Tractoid in 13.6 Hyperboloid of One Sheet in 13.7 Hyperboloid of Two Sheets in 13.8 Torus in 13.9 Pseudosphere in 13.10 Hyperbolic Space in
Part III: Smooth Manifolds and Semi‐Riemannian Manifolds
Chapter 14: Smooth Manifolds
14.1 Smooth Manifolds14.2 Functions and Maps14.3 Tangent Spaces14.4 Differential of Maps14.5 Differential of Functions14.6 Immersions and Diffeomorphisms14.7 Curves14.8 Submanifolds14.9 Parametrized Surfaces
Chapter 15: Fields on Smooth Manifolds
15.1 Vector Fields15.2 Representation of Vector Fields15.3 Lie Bracket15.4 Covector Fields15.5 Representation of Covector Fields15.6 Tensor Fields15.7 Representation of Tensor Fields15.8 Differential Forms15.9 Pushforward and Pullback of Functions15.10 Pushforward and Pullback of Vector Fields15.11 Pullback of Covector Fields15.12 Pullback of Covariant Tensor Fields15.13 Pullback of Differential Forms15.14 Contraction of Tensor Fields
Chapter 16: Differentiation and Integration on Smooth Manifolds
16.1 Exterior Derivatives16.2 Tensor Derivations16.3 Form Derivations16.4 Lie Derivative16.5 Interior Multiplication16.6 orientation16.7 Integration of Differential Forms16.8 Line Integrals16.9 Closed and Exact Covector Fields16.10 Flows
Chapter 17: Smooth Manifolds with Boundary
17.1 Smooth Manifolds with Boundary17.2 Inward‐Pointing and Outward‐Pointing Vectors17.3 orientation of Boundaries17.4 Stokes's Theorem
Chapter 18: Smooth Manifolds with a Connection
18.1 Covariant Derivatives18.2 Christoffel Symbols18.3 Covariant Derivative on Curves18.4 Total Covariant Derivatives18.5 Parallel Ranslation18.6 Torsion Tensors18.7 Curvature Tensors18.8 Geodesics18.9 Radial Geodesics and Exponential Maps18.10 Normal Coordinates18.11 Jacobi Fields
Chapter 19: Semi‐Riemannian Manifolds
19.1 Semi‐Riemannian Manifolds19.2 Curves19.3 Fundamental Theorem of Semi‐Riemannian Manifolds19.4 Flat Maps and Sharp Maps19.5 Representation of Tensor Fields19.6 Contraction of Tensor Fields19.7 Isometries19.8 Riemann Curvature Tensor19.9 Geodesics19.10 Volume Forms19.11 orientation of Hypersurfaces19.12 Induced Connections
Chapter 20: Differential Operators on Semi‐Riemannian Manifolds
20.1 Hodge Star20.2 Codifferential20.3 Gradient20.4 Divergence of Vector Fields20.5 Curl20.6 Hesse Operator20.7 Laplace Operator20.8 Laplace–de Rham Operator20.9 Divergence of Symmetric 2‐Covariant Tensor Fields
Chapter 21: Riemannian Manifolds
21.1 Geodesics and Curvature on Riemannian Manifolds21.2 Classical Vector Calculus Theorems
Chapter 22: Applications to Physics
22.1 Linear Isometries on Lorentz Vector Spaces22.2 Maxwell's Equations22.3 Einstein Tensor
Part IV: Appendices
Appendix A: Notation and Set Theory
Appendix B: Abstract Algebra
B.1. GroupsB.2. Permutation GroupsB.3. RingsB.4. FieldsB.5. ModulesB.6. Vector SpacesB.7. Lie Algebras
Further Reading
Index
End User License Agreement

Content preview from Semi-Riemannian Geometry

Part I Preliminaries

Differential geometry rests on the twin pillars of linear algebra‐multilinear algebra and topology‐analysis. Part I of the book provides an overview of selected topics from these areas of mathematics. Most of the linear algebra presented here is likely familiar to the reader, but the same may not be true of the multilinear algebra, with the exception of the material on determinants. Topology and analysis are vast subjects, and only the barest of essentials are touched on here. In order to keep the book to a manageable size, not all theorems are provided with a proof, a remark that also applies to Part II and Part III.

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Semi-Riemannian Geometry

by Stephen C. Newman

Part I Preliminaries

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.