The geometry of physics: an introduction

Cover ......Page 1
Title page ......Page 2
Date-line ......Page 3
Dedication ......Page 4
Contents ......Page 6
Preface to the Revised Printing ......Page 18
Preface ......Page 20
I. Manifolds, Tensors, and Exterior Forms ......Page 24
1.1. Submanifolds of Euclidean Space ......Page 26
1.1a. Submanifolds of $\mathbb{R}^n$ ......Page 27
1.1b. The Geometry of Jacobian Matrices: The "Differential" ......Page 30
1.1c. The Main Theorem on Submanifolds of $\mathbb{R}^n$ ......Page 31
1.1d. A Nontrivial Example: The Configuration Space of a Rigid Body ......Page 32
1.2a. Some Notions from Point Set Topology ......Page 34
1.2b. The Idea of a Manifold ......Page 36
1.2c. A Rigorous Definition of a Manifold ......Page 42
1.2d. Complex Manifolds: The Riemann Sphere ......Page 44
1.3. Tangent Vectors and Mappings ......Page 45
1.3a. Tangent or "Contravariant" Vectors ......Page 46
1.3b. Vectors as Differential Operators ......Page 47
1.3c. The Tangent Space to $M^n$ at a Point ......Page 48
1.3d. Mappings and Submanifolds of Manifolds ......Page 49
1.3e. Change of Coordinates ......Page 52
1.4a. Vector Fields and Flows on $\mathbb{R}^n$ ......Page 53
1.4b. Vector Fields on Manifolds ......Page 56
1.4c. Straightening Flows ......Page 57
2.1a. Linear Functional and the Dual Space ......Page 60
2.1b. The Differential of a Function ......Page 63
2.1c. Scalar Products in Linear Algebra ......Page 65
2.1d. Riemannian Manifolds and the Gradient Vector ......Page 68
2.1e. Curves of Steepest Ascent ......Page 69
2.2a. The Tangent Bundle ......Page 71
2.2b. The Unit Tangent Bundle ......Page 73
2.3b. The Pull-Back of a Covector ......Page 75
2.3c. The Phase Space in Mechanics ......Page 77
2.3d. The Poincare 1-Forai ......Page 79
2.4a. Covariant Tensors ......Page 81
2.4b. Contravariant Tensors ......Page 82
2.4c. Mixed Tensors ......Page 83
2.4d. Transformation Properties of Tensors ......Page 85
2.4e. Tensor Fields on Manifolds ......Page 86
2.5b. The Grassmann or Exterior Algebra ......Page 89
2.5d. Special Cases of the Exterior Product ......Page 93
2.5e. Computations and Vector Analysis ......Page 94
2.6a. The Exterior Differential ......Page 96
2.6b. Examples in $\mathbb{R}^3$ ......Page 98
2.6c. A Coordinate Expression for $d$ ......Page 99
2.7a. The Pull-Back of a Covariant Tensor ......Page 100
2.7b. The Pull-Back in Elasticity ......Page 103
2.8a. Orientation of a Vector Space ......Page 105
2.8b. Orientation of a Manifold ......Page 106
2.8c. Orientability and 2-Sided Hypersurfaces ......Page 107
2.8e. Pseudoforms and the Volume Form ......Page 108
2.8f. The Volume Form in a Riemannian Manifold ......Page 110
2.9a. Interior Products and Contractions ......Page 112
2.9b. Interior Product in $\mathbb{R}^3$ ......Page 113
2.9c. Vector Analysis in $\mathbb{R}^3$ ......Page 115
2.10. Dictionary ......Page 117
3.1a. Integration of a $p$-Form in $\mathbb{R}^p$ ......Page 118
3.1b. Integration over Parameterized Subsets ......Page 119
3.1c. Line Integrals ......Page 120
3.1d. Surface Integrals ......Page 122
3.1e. Independence of Parameterization ......Page 124
3.1g. Concluding Remarks ......Page 125
3.2. Integration over Manifolds with Boundary ......Page 127
3.2a. Manifolds with Boundary ......Page 128
3.2b. Partitions of Unity ......Page 129
3.2c. Integration over a Compact Oriented Submanifold ......Page 131
3.2d. Partitions and Riemannian Metrics ......Page 132
3.3a. Orienting the Boundary ......Page 133
3.3b. Stokes's Theorem ......Page 134
3.4. Integration of Pseudoforms ......Page 137
3.4b. Submanifolds with Transverse Orientation ......Page 138
