Differential Geometry
Manifold, Differential Geometry, Fibre Bundle
Manifold and differential geometry is mathematical structure of physics.
Munkres Topology, chapter 36: Embedding of Manifold
Munkres Topology, chapter 75: Homology of Surfaces
[Lee] Smooth Manifolds
1. Smooth Manifolds
2. Smooth Maps
3. Tangent Vectors
4. Submersions, Immersions, and Embeddings
5. Submanifolds
6. Sard's Theorem
7. Lie Groups
8. Vector Fields
9. Integral Curves and Flows
10. Vector Bunudles
11. The Cotangent Bundle
12. Tensors
13. Riemannian Metrics
14. Differential Forms
15. Orientations
16. Integration on Manifolds
17. De Rham Cohomology
18. The de Rham Theorem
19. DIstributions and Foliations
20. The Exponential Map
21. Quotien Manifolds
22. Symplectic Manifolds
[Tu] Differential Geometry
Curvature and Vector Fields
2. Curves
4. Directional Derivatives in Euclidean Space
5. The Shape Operator
8. Gauss's Theorema Egregium
9. Generalizations to Hypersurfaces in $\mathbb{R}^{n+1}$
Curvature and Differential Forms
10. Connections on a Vector Bundle
11. Connection, Curvature, and Torsion Forms
12. The Theorema Egregium Using Forms
Geodesics
13. More on Affine Connections
14. Geodesics
15. Exponential Maps
Tools from Algebra and Topology
18. The Tensor Product and the Dual Module
19. The Exterior Power
20. Operations on Vector Bundles
21. Vector-Valued Forms
Vector Bundles and Characteristic Classes
22. Connections and Curvature Again
23. Characteristic Classes
24. Pontrjagin Classes
25. The Euler Class and Chern Classes
26. Some Applications of Characteristic Classes
Principal Bundles and Characteristic Classes
27. Principal Bundles
28. Connections on a Principal Bundle
29. Horizontal Distributions on a Frame Bundle
30. Curvature on a Principal Bundle
31. Covariant Derivative on a Principal Bundle
32. Characteristic Classes of Principal Bundles
[Nakahara] Geometry, Topology And Physics
11. Characteristic Class
11.1 Invariant polynomials and the Chern-Weil homomorphism
11.2 Chern classes
11.3 Chern characters
11.4 Pontrjagin and Euler classes
11.5 Chern-Simons forms
11.6 Stiefel-Whitney class
12. Index Theorems
12.1 Elliptic operators and Fredholm operators
12.2 The Atiyah–Singer index theorem
12.3 The de Rham complex
12.4 The Dolbeault complex
12.5 The signature complex
12.6 Spin complexes
12.7 The heat kernel and generalized $\zeta$-functions
12.8 The Atiyah–Patodi–Singer index theorem
12.9 Supersymmetric quantum mechanics
12.10 Supersymmetric proof of index theorem
13. Anomalies in Gauge Field Theories
13.1 Introduction
13.2 Abelian anomalies
13.4 The Wess–Zumino consistency conditions
13.5 Abelian anomalies versus non-Abelian anomalies
13.6 The parity anomaly in odd-dimensional spaces
[Husemoller] Fibre Bundles
1. Preliminaries on Homotopy Theory
The General Theory of Fibre Bundles
2. Generalities on Bundles
3. Vector Bundles
4. General Fibre Bundles
5. Local Coordinate Description of Fibre Bundles
6. Change of Structure Group in Fibre Bundles
7. The Gauge Group of a Principal Bundle
8. Calculations Involving the Classical Groups
Elements of $K$-Theory
9. Stability Properties of Vector Bundles
10. Relative K-Theory
11. Bott Periodicity in the Complex Case
12. Clifford Algebras
13. The Adams Operations and Representations
14. Representation Rings of Classical Groups
15. The Hopf Invariant
16. Vector Fields on the Sphere
Characteristic Classes
17. Chern Classes and Stiefel-Whitney Classes
18. Differentiable Manifolds
19. Characteristic Classes and Connections
20. General Theory of Characteristic Classes
Reference
Tu - An Introduction to Manifolds
Lee - Introduction to Smooth Manifolds
Nakahara - Geometry, Topology and Physics