物理学家的微分几何DIFFERENTIAL GEOMETRY FOR PHYSICISTS

This book is divided into 14 chapters, with 18 appendices as introduction　to prerequisite topological and algebraic knowledge, etc. The first seven　chapters focus on local analysis. This can be used as a fundamental textbook　for graduate students of theoretical physics. Chapters 8-10 discuss geometry　on fibre bundles, which facilitates further reference for researchers. The last　four chapters deal with the Atiyah-Singer index theorem, its generalization　and its application, quantum anomaly, cohomology field theory and noncom-mutative geometry, giving the reader a glimpse of the frontier of current　research in theoretical physics.

Preface
1 Differentiable Manifolds and Differential Forms
1.1 Manifold
1.2 Differentiable manifold
1.3 Tangent space and tangent vector field
1.4 Cotangent vector field
1.5 Tensor product, exterior product and various higher order tensor fields
1.6 Exterior differentiation
1.7 Orientation and Stokes formula
　Notations and formulae
　Exercises
2 Transformation of Manifold, Manifolds with Given Vector Fields and Lie Group Manifold
　2.1 Continuous mapping between manifolds and its induced mapping
　2.2 Integral submanifold and Frobenius theorem
　2.3 Integrability of differential equations and Frobenius theorem in terms of differential forms
　2.4 The flow of vector fields, one parameter local Lie transformation groups and Lie derivative
　2.5 Lie group, Lie algebra and exponential map
　2.6 Lie transformation groups, orbit and the space of orbits
　Notations and Formulae
　Exercises
3 Affine Connection and Covariant Differentiation
　3.1 Moving frame approach to tensor field
　3.2 Affine connection and covariant differentiation
　3.3 The curvature 2-form and the curvature tensor
　3.4 Torsion tensor
　3.5 Covariant exterior differential
　3.6 Holonomy group of connections
　3.7 Berry phase, holonomy in physical system
　Notations and Formulea
　Exercises
4 Riemannian Manifold
　4.1 Metric tensor field, Hodge star and codifferentiation
　4.2 Riemannian connection
　4.3 Riemannian curvature
　4.4 Bianchi identity and Einstein field equation of gravity
　4.5 Isometry, conformal transformation and constant curvature space
　4.6 Orthogonal frame field and spin connection
　4.7 Surfaces and curves in 3-dimensional Euclidean space
　4.8 The computation of Riemannian curvature tensor
　4.9 Pseudosphere and Backlund transformation
　Notations and Formulae
　Exercises
5 Sympleetic Manifold and Contact Manifold
　5.1 Symplectic manifold
　5.2 Special submanifolds of symplectic manifold
　5.3 Symplectic and Hamiltonian vector fields, Poisson bracket
　5.4 Poission manifold and symplectic leaves
　5.5 Homogeneous symplectic manifold and the reduced phase space
　5.6 Contact manifold
　Notations and Formulae
　Exercises
6 Complex Manifolds
　6.1 Complex structure of manifolds, almost complex manifolds
　6.2 Integrable condition of almost complex structure
　6.3 Hermitian manifold
　6.4 Kahler manifold
　6.5 Connections on complex manifold
　6.6 Riemannian symmetric space, its Kahler structure and nonlinear real ization
　6.7 Nonlinear a-models, soliton solutions and their geometric meaning
　Notations and Formulae
　Exercises
7 Homology of Manifolds
7.1 Homotopic mapping and manifolds with the same homotopy type
7.2 Singular homology group
7.3 General homology group and universal coefficient theorem
……
8　Homotopy of Manifold,Fibre Bundle,Classification of Fibre Bun-dles
9 Differential Geometry of Fibre Bundle,Yang-Mills Gauge Theory
10 Characteristic Classes
11 The Atiyah-Singer Index Theorem
12 Index Theorem on Manifold With Boundary and on Open Infinite Manifold
13 Family Index Theorem,Topological properties of Quantum Gauge Theory
14 Noncommutative Geomitry,Quantum Group,and q-deformation of Chern-Characters
Appendix
Refernces
Index