丛、联络、度量与曲率是现代微分几何和理论物理的通用基础语言。本书旨在为数学或理论物理专业研究生提供上述核心研究对象的基本理论知识。
本书系统讲解微分流形、光滑映射、微分形式、向量场、李群与格拉斯曼流形的大量基础结论，涵盖微分拓扑的核心工具。微分几何相关内容包括测地线与雅可比场基本定理、平坦联络分类定理、示性类定义，同时介绍复几何与凯勒几何。
书中收录大量相关的经典实例与应用案例，依托实例阐释抽象理论。全书内容自洽完备，绝大多数结论配有完整证明，证明或置于正文内，或附于各章附录。
- 目录
- 前辅文
- 1 Smooth manifolds
- 1.1 Smooth manifolds
- 1.2 The inverse function theorem and implicit function theorem
- 1.3 Submanifolds of Rm
- 1.4 Submanifolds of manifolds
- 1.5 More constructions of manifolds
- 1.6 More smooth manifolds: The Grassmannians
- Appendix 1.1 How to prove the inverse function and implicit function theorems
- Appendix 1.2 Partitions of unity
- Additional reading
- 2 Matrices and Lie groups
- 2.1 The general linear group
- 2.2 Lie groups
- 2.3 Examples of Lie groups
- 2.4 Some complex Lie groups
- 2.5 The groups Sl(n; C), U(n) and SU(n)
- 2.6 Notation with regards to matrices and differentials
- Appendix 2.1 The transition functions for the Grassmannians
- Additional reading
- 3 Introduction to vector bundles
- 3.1 The definition
- 3.2 The standard definition
- 3.3 The first examples of vector bundles
- 3.4 The tangent bundle
- 3.5 Tangent bundle examples
- 3.6 The cotangent bundle
- 3.7 Bundle homomorphisms
- 3.8 Sections of vector bundles
- 3.9 Sections of TM and T*M
- Additional reading
- 4 Algebra of vector bundles
- 4.1 Subbundles
- 4.2 Quotient bundles
- 4.3 The dual bundle
- 4.4 Bundles of homomorphisms
- 4.5 Tensor product bundles
- 4.6 The direct sum
- 4.7 Tensor powers
- Additional reading
- 5 Maps and vector bundles
- 5.1 The pull-back construction
- 5.2 Pull-backs and Grassmannians
- 5.3 Pull-back of differential forms and push-forward of vector fields
- 5.4 Invariant forms and vector fields on Lie groups
- 5.5 The exponential map on a matrix group
- 5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups
- 5.7 Immersion, submersion and transversality
- Additional reading
- 6 Vector bundles with Cn as fiber
- 6.1 Definitions
- 6.2 Comparing definitions
- 6.3 Examples: The complexification
- 6.4 Complex bundles over surfaces in R3
- 6.5 The tangent bundle to a surface in R3
- 6.6 Bundles over 4-dimensional submanifolds in R5
- 6.7 Complex bundles over 4-dimensional manifolds
- 6.8 Complex Grassmannians
- 6.9 The exterior product construction
- 6.10 Algebraic operations
- 6.11 Pull-back
- Additional reading
- 7 Metrics on vector bundles
- 7.1 Metrics and transition functions for real vector bundles
- 7.2 Metrics and transition functions for complex vector bundles
- 7.3 Metrics, algebra and maps
- 7.4 Metrics on TM
- Additional reading
- 8 Geodesics
- 8.1 Riemannian metrics and distance
- 8.2 Length minimizing curves
- 8.3 The existence of geodesics
- 8.4 First examples
- 8.5 Geodesics on SO(n)
- 8.6 Geodesics on U(n) and SU(n)
- 8.7 Geodesics and matrix groups
- Appendix 8.1 The proof of the vector field theorem
- Additional reading
- 9 Properties of geodesics
- 9.1 The maximal extension of a geodesic
- 9.2 The exponential map
- 9.3 Gaussian coordinates
- 9.4 The proof of the geodesic theorem
- Additional reading
- 10 Principal bundles
- 10.1 The definition
- 10.2 A cocycle definition
- 10.3 Principal bundles constructed from vector bundles
- 10.4 Quotients of Lie groups by subgroups
- 10.5 Examples of Lie group quotients
- 10.6 Cocycle construction examples
- 10.7 Pull-backs of principal bundles
- 10.8 Reducible principal bundles
- 10.9 Associated vector bundles
- Appendix 10.1 Proof of Proposition 10.1
- Additional reading
- 11 Covariant derivatives and connections
- 11.1 Covariant derivatives
- 11.2 The space of covariant derivatives
- 11.3 Another construction of covariant derivatives
- 11.4 Principal bundles and connections
- 11.5 Connections and covariant derivatives
- 11.6 Horizontal lifts
