物理学家的微分几何DIFFERENTIAL GEOMETRY FOR PHYSICISTS
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作者： Bo-Yuan Hou著
出 版 社： 东南大学出版社
出版时间： 1997-12-1字数：版次： 1页数： 546印刷时间： 1997/04/01开本：印次： 1纸张： 胶版纸I S B N ： 9789810231057包装： 精装内容简介
This book is divided into 14 chapters, with 18 appendices as introductionto prerequisite topological and algebraic knowledge, etc. The first sevenchapters focus on local analysis. This can be used as a fundamental textbookfor graduate students of theoretical physics. Chapters 8-10 discuss geometryon fibre bundles, which facilitates further reference for researchers. The lastfour chapters deal with the Atiyah-Singer index theorem, its generalizationand its application, quantum anomaly, cohomology field theory and noncom-mutative geometry, giving the reader a glimpse of the frontier of currentresearch in theoretical physics.
目录
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
……
8Homotopy 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
