分析、流形和物理学 第2卷 英文PDF|Epub|txt|kindle电子书版本网盘下载

分析、流形和物理学 第2卷 英文PDF格式电子书版下载

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Ⅰ．REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS1

1．Graded algebras1

2．Berezinian3

3．Tensor product of algebras5

4．Clifford algebras6

5．Clifford algebra as a coset of the tensor algebra14

6．Fierz identity15

7．Pin and Spin groups17

8．Weyl spinors,helicity operator;Majorana pinors,charge conjugation27

9．Representations of Spin(n,m),n+m odd33

10．Dirac adjoint36

11．Lie algebra of Pin(n,m)and Spin(n,m)37

12．Compact spaces39

13．Compactness in weak star topology40

14．Homotopy groups,general properties42

15．Homotopy of topological groups46

16．Spectrum of closed and self-adjoint linear operators47

Ⅱ．DIFFERENTIAL CALCULUS ON BANACH SPACES51

1．Supersmooth mappings51

2．Berezin integration;Gaussian integrals57

3．Noether's theorems Ⅰ64

4．Noether's theorems Ⅱ71

5．Invariance of the equations of motion79

6．String action82

7．Stress-energy tensor;energy with respect to a timelike vector field83

Ⅲ．DIFFERENTIABLE MANIFOLDS91

1．Sheaves91

2．Differentiable submanifolds91

3．Subgroups of Lie groups.When are they Lie subgroups?92

4．Cartan-Killing form on the Lie algebra？of a Lie group G93

5．Direct and semidirect products of Lie groups and their Lie algebra95

6．Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields102

7．Homogeneous spaces;symmetric spaces103

8．Examples of homogeneous spaces,Stiefel and Grassmann manifolds108

9．Abelian representations of nonabelian groups110

10．Irreducibility and reducibility111

11．Characters114

12．Solvable Lie groups114

13．Lie algebras of linear groups115

14．Graded bundles118

Ⅳ .INTEGRATION ON MANIFOLDS127

1．Cohomology.Definitions and exercises127

2．Obstruction to the construction of Spin and Pin bundles;Stiefel-Whitney classes134

3．Inequivalent spin structures150

4．Cohomology of groups158

5．Lifting a group action161

6．Short exact sequence;Weyl Heisenberg group163

7．Cohomology of Lie algebras167

8．Quasi-linear first-order partial differential equation171

9．Exterior differential systems(contributed by B.Kent Harrison)173

10．B？cklund transformations for evolution equations(contributed by N.H.Ibragimov)181

11．Poisson manifolds Ⅰ184

12．Poisson manifolds Ⅱ(contributed by C.Moreno)200

13．Completely integrable systems(contributed by C.Moreno)219

Ⅴ．RIEMANNIAN MANIFOLDS.K？HLERIAN MANIFOLDS235

1．Necessary and sufficient conditions for Lorentzian signature235

2．First fundamental form(induced metric)238

3．Killing vector fields239

4．Sphere Sn240

5．Curvature of Einstein cylinder244

6．Conformal transformation of Yang-Mills,Dirac and Higgs operators in d dimensions244

7．Conformal system for Einstein equations249

8．Conformal transformation of nonlinear wave equations256

9．Masses of"homothetic"space-time262

10．Invariant geometries on the squashed seven spheres263

11．Harmonic maps274

12．Composition of maps281

13．Kaluza-Klein theories286

14．K？hler manifolds;Calabi-Yau spaces294

V BIS.CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE303

1．An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base303

2．Gauge transformations305

3．Hopf fibering S3→S2307

4．Subbundles and reducible bundles308

5．Broken symmetry and bundle reduction,Higgs mechanism310

6．The Euler-Poincaré characteristic321

7．Equivalent bundles334

8．Universal bundles.Bundle classification335

9．Generalized Bianchi identity340

10．Chem-Simons classes340

11．Cocycles on the Lie algebra of a gauge group;Anomalies349

12．Virasoro representation of ？(Diff S1).Ghosts.BRST operator363

Ⅵ．DISTRIBUTIONS373

1．Elementary solution of the wave equation in d-dimensional spacetime373

2．Sobolev embedding theorem377

3．Multiplication properties of Sobolev spaces386

4．The best possible constant for a Sobolev inequality on Rn,n？3(contributed by H.Grosse)389

5．Hardy-Littlewood-Sobolev inequality(contributed by H.Grosse)391

6．Spaces Hs,δ(Rn)393

7．Spaces Hs(Sn)and Hs,δ(Rn)396

8．Completeness of a ball on Wps in Wps-1398

9．Distribution with laplacian in L2(Rn)399

10．Nonlinear wave equation in curved spacetime400

11．Harmonic coordinates in general relativity405

12．Leray theory of hyperbolic systems.Temporal gauge in general relativity407

13．Einstein equations with sources as a hyperbolic system413

14．Distributions and analyticity:Wightman distributions and Schwinger functions(contributed by C.Doering)414

15．Bounds on the number of bound states of the Schr？dinger operator425

16．Sobolev spaces on Riemannian manifolds428

SUPPLEMENTS AND ADDITIONAL PROBLEMS433

1．The isomorphism H？H？M4(R).A supplement to Problem Ⅰ.4(Ⅰ.17)435

2．Lie derivative of spinor fields(Ⅲ.15)437

3．Poisson-Lie groups,Lie bialgebras,and the generalized classical Yang-Baxter equation(Ⅳ.14)(contributed by Carlos Moreno and Luis Valero)443

4．Volume of the sphere Sn.A supplement to Problem Ⅴ.4(Ⅴ.15)476

5．Teichmuller spaces(Ⅴ.16)478

6．Yamabe property on compact manifolds(Ⅴ.17)483

7．The Euler class.A supplement to Problem Vbis.6(Vbis.13)495

8．Formula for laplacians at a point of the frame bundle(Vbis.14)496

9．The Berry and Aharanov-Anandan phases(Vbis.15)500

10．A density theorem.A supplement to Problem Ⅵ.6"Spaces Hs,δ(Rn)"(Ⅵ.17)512

11．Tensor distributions on submanifolds,multiple layers,and shocks(Ⅵ.18)513

12．Discrete Boltzmann equation(Ⅵ.19)521

Subject Index525

Errata to Part I531