- 11.7 An application to the classification of principal G-bundles up to isomorphism
- 11.8 Connections, covariant derivatives and pull-back bundles
- Additional reading
- 12 Covariant derivatives, connections and curvature
- 12.1 Exterior derivative
- 12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology
- 12.3 Lie derivative
- 12.4 Curvature and covariant derivatives
- 12.5 An example
- 12.6 Curvature and commutators
- 12.7 Connections and curvature
- 12.8 The horizontal subbundle revisited
- Additional reading
- 13 Flat connections and holonomy
- 13.1 Flat connections
- 13.2 Flat connections on bundles over the circle
- 13.3 Foliations
- 13.4 Automorphisms of a principal bundle
- 13.5 The fundamental group
- 13.6 The flat connections on bundles over M
- 13.7 The universal covering space
- 13.8 Holonomy and curvature
- 13.9 Proof of the classification theorem for flat connections
- Appendix 13.1 Smoothing maps
- Appendix 13.2 The proof of the Frobenius theorem
- Additional reading
- 14 Curvature polynomials and characteristic classes
- 14.1 The Bianchi Identity
- 14.2 Characteristic forms
- 14.3 Characteristic classes: Part
- 14.4 Characteristic classes: Part
- 14.5 Characteristic classes for complex vector bundles and the Chern classes
- 14.6 Characteristic classes for real vector bundles and the Pontryagin classes
- 14.7 Examples of bundles with nonzero Chern classes
- 14.8 The degree of the map g → gm from SU(2) to itself
- 14.9 A Chern–Simons form
- Appendix 14.1 The ad-invariant functions on M(n; C)
- Appendix 14.2 Integration on manifolds
- Appendix 14.3 The degree of a map
- Additional reading
- 15 Covariant derivatives and metrics
- 15.1 Metric compatible covariant derivatives
- 15.2 Torsion free covariant derivatives on T*M
- 15.3 The Levi-Civita connection/covariant derivative
- 15.4 A formula for the Levi-Civita connection
- 15.5 Covariantly constant sections
- 15.6 An example of the Levi-Civita connection
- 15.7 The curvature of the Levi-Civita connection
- Additional reading
- 16 The Riemann curvature tensor
- 16.1 Spherical metrics, flat metrics and hyperbolic metrics
- 16.2 The Schwarzchild metric
- 16.3 Curvature conditions
- 16.4 Manifolds of dimension 2: The Gauss–Bonnet formula
- 16.5 Metrics on manifolds of dimension
- 16.6 Conformal changes
- 16.7 Sectional curvatures and universal covering spaces
- 16.8 The Jacobi field equation
- 16.9 Constant sectional curvature and the Jacobi field equation
- 16.10 Manifolds of dimension
- 16.11 The Riemannian curvature of a compact matrix group
- Additional reading
- 17 Complex manifolds
- 17.1 Some basics concerning holomorphic functions on Cn
- 17.2 The definition of a complex manifold
- 17.3 First examples of complex manifolds
- 17.4 The Newlander–Nirenberg theorem
- 17.5 Metrics and almost complex structures on TM
- 17.6 The almost Kähler 2-form
- 17.7 Symplectic forms
- 17.8 Kähler manifolds
- 17.9 Complex manifolds with closed almost Kähler form
- 17.10 Examples of Kähler manifolds
- Appendix 17.1 Compatible almost complex structures
- Additional reading
- 18 Holomorphic submanifolds, holomorphic sections and curvature
- 18.1 Holomorphic submanifolds of a complex manifold
- 18.2 Holomorphic submanifolds of projective spaces
- 18.3 Proof of Proposition 18.2, about holomorphic submanifolds in CPn
- 18.4 The curvature of a Kähler metric
- 18.5 Curvature with no (0, 2) part
- 18.6 Holomorphic sections
- 18.7 Example on CPn
- Additional reading
- 19 The Hodge star
- 19.1 Definition of the Hodge star
- 19.2 Representatives of De Rham cohomology
- 19.3 A fairy tale
- 19.4 The Hodge theorem
- 19.5 Self-duality
- Additional reading
- List of lemmas, propositions, corollaries and theorems
- List of symbols
- Index
