Introduction
References
Topology
2.1 Basic Definitions
2.2 Base of Topology， Metric， Norm
2.3 Derivatives
2.4 Compactness
2.5 Connectedness， Homotopy
2.6 Topological Charges in Physics
References
Manifolds
3.1 Charts and Atlases
3.2 Smooth Manifolds
3.3 Tangent Spaces
3.4 Vector Fields
3.5 Mappings of Manifolds， Submanifolds
3.6 Frobenius' Theorem
3.7 Examples from Physics
3.7.1 Classical Point Mechanics
3.7.2 Classical and Quantum Mechanics
3.7.3 Classical Point Mechanics Under
Momentum Constraints
3.7.4 Classical Mechanics Under Velocity Constraints
3.7.5 Thermodynamics
References
4 Tensor Fields
4.1 Tensor Algebras
4.2 Exterior Algebras
4.3 Tensor Fields and Exterior Forms
4.4 Exterior Differential Calculus
References
5 Integration， Homology and Cohomology
5.1 Prelude in Euclidean Space
5.2 Chains of Simplices
5.3 Integration of Differential Forms
5.4 De Rham Cohomology
5.5 Homology and Homotopy
5.6 Homology and Cohomology of Complexes
5.7 Euler's Characteristic
5.8 Critical Points
5.9 Examples from Physics
References
Lie Groups
6.1 Lie Groups and Lie Algebras
6.2 Lie Group Homomorphisms and Representations
6.3 Lie Subgroups
6.4 Simply Connected Covering Group
6.5 The Exponential Mapping
6.6 The General Linear Group Gl（n，K）
6.7 Example from Physics： The Lorentz Group
6.8 The Adjoint Representation
References
Bundles and Connections
7.1 Principal Fiber Bundles
7.2 Frame Bundles
7.3 Connections on Principle Fiber Bundles
7.4 Parallel Transport and Holonomy
7.5 Exterior Covariant Derivative and Curvature Form
7.6 Fiber Bundles
7.7 Linear and Affine Connections
7.8 Curvature and Torsion Tensors
7.9 Expressions in Local Coordinates on M
References
Parallelism， Holonomy， Homotopy and （Co）homology
8.1 The Exact Homotopy Sequence
8.2 Homotopy of Sections
8.3 Gauge Fields and Connections on R4
8.4 Gauge Fields and Connections on Manifolds
8.5 Characteristic Classes
8.6 Geometric Phases in Quantum Physics
8.6.1 Berry—Simon Connection
8.6.2 Degenerate Case
8.6.3 Electrical Polarization
8.6.4 Orbital Magnetism
8.6.5 Topological Insulators
8.7 Gauge Field Theory of Molecular Physics
References
Riemannian Geometry
9.1 Riemannian Metric
9.2 Homogeneous Manifolds
9.3 Riemannian Connection
9.4 Geodesic Normal Coordinates
9.5 Sectional Curvature
9.6 Gravitation
9.7 Complex， Hermitian and Kaihlerian Manifolds
References
Compendium
List of Symbols
Index
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