《分析流形和物理学》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。

肖凯-布吕埃编写的《分析流形和物理学(第2卷修订版)(英文版)》为第2卷。

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-Killing form on the Lie algebraof 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.Bicklund 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

1.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.Conformal system for Einstein equations

8.Conformal 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.Kihler 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 S3S2

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.Chern-Simons classes

11.Cocycles on the Lie algebra of a gauge group; Anomalies

12.Virasoro representation of(Diff Sl ). 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 H,.

7.Spaces Hs(Sn) and H,.,

8.Completeness of a ball on Wp in Ws

9.Distribution with laplacian in L2(In)

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 Schr6dinger

operator

16.Sobolev spaces on Riemannian manifolds

SUPPLEMENTS AND ADDITIONAL PROBLEMS

1.The isomorphism HHM4. 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.Teichmuller 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

11.Tensor distributions on submanifolds, multiple layers, and

shocks (VI. 18)

12.Discrete Boltzrnann equation (VI. 19)

Subject Index

Errata to Part I