Elsevier S & T

Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition

Choquet-Bruhat, Y.

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Table of contents
  • Cover
  • Preface to the second editionv
  • Prefacevii
  • Contentsix
  • Conventionsxv
  • CHAPTER 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
  • CHAPTER 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
  • CHAPTER 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 g of a Lie group G93
  • 5. Direct and semidirect products of Lie groups and their Lie algebra95
  • 6. Homomorphisma and anthihomomorphisms of a life 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
  • CHAPTER 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 systems173
  • 10. Bäcklund transformations for evolution equations181
  • 11. Poisson manifolds I184
  • 12. Poisson manifolds II200
  • 13. Completely integrable systems219
  • CHAPTER V. RIEMANNIAN MANIFOLDS. KÄHLERIAN 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. Kähler manifolds; Calabi–Yau spaces294
  • CHAPTER V BIS. CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE303
  • 1. An explicit proof of the existence of infinitely many connections on a principal bundle with para303
  • 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. Chern–Simons classes340
  • 11. Cocycles on the Lie algebra of a gauge group; Anomalies349
  • 12. Virasoro representation of L (Diff S1) ghosts. brst operator363
  • CHAPTER 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 R n, n >= 3389
  • 5. Hardy–Littlewood-Sobolev inequality391
  • 6. Spaces Hs,a (Rn)393
  • 7. Spaces Hs(Sn) and Hs,δ(Rn)396
  • 8. Completeness of a ball on W p s in W p s-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 functions414
  • 15. Bounds on the number of bound states of the Schrödinger operator425
  • 16. Sobolev spaces on Riemannian manifolds428
  • SUPPLEMENTS AND ADDITIONAL PROBLEMS433
  • 1. The isomorphism H × H = M4(R). A supplement to Problem 1.4 (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)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
Book details
  • Vendor Elsevier S & T
  • SKU 9780444504739
  • ISBN-13 9780080527154
  • Author Choquet-Bruhat, Y.
  • Category Mathematics
  • Subject Calculus

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Description eBook delivery & access Reviews
Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.