Analysis, manifolds and physics.Part II

Analysis, Manifolds and Physics......Page 4
Copyright Page......Page 5
Preface to the second edition......Page 6
Preface......Page 8
Contents......Page 10
Conventions......Page 16
1. Graded algebras......Page 18
2. Berezinian......Page 20
3. Tensor product of algebras......Page 22
4. Clifford algebras......Page 23
5. Clifford algebra as a coset of the tensor algebra......Page 31
6. Fierz identity......Page 32
7. Pin and Spin groups......Page 34
8. Weyl spinors, helicity operator; Majorana pinors, charge conjugation......Page 44
9. Representations of Spin(n, m), n + m odd......Page 50
10. Dirac adjoint......Page 53
11. Lie algebra of Pin(n, m) and Spin(n, m)......Page 54
12. Compact spaces......Page 56
13. Compactness in weak star topology......Page 57
14. Homotopy groups, general properties......Page 59
15. Homotopy of topological groups......Page 63
16. Spectrum of closed and self-adjoint linear operators......Page 64
1. Supersmooth mappings......Page 68
2. Berezin integration; Gaussian integrals......Page 74
3. Noether\'s theorems I......Page 81
4. Noether\'s theorems II......Page 88
5. Invariance of the equations of motion......Page 96
6. String action......Page 99
7. Stress-energy tensor; energy with respect to a timelike vector field......Page 100
2. Differentiable submanifolds......Page 108
3. Subgroups of Lie groups. When are they Lie subgroups?......Page 109
4. Cartan-Killing form on the Lie algebra g of a Lie group G......Page 110
5. Direct and semidirect products of Lie groups and their Lie algebra......Page 112
6. Homomorphisma and anthihomomorphisms of a life algebra into spaces of vector fields......Page 119
7. Homogeneous spaces; symmetric spaces......Page 120
8. Examples of homogeneous spaces, Stiefel and Grassmann manifolds......Page 125
9. Abelian representations of nonabelian groups......Page 127
10. Irreducibility and reducibility......Page 128
12. Solvable Lie groups......Page 131
13. Lie algebras of linear groups......Page 132
14. Graded bundles......Page 135
1. Cohomology. Definitions and exercises......Page 144
2. Obstruction to the construction of Spin and Pin bundles; Stiefel–Whitney classes......Page 151
3. Inequivalent spin structures......Page 167
4. Cohomology of groups......Page 175
5. Lifting a group action......Page 178
6. Short exact sequence; Weyl Heisenberg group......Page 180
7. Cohomology of Lie algebras......Page 184
8. Quasi-linear first-order partial differential equation......Page 188
9. Exterior differential systems......Page 190
10. Bäcklund transformations for evolution equations......Page 198
11. Poisson manifolds I......Page 201
12. Poisson manifolds II......Page 217
13. Completely integrable systems......Page 236
1. Necessary and sufficient conditions for Lorentzian signature......Page 252
2. First fundamental form (induced metric)......Page 255
3. Killing vector fields......Page 256
4. Sphere Sn......Page 257
6. Conformal transformation of Yang–Mills, Dirac and Higgs operators in d dimensions......Page 261
7. Conformal system for Einstein equations......Page 266
8. Conformal transformation of nonlinear wave equations......Page 273
9. Masses of \"homothetic\" space-time......Page 279
10. Invariant geometries on the squashed seven spheres......Page 280
11. Harmonic maps......Page 291
12. Composition of maps......Page 298
13. Kaluza–Klein theories......Page 303
14. Kähler manifolds; Calabi–Yau spaces......Page 311
1. An explicit proof of the existence of infinitely many connections on a principal bundle with paracompact base......Page 320
2. Gauge transformations......Page 322
3. Hopf fibering S3 –> S2......Page 324
4. Subbundles and reducible bundles......Page 325
5. Broken symmetry and bundle reduction, Higgs mechanism......Page 327
6. The Euler–Poincaré characteristic......Page 338
7. Equivalent bundles......Page 351
8. Universal bundles. Bundle classification......Page 352
10. Chern–Simons classes......Page 357
11. Cocycles on the Lie algebra of a gauge group; Anomalies......Page 366
12. Virasoro representation of L (Diff S1) ghosts. brst operator......Page 380
1. Elementary solution of the wave equation in d-dimensional spacetime......Page 390
2. Sobolev embedding theorem......Page 394
3. Multiplication properties of Sobolev spaces......Page 403
4. The best possible constant for a Sobolev inequality on R n, n >= 3......Page 406
5. Hardy–Littlewood-Sobolev inequality......Page 408
6. Spaces Hs,a (Rn)......Page 410
7. Spaces Hs(Sn) and Hs,18 δ(Rn)......Page 413
8. Completeness of a ball on W p s in W p s-1......Page 415
9. Distribution with laplacian in L2 (Rn)......Page 416
10. Nonlinear wave equation in curved spacetime......Page 417
11. Harmonic coordinates in general relativity......Page 422
12. Leray theory of hyperbolic systems. Temporal gauge in general relativity......Page 424
13. Einstein equations with sources as a hyperbolic system......Page 430
14. Distributions and analyticity: Wightman distributions and Schwinger functions......Page 431
15. Bounds on the number of bound states of the Schrödinger operator......Page 442
16. Sobolev spaces on Riemannian manifolds......Page 445
SUPPLEMENTS AND ADDITIONAL PROBLEMS......Page 450
1. The isomorphism H × H = M4(R). A supplement to Problem 1.4 (I. 17)......Page 452
2. Lie derivative of spinor fields (III. 15)......Page 454
3. Poisson–Lie groups, Lie bialgebras, and the generalized classical Yang-Baxter equation (IV. 14)......Page 460
4. Volume of the sphere Sn. A supplement to Problem V.4 (V. 15)......Page 493
5. Teichmuller spaces (V.16)......Page 495
6. Yamabe property on compact manifolds (V. 17)......Page 500
7. The Euler class. A supplement to Problem Vbis.6 (Vbis.13)......Page 512
8. Formula for laplacians at a point of the frame bundle (Vbis. 14)......Page 513
9. The Berry and Aharanov–Anandan phases (Vbis. 15)......Page 517
10. A density theorem. A supplement to Problem VI.6 \"Spaces Hs,δ(Rn) \'\' (VI.17)......Page 529
11. Tensor distributions on submanifolds, multiple layers, and shocks (VI. 18)......Page 530
12. Discrete Boltzmann equation (VI. 19)......Page 538
Index......Page 542
Errata to Analysis, Manifolds, and Physics, Part I......Page 548