Geometry of Classical Fields (eBook)

Front Cover 1
Geometry of Classical Fields 4
Copyright Page 5
Table of Contents 6
Introduction 10
PART I: DIFFERENTIAL GEOMETRIC PRELIMINARIES 20
Chapter 1. MANIFOLDS AND LIE GROUPS 22
1.1. Manifolds, tangent Manifolds 22
1.2. Flows, the theorem of Frobenius 32
1.3. Lie groups 39
1.4. Immersed Lie groups 53
1.5. Examples 56
1.6. Aut G for a connected G 74
1.7. The semidirect product 76
References, Chapter 1 80
Chapter 2. VECTOR BUNDLES 82
2.1. Fibre bundles 82
2.2. Vector bundles 84
2.3. Construction of vector bundles, the pull back 86
2.4. Homotopy 88
2.5. E @ F, E Q F, L(E,F) 90
2.6. A"E 92
2.7. Section modules of E, orientation in E 94
2.8. The jet bundle 96
2.9. The canonical 1–form on J N 99
2.10. Vertical and horizontal bundles, connections 104
2.11. Riemannian structures on vector bundles 107
References, Chapter 2 110
Chapter 3. ELEMENTARY DIFFERENTIAL GEOMETRY 112
3.1. The Lemma of PoincarC 112
3.2. Induced Riemannian metrics, covariant derivatives and second fundamental tensors on submanifolds of Euclidean spaces 117
3.3. Linear connections, sprays, geodesics and the exponential map 122
3.4. The canonical one–and twe–form on T M, Riemannian spray and the Levi–Civita connection * 129
3.5. Curvature tensors and the Bianchi identity 135
3.6. Embedding, the Weingarten map and the second fundamental form, the equations of Gauss and Codazzi, the mean and the Gaussian curvature 139
3.7. Geodesic spray of a right resp. left invariant metric on a Lie group 145
References, Chapter 3 151
Chapter 4. PRINCIPAL BUNDLES AND CONNECTIONS 152
4.0. Preliminaries 152
4.1. Principal bundles 157
4.2. Examples 170
4.3. Associated bundles 178
4.4. Connections 187
4.5. The special case G —G/H 206
4.6. Invariant connections on principal bundles 211
4.7. Linear connections in vector bundles 215
4.8. Connection forms and linear connections 222
References, Chapter 4 229
Chapter 5. FUNCTION SPACE 230
5.1. Space of functions and distributions 230
5.2. Globally defined function spaces such as Appendix: Currents c(M,E), ck(M,N), c(M,N), ro(E), rE, HKE 238
5.3. Remarks on calculus 247
5.4. Ck(M,N) as a manifold 254
5.5. Examples of manifolds of maps and some tangent mappings 261
5.6. Gauge groups 274
5.7. On the deformation of differentials of immersions 284
References, Chapter 5 293
PART II: COVARIANT HAMILTONIAN DYNAMICS 296
Chapter 6. NON–RELATIVISTIC DYNAMICS 298
6.1. Action principle 299
6.2. Canonical Hamiltonian formalism 303
6.3. Symplectic manifolds and Poisson algebras 307
6.4. Degenerate Lagrangians and constraints 315
6.5. Cartan equations and symmetries 321
6.6. Generalized Hamiltonian dynamics 327
6.7. Constraints and reduction 337
References, Chapter 6 341
Chapter 7. DYNAMICS OF CLASSICAL FIELDS 344
7.1. Action principle and field equations 345
7.2. Boundary conditions, symmetries and conservation laws 352
7.3. Cartan formalism in the space of Cauchy data 360
7.4. Hamiltonian formulation 368
7.5. Constraints and reduction 376
References, Chapter 7 382
Chapter 8. YANG–MILLS THEORY 384
8.1. Field equations 385
8.2. Gauge transformations and conservation laws 390
8.3. Hamiltonian formulation 397
8.4. Minkowski space 402
8.5. Maxwell–Dirac theory 408
8.6. Yang–Mills charges 418
References, Chapter 8 423
Chapter 9. GENERAL RELATIVITY 426
9.1. Field equations 427
9.2. Conservation laws and constraints 432
9.3. Hamiltonian formulation 441
9.4. Asymptotically flat space times 448
References, Chapter 9 460
Index, Part II 462
Glossary of symbols, Part II 466