物理学中的拓扑与几何(英文影印版)

书籍目录

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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