分析、流形和物理学（第2卷 英文修订版）

分析、流形和物理学（第2卷 英文修订版）
出版时间：2015年版
内容简介
《分析、流形和物理学 （第2卷 修订版）》分为2卷,第1卷1977年初版，之后7次重印或修订。第2卷也在原来的基础上做了不少改进，增加了一部分内容讲述主纤维丛上的连通，包括完整，协变倒数，曲率，线性连通，示性类和不变曲率积分。书中有部分内容完全重写，增加了不少例子和练习，使得内容更加容易理解。目次：分析基本观点；Banach空间上的微积分；微分流行、有限维的例子；流形上的积分；Riemannian流形，K?hlerian流形；分布；微分流形，无限维的例子。读者对象：适用于物理、数学专业研究人员和学生.
目录
Preface to the second edition
Preface
Contents
Conventions
Ⅰ.REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS
1.Graded algebras
2.Berezinian
3.Tensor product of algebras
4.Clifford algebras
5.Clifford algebra as a coset of the tensor algebra
6.Fierz identity
7.Pin and Spin groups
8.Weyl spinors， helicity operator； Majorana pinors， charge conjugation
9.Representations of Spin（n， m）， n + m odd
10.Dirac adjoint
11.Lie algebra of Pin（n， m） and Spin（n， m）
12.Compact spaces
13.Compactness in weak star topology
14.Homotopy groups， general properties
15.Homotopy of topological groups
16.Spectrum of closed and self—adjoint linear operators
Ⅱ.DIFFERENTIAL CALCULUS ON BANACH SPACES
1.Supersmooth mappings
2.Berezin integration； Gaussian integrals
3.Noether's theorems Ⅰ
4.Noether's theorems Ⅱ
5.Invariance of the equations of motion
6.String action
7.Stress—energy tensor； energy with respect to a timelike vector field
Ⅲ.DIFFERENTIABLE MANIFOLDS
1.Sheaves
2.Differentiable submanifolds
3.Subgroups of Lie groups.When are they Lie subgroups？
4.Cartan—Kiiling form on the Lie algebra □ of a Lie group G
5.Direct and semidirect products of Lie groups and their Lie algebra
6.Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields
7.Homogeneous spaces； symmetric spaces
8.Examples of homogeneous spaces， Stiefel and Grassmann manifolds
9.Abelian representations of nonabelian groups
10.Irreducibility and reducibility
11.Characters
12.Solvable Lie groups
13.Lie algebras of linear groups
14.Graded bundles
Ⅳ.INTEGRATION ON MANIFOLDS
1.Cohomology.Definitions and exercises
2.Obstruction to the construction of Spin and Pin bundles； Stiefel—Whitney classes
3.Inequivalent spin structures
4.Cohomology of groups
5.Lifting a group action
6.Short exact sequence； Weyl Heisenberg group
7.Cohomology of Lie algebras
8.Quasi—linear first—order partial differential equation
9.Exterior differential systems （contributed by B.Kent Harrison）
10.Backlund transformations for evolution equations （contributed by N.H.Ibragimov）
11.Poisson manifolds Ⅰ
12.Poisson manifolds Ⅱ （contributed by C.Moreno）
13.Completely integrable systems （contributed by C.Moreno）
Ⅴ.RIEMANNIAN MANIFOLDS.KAHLERIAN MANIFOLDS
I.Necessary and sufficient conditions for Lorentzian signature
2.First fundamental form （induced metric）
3.Killing vector fields
4， Sphere Sn
5.Curvature of Einstein cylinder
6.Conformal transformation of Yang—Mills， Dirac and Higgs operators in d dimensions
7.Conforrnal system for Einstein equations
8.Conforrnal transformation of nonlinear wave equations
9.Masses of"homothetic" space—time
10.Invariant geometries on the squashed seven spheres
11.Harmonic maps
12.Composition of maps
13.Kaluza—Klein theories
14.Kahler manifolds； Calabi—Yau spaces
V BIS.CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE
1.An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base
2.Gauge transformations
3.Hopf fibering S3 → S2
4.Subbundles and reducible bundles
5.Broken symmetry and bundle reduction， Higgs mechanism
6.The Euler—Poincare characteristic
7.Equivalent bundles
8.Universal bundles.Bundle classification
9.Generalized Bianchi identity
10.Chem—Simons classes
11.Cocycles on the Lie algebra of a gauge group； Anomalies
12.Virasoro representation of □ （Diff S1 ）.Ghosts.BRST operator
Ⅵ.DISTRIBUTIONS
1.Elementary solution of the wave equation in d—dimensional spacetime
2.Sobolev embedding theorem
3.Multiplication properties of Sobolev spaces
4.The best possible constant for a Sobolev inequality on Rn， n ≥3 （contributed by H.Grosse）
5.Hardy—Littlewood—Sobolev inequality （contributed by H.Grosse）
6.Spaces Hs，δ（Rn）
7.Spaces Hs（Sn） and Hs，δ（Rn）
8.Completeness of a ball on Wsp in Ws—1p
9.Distribution with laplacian in L2（Rn）
10.Nonlinear wave equation in curved spacetime
11.Harmonic coordinates in general relativity
12.Leray theory of hyperbolic systems.Temporal gauge in general relativity
13.Einstein equations with sources as a hyperbolic system
14.Distributions and analyticity： Wightman distributions and Schwinger functions （contributed by C.Doering）
15.Bounds on the number of bound states of the SchrOdinger operator
16.Sobolev spaces on Riemannian manifolds
SUPPLEMENTS AND ADDITIONAL PROBLEMS
1.The isomorphism H □ H □ M4（R）.A supplement to Problem 1.4 （Ⅰ.17）
2.Lie derivative of spinor fields （Ⅲ.15）
3.Poisson—Lie groups， Lie bialgebras， and the generalized classical Yang—Baxter equation （Ⅳ.14） （contributed by Carlos Moreno and Luis Valero）
4.Volume of the sphere Sn.A supplement to Problem Ⅴ.4 （Ⅴ， 15）
5.TeichmuUer spaces （Ⅴ.16）
6.Yamabe property on compact manifolds （Ⅴ.17）
7.The Euler class.A supplement to Problem Vbis.6 （Vbis.13）
8.Formula for laplacians at a point of the frame bundle （Vbis.14）
9.The Berry and Aharanov—Anandan phases （Vbis.15）
10.A density theorem.A supplement to Problem Ⅵ.6 "Spaces HS，δ （Rn）" （Ⅵ.17）
11.Tensor distributions on submanifolds， multiple layers， and shocks （Ⅵ， 18）
12.Discrete Boltzmann equation （Ⅵ.19）
Subject Index
Errata to Part Ⅰ