包邮物理学家用的微分几何和李群

Preface
Introduction
1 The concept of a manifold
1.1 Topology and continuous maps
1.2 Classes of smoothness of maps of Cartesian spaces
1.3 Smooth structure, smooth manifold
1.4 Smooth maps of manifolds
1.5 A technical description of smooth surfaces in Rn
Summary of Chapter 1

2 Vector and tensor fields
2.1 Curves and functions on M
2.2 Tangent space, vectors and vector fields
2.3 Integral curves of a vector field
2.4 Linear algebra of tensors (multilinear algebra)
2.5 Tensor fields on M
2.6 Metric tensor on a manifold
Summary of Chapter 2

3 Mappings of tensors induced by mappings of manifolds
3.1 Mappings of tensors and tensor fields
3.2 Induced metric tensor
Summary of Chapter 3

4 Lie derivative
4.1 Local flow of a vector field
4.2 Lie transport and Lie derivative
4.3 Properties of the Lie derivative
4.4 Exponent of the Lie derivative
4.5 Geometrical interpretation of the commutator [V, W], non-holonomic frames
4.6 Isometries and conformal transformations, Killing equations
Summary of Chapter 4

5 Exterior algebra
5.1 Motivation: volumes of paraUelepipeds
5.2 p-forms and exterior product
5.3 Exterior algebra AL*
5.4 Interior product iv
5.5 Orientation in L
5.6 Determinant and generalized Kronecker symbols
5.7 The metric volume form
5.8 Hodge (duality) operator*
Summary of Chapter 5

6 Differential calculus of forms
6.1 Forms on a manifold
6.2 Exterior derivative
6.3 Orientability, Hodge operator and volume form on M
6.4 V-valued forms
Summary of Chapter 6

7 Integral calculus of forms
7.1 Quantities under the integral sign regarded as differential forms
7.2 Euclidean simplices and chains
7.3 Simplices and chains on a manifold
7.4 Integral of a form over a chain on a manifold
7.5 Stokes' theorem
7.6 Integral over a domain on an orientable manifold
7.7 Integral over a domain on an orientable Riemannian manifold
7.8 Integral and maps of manifolds
Summary of Chapter 7

8 Particular cases and applications of Stokes' theorem
8.1 Elementary situations
8.2 Divergence of a vector field and Gauss' theorem
8.3 Codifferential and LaPlace-deRhana operator
8.4 Green identities
8.5 Vector analysis in E3
8.6 Functions of complex variables
Summary of Chapter 8

9 Poincare lemma and cohomologies
9.1 Simple examples of closed non-exact forms
9.2 Construction of a potential on contractible manifolds
9.3* Cohomologies and deRham complex
Summary of Chapter 9

10 Lie groups: basic facts
10.1 Automorphisms of various structures and groups
10.2 Lie groups: basic concepts
Summary of Chapter 10

11 Differential geometry on Lie groups
11.1 Left-invariant tensor fields on a Lie group
11.2 Lie algebra g of a group G
11.3 One-parameter subgroups
11.4 Exponential map
11.5 Derived homomorphism of Lie algebras
11.6 Invariant integral on G
11.7 Matrix Lie groups: enjoy simplifications
Summary of Chapter 11

12 Representations of Lie groups and Lie algebras
12.1 Basic concepts
12.2 Irreducible and equivalent representations, Schur's lemma
12.3 Adjoint representation, Killing-Cartan metric
12.4 Basic constructions with groups, Lie algebras and their representations
12.5 Invariant tensors and intertwining operators
12.6* Lie algebra cohomologies
Summary of Chapter 12

13 Actions of Lie groups and Lie algebras on manifolds
13.1 Action of a group, orbit and stabilizer
13.2 The structure of homogeneous spaces, G/H
13.3 Covering homomorphism, coverings SU(2) →SO(3) andSL(2, C)→ L↑+
13.4 Representations of G and g in the space of functions on a G-space, fundamental fields
13.5 Representations of G and g in the space of tensor fields of type p
Summary of Chapter 13

14 Hamiltonian mechanics and symplectic manifolds
14.1 Poisson and sympl