3.4c. Integration over a Submanifold with Transverse Orientation ......Page 139
3.4d. Stokes's Theorem for Pseudoforms ......Page 140
3.5a. Charge and Current in Classical Electromagnetism ......Page 141
3.5b. The Electric and Magnetic Fields ......Page 142
3.5c. Maxwell's Equations ......Page 143
3.5d. Forms and Pseudoforms ......Page 145
4.1a. The Lie Bracket ......Page 148
4.1b. Jacobi's Variational Equation ......Page 150
4.1c. The Flow Generated by $[X,Y]$ ......Page 152
4.2a. Lie Derivatives of Forms ......Page 155
4.2b. Formulas Involving the Lie Derivative ......Page 157
4.2c. Vector Analysis Again ......Page 159
4.3a. The Autonomous (Time-Independent) Case ......Page 161
4.3b. Time-Dependent Fields ......Page 163
4.3c. Differentiating Integrals ......Page 165
4.4. A Problem Set on Hamiltonian Mechanics ......Page 168
4.4a. Time-Independent Hamiltonians ......Page 170
4.4b. Time-Dependent Hamiltonians and Hamilton's Principle ......Page 174
4.4c. Poisson brackets ......Page 177
5.1. A More General Stokes's Theorem ......Page 178
5.2. Closed Forms and Exact Forms ......Page 179
5.3. Complex Analysis ......Page 181
5.4. The Converse to the Poincare Lemma ......Page 183
5.5. Finding Potentials ......Page 185
6.1a. Planes in $\mathbb{R}^3$ ......Page 188
6.1c. Distributions and 1-Forms ......Page 190
6.1d. The Frobenius Theorem ......Page 192
6.2a. Foliations and Maximal Leaves ......Page 195
6.2b. Systems of Mayer-Lie ......Page 197
6.2c. Holonomic and Nonholonomic Constraints ......Page 198
6.3a. Introduction ......Page 201
6.3b. The First Law of Thermodynamics ......Page 202
6.3c. Some Elementary Changes of State ......Page 203
6.3d. The Second Law of Thermodynamics ......Page 204
6.3e. Entropy ......Page 206
6.3f. Increasing Entropy ......Page 208
6.3g. Chow's Theorem on Accessibility ......Page 210
II. Geometry and Topology ......Page 212
7.1a. Curvature of a Space Curve in $\mathbb{R}^3$ ......Page 214
7.1b. Minkowski Space and Special Relativity ......Page 215
7.2a. Minkowski's Electromagnetic Field Tensor ......Page 219
7.2b. Maxwell's Equations ......Page 221
8.1a. The First Fundamental Form, or Metric Tensor ......Page 224
8.1b. The Second Fundamental Form ......Page 226
8.2a. Symmetry and Self-Adjointness ......Page 228
8.2b. Principal Normal Curvatures ......Page 229
8.2c. Gauss and Mean Curvatures: The Gauss Normal Map ......Page 230
8.3a. The Brouwer Degree ......Page 233
8.3b. Complex Analytic (Holomorphic) Maps ......Page 237
8.3d. The Kronecker Index of a Vector Field ......Page 238
8.3e. The Gauss Looping Integral ......Page 241
8.4a. The First Variation of Area ......Page 244
8.4b. Soap Bubbles and Minimal Surfaces ......Page 249
8.5a. The Equations of Gauss and Codazzi ......Page 251
8.5b. The Theorema Egregium ......Page 253
8.6a. The First Variation of Arc Length ......Page 255
8.6b. The Intrinsic Derivative and the Geodesic Equation ......Page 257
8.7. The Parallel Displacement of Levi-Civita ......Page 259
9.1a. Covariant Derivative ......Page 264
9.1b. Curvature of an Affine Connection ......Page 267
9.1c. Torsion and Symmetry ......Page 268
9.2. The Riemannian Connection ......Page 269
9.3a. Vector-Valued Forms ......Page 270
9.3b. The Covariant Differential of a Vector Field ......Page 271
9.3c. Cartan's Structural Equations ......Page 272
9.3d. The Exterior Covariant Differential of a Vector-Valued Form ......Page 273
9.3e. The Curvature 2-Forms ......Page 274
9.4b. Change of Frame ......Page 276
9.5a. The Riemannian Connection ......Page 278
9.5c. An Example ......Page 280
9.6. Parallel Displacement and Curvature on a Surface ......Page 282
9.7b. The Horizontