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[交流] Differential Geometry And Lie Groups For Physicists已有4人参与

Preface page xi
Introduction 1
1 The concept of a manifold 4
1.1 Topology and continuous maps 4
1.2 Classes of smoothness of maps of Cartesian spaces 6
1.3 Smooth structure, smooth manifold 7
1.4 Smooth maps of manifolds 11
1.5 A technical description of smooth surfaces in Rn 16
Summary of Chapter 1 20
2 Vector and tensor fields 21
2.1 Curves and functions on M 22
2.2 Tangent space, vectors and vector fields 23
2.3 Integral curves of a vector field 30
2.4 Linear algebra of tensors (multilinear algebra) 34
2.5 Tensor fields on M 45
2.6 Metric tensor on a manifold 48
Summary of Chapter 2 53
3 Mappings of tensors induced by mappings of manifolds 54
3.1 Mappings of tensors and tensor fields 54
3.2 Induced metric tensor 60
Summary of Chapter 3 63
4 Lie derivative 65
4.1 Local flow of a vector field 65
4.2 Lie transport and Lie derivative 70
4.3 Properties of the Lie derivative 72
4.4 Exponent of the Lie derivative 75
4.5 Geometrical interpretation of the commutator [V,W],
non-holonomic frames 77
4.6 Isometries and conformal transformations, Killing
equations 81
Summary of Chapter 4 91
v
vi Contents
5 Exterior algebra 93
5.1 Motivation: volumes of parallelepipeds 93
5.2 p-forms and exterior product 95
5.3 Exterior algebra L∗ 102
5.4 Interior product iv 105
5.5 Orientation in L 106
5.6 Determinant and generalized Kronecker symbols 107
5.7 The metric volume form 112
5.8 Hodge (duality) operator ∗ 118
Summary of Chapter 5 125
6 Differential calculus of forms 126
6.1 Forms on a manifold 126
6.2 Exterior derivative 128
6.3 Orientability, Hodge operator and volume form on M 133
6.4 V-valued forms 139
Summary of Chapter 6 143
7 Integral calculus of forms 144
7.1 Quantities under the integral sign regarded as
differential forms 144
7.2 Euclidean simplices and chains 146
7.3 Simplices and chains on a manifold 149
7.4 Integral of a form over a chain on a manifold 150
7.5 Stokes’ theorem 151
7.6 Integral over a domain on an orientable manifold 153
7.7 Integral over a domain on an orientable Riemannian manifold 159
7.8 Integral and maps of manifolds 161
Summary of Chapter 7 163
8 Particular cases and applications of Stokes’ theorem 164
8.1 Elementary situations 164
8.2 Divergence of a vector field and Gauss’ theorem 166
8.3 Codifferential and Laplace–deRham operator 171
8.4 Green identities 177
8.5 Vector analysis in E3 178
8.6 Functions of complex variables 185
Summary of Chapter 8 188
9 Poincar´e lemma and cohomologies 190
9.1 Simple examples of closed non-exact forms 191
9.2 Construction of a potential on contractible manifolds 192
9.3∗ Cohomologies and deRham complex 198
Summary of Chapter 9 203
10 Lie groups: basic facts 204
10.1 Automorphisms of various structures and groups 204
Contents vii
10.2 Lie groups: basic concepts 210
Summary of Chapter 10 213
11 Differential geometry on Lie groups 214
11.1 Left-invariant tensor fields on a Lie group 214
11.2 Lie algebra G of a group G 222
11.3 One-parameter subgroups 225
11.4 Exponential map 227
11.5 Derived homomorphism of Lie algebras 230
11.6 Invariant integral on G 231
11.7 Matrix Lie groups: enjoy simplifications 232
Summary of Chapter 11 243
12 Representations of Lie groups and Lie algebras 244
12.1 Basic concepts 244
12.2 Irreducible and equivalent representations, Schur’s lemma 252
12.3 Adjoint representation, Killing–Cartan metric 259
12.4 Basic constructions with groups, Lie algebras and their representations 269
12.5 Invariant tensors and intertwining operators 278
12.6∗ Lie algebra cohomologies 282
Summary of Chapter 12 287
13 Actions of Lie groups and Lie algebras on manifolds 289
13.1 Action of a group, orbit and stabilizer 289
13.2 The structure of homogeneous spaces, G/H 294
13.3 Covering homomorphism, coverings SU(2) → SO(3) and
SL(2,C) → L↑
+
299
13.4 Representations of G and G in the space of functions on a G-space,
fundamental fields 310
