Table of Contents

Chapter 0/Set Theory and Topology
0.1. SET THEORY
0.1.1. Sets
0.1.2. Set Operations
0.1.3. Cartesian Products
0.1.4. Functions
0.1.5. Functions and Set Operations
0.1.6. Equivalence Relations
0.2. TOPOLOGY
0.2.1. Topologies
0.2.2. Metric Spaces
0.2.3. Subspaces
0.2.4. Product Topologies
0.2.5. Hausdorff Spaces
0.2.6. Continuity
0.2.7. Connectedness
0.2.8. Compactness
0.2.9. Local Compactness
0.2.10. Separability
0.2.11 Paracompactness
Chapter 1/Manifolds
1.1. Definition of a Mainifold
1.2. Examples of Manifolds
1.3. Differentiable Maps
1.4. Submanifolds
1.5. Differentiable Maps
1.6. Tangents
1.7. Coordinate Vector Fields
1.8. Differential of a Map
Chapter 2/Tensor Algebra
2.1. Vector Spaces
2.2. Linear Independence
2.3. Summation Convention
2.4. Subspaces
2.5. Linear Functions
2.6. Spaces of Linear Functions
2.7. Dual Space
2.8. Multilinear Functions
2.9. Natural Pairing
2.10. Tensor Spaces
2.11. Algebra of Tensors
2.12. Reinterpretations
2.13. Transformation Laws
2.14. Invariants
2.15. Symmetric Tensors
2.16. Symmetric Algebra
2.17. Skew-Symmetric Tensors
2.18. Exterior Algebra
2.19. Determinants
2.20. Bilinear Forms
2.21. Quadratic Forms
2.22. Hodge Duality
2.23. Symplectic Forms
Chapter 3/Vector Analysis on Manifolds
3.1. Vector Fields
3.2. Tensor Fields
3.3. Riemannian Metrics
3.4. Integral Curves
3.5. Flows
3.6. Lie Derivatives
3.7. Bracket
3.8. Geometric Interpretation of Brackets
3.9. Action of Maps
3.10. Critical Point Theory
3.11. First Order Partial Differential Equations
3.12. Frobenius' Theorem
Appendix to Chapter 3
3A. Tensor Bundles
3B. Parallelizable Manifolds
3C. Orientability
Chapter 4/Integration Theory
4.1. Introduction
4.2. Differential Forms
4.3. Exterior Derivatives
4.4. Interior Products
4.5. Converse of the Poincaré Lemma
4.6. Cubical Chains
4.7. Integration on Euclidean Spaces
4.8. Integration of Forms
4.9. Strokes' Theorem
4.10. Differential Systems
Chapter 5/Riemannian and Semi-riemannian Manifolds
5.1. Introduction
5.2. Riemannian and Semi-riemannian Metrics
5.3. "Lengeth, Angle, Distance, and Energy"
5.4. Euclidean Space
5.5. Variations and Rectangles
5.6. Flat Spaces
5.7. Affine connexions
5.8 Parallel Translation
5.9. Covariant Differentiation of Tensor Fields
5.10. Curvature and Torsion Tensors
5.11. Connexion of a Semi-riemannian Structure
5.12. Geodesics
5.13. Minimizing Properties of Geodesics
5.14. Sectional Curvature
Chapter 6/Physical Application
6.1 Introduction
6.2. Hamiltonian Manifolds
6.3. Canonical Hamiltonian Structure on the Cotangent Bundle
6.4. Geodesic Spray of a Semi-riemannian Manifold
6.5. Phase Space
6.6. State Space
6.7. Contact Coordinates
6.8. Contact Manifolds
Bibliography
Index