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Ⅰ．REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS 1

1．Graded algebras 1

2．Berezinian 3

3．Tensor product of algebras 5

4．Clifford algebras 6

5．Clifford algebra as a coset of the tensor algebra 14

6．Fierz identity 15

7．Pin and Spin groups 17

8．Weyl spinors,helicity operator;Majorana pinors,charge conjugation 27

9．Representations of Spin(n,m),n+m odd 33

10．Dirac adjoint 36

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

12．Compact spaces 39

13．Compactness in weak star topology 40

14．Homotopy groups,general properties 42

15．Homotopy of topological groups 46

16．Spectrum of closed and self-adjoint linear operators 47

Ⅱ．DIFFERENTIAL CALCULUS ON BANACH SPACES 51

1．Supersmooth mappings 51

2．Berezin integration;Gaussian integrals 57

3．Noether's theorems Ⅰ 64

4．Noether's theorems Ⅱ 71

5．Invariance of the equations of motion 79

6．String action 82

7．Stress-energy tensor;energy with respect to a timelike vector field 83

Ⅲ．DIFFERENTIABLE MANIFOLDS 91

1．Sheaves 91

2．Differentiable submanifolds 91

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

4．Cartan-Killing form on the Lie algebra？of a Lie group G 93

5．Direct and semidirect products of Lie groups and their Lie algebra 95

6．Homomorphisms and antihomomorphisms of a Lie algebra into spaces of vector fields 102

7．Homogeneous spaces;symmetric spaces 103

8．Examples of homogeneous spaces,Stiefel and Grassmann manifolds 108

9．Abelian representations of nonabelian groups 110

10．Irreducibility and reducibility 111

11．Characters 114

12．Solvable Lie groups 114

13．Lie algebras of linear groups 115

14．Graded bundles 118

Ⅳ .INTEGRATION ON MANIFOLDS 127

1．Cohomology.Definitions and exercises 127

2．Obstruction to the construction of Spin and Pin bundles;Stiefel-Whitney classes 134

3．Inequivalent spin structures 150

4．Cohomology of groups 158

5．Lifting a group action 161

6．Short exact sequence;Weyl Heisenberg group 163

7．Cohomology of Lie algebras 167

8．Quasi-linear first-order partial differential equation 171

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 Ⅰ 184

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

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

Ⅴ．RIEMANNIAN MANIFOLDS.K？HLERIAN MANIFOLDS 235

1．Necessary and sufficient conditions for Lorentzian signature 235

2．First fundamental form(induced metric) 238

3．Killing vector fields 239

4．Sphere Sn 240

5．Curvature of Einstein cylinder 244

6．Conformal transformation of Yang-Mills,Dirac and Higgs operators in d dimensions 244

7．Conformal system for Einstein equations 249

8．Conformal transformation of nonlinear wave equations 256

9．Masses of"homothetic"space-time 262

10．Invariant geometries on the squashed seven spheres 263

11．Harmonic maps 274

12．Composition of maps 281

13．Kaluza-Klein theories 286

14．K？hler manifolds;Calabi-Yau spaces 294

V BIS.CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE 303

1．An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base 303

2．Gauge transformations 305

3．Hopf fibering S3→S2 307

4．Subbundles and reducible bundles 308

5．Broken symmetry and bundle reduction,Higgs mechanism 310

6．The Euler-Poincaré characteristic 321

7．Equivalent bundles 334

8．Universal bundles.Bundle classification 335

9．Generalized Bianchi identity 340

10．Chem-Simons classes 340

11．Cocycles on the Lie algebra of a gauge group;Anomalies 349

12．Virasoro representation of ？(Diff S1).Ghosts.BRST operator 363

Ⅵ．DISTRIBUTIONS 373

1．Elementary solution of the wave equation in d-dimensional spacetime 373

2．Sobolev embedding theorem 377

3．Multiplication properties of Sobolev spaces 386

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-1 398

9．Distribution with laplacian in L2(Rn) 399

10．Nonlinear wave equation in curved spacetime 400

11．Harmonic coordinates in general relativity 405

12．Leray theory of hyperbolic systems.Temporal gauge in general relativity 407

13．Einstein equations with sources as a hyperbolic system 413

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 operator 425

16．Sobolev spaces on Riemannian manifolds 428

SUPPLEMENTS AND ADDITIONAL PROBLEMS 433

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 Index 525

Errata to Part I 531