13.5 Representations of G and G in the space of tensor fields of type ρˆ 319
Summary of Chapter 13 325
14 Hamiltonian mechanics and symplectic manifolds 327
14.1 Poisson and symplectic structure on a manifold 327
14.2 Darboux theorem, canonical transformations and symplectomorphisms 336
14.3 Poincar´e–Cartan integral invariants and Liouville’s theorem 341
14.4 Symmetries and conservation laws 346
14.5∗ Moment map 349
14.6∗ Orbits of the coadjoint action 354
14.7∗ Symplectic reduction 360
Summary of Chapter 14 368
15 Parallel transport and linear connection on M 369
15.1 Acceleration and parallel transport 369
15.2 Parallel transport and covariant derivative 372
15.3 Compatibility with metric, RLC connection 382
15.4 Geodesics 389
viii Contents
15.5 The curvature tensor 401
15.6 Connection forms and Cartan structure equations 406
15.7 Geodesic deviation equation (Jacobi’s equation) 418
15.8∗ Torsion, complete parallelism and flat connection 422
Summary of Chapter 15 428
16 Field theory and the language of forms 429
16.1 Differential forms in the Minkowski space E1,3 430
16.2 Maxwell’s equations in terms of differential forms 436
16.3 Gauge transformations, action integral 441
16.4 Energy–momentum tensor, space-time symmetries and conservation
laws due to them 448
16.5∗ Einstein gravitational field equations, Hilbert and Cartan action 458
16.6∗ Non-linear sigma models and harmonic maps 467
Summary of Chapter 16 476
17 Differential geometry on TM and T ∗M 478
17.1 Tangent bundle TM and cotangent bundle T ∗M 478
17.2 Concept of a fiber bundle 482
17.3 The maps T f and T ∗ f 485
17.4 Vertical subspace, vertical vectors 487
17.5 Lifts on TM and T ∗M 488
17.6 Canonical tensor fields on TM and T ∗M 494
17.7 Identities between the tensor fields introduced here 497
Summary of Chapter 17 497
18 Hamiltonian and Lagrangian equations 499
18.1 Second-order differential equation fields 499
18.2 Euler–Lagrange field 500
18.3 Connection between Lagrangian and Hamiltonian mechanics,
Legendre map 505
18.4 Symmetries lifted from the base manifold (configuration space) 508
18.5 Time-dependent Hamiltonian, action integral 518
Summary of Chapter 18 522
19 Linear connection and the frame bundle 524
19.1 Frame bundle π : LM → M 524
19.2 Connection form on LM 527
19.3 k-dimensional distribution D on a manifoldM 530
19.4 Geometrical interpretation of a connection form: horizontal
distribution on LM 538
19.5 Horizontal distribution on LM and parallel transport on M 543
19.6 Tensors on M in the language of LM and their parallel transport 545
Summary of Chapter 19 550
20 Connection on a principal G-bundle 551
20.1 Principal G-bundles 551
Contents ix
20.2 Connection form ω ∈ 1(P, Ad) 559
20.3 Parallel transport and the exterior covariant derivative D 563
20.4 Curvature form ∈ 2(P, Ad) and explicit expressions of D 567
20.5∗ Restriction of the structure group and connection 576
Summary of Chapter 20 585
21 Gauge theories and connections 587
21.1 Local gauge invariance: “conventional” approach 587
21.2 Change of section and a gauge transformation 594
21.3 Parallel transport equations for an object of type ρ in a gauge σ 600
21.4 Bundle P ×ρ V associated to a principal bundle π : P → M 606
21.5 Gauge invariant action and the equations of motion 607
21.6 Noether currents and Noether’s theorem 618
21.7∗ Once more (for a while) on LM 626
Summary of Chapter 21 633
22∗ Spinor fields and the Dirac operator 635
22.1 Clifford algebras C(p, q) 637
22.2 Clifford groups Pin (p, q) and Spin (p, q) 645
22.3 Spinors: linear algebra 650
22.4 Spin bundle π : SM → M and spinor fields on M 654
22.5 Dirac operator 662
Summary of Chapter 22 670
Appendix A Some relevant algebraic structures 673
A.1 Linear spaces 673
A.2 Associative algebras 676
A.3 Lie algebras 676
A.4 Modules 679
A.5 Grading 680
A.6 Categories and functors 681
Appendix B Starring 683
Bibliography 685
Index of (frequently used) symbols 687
Index 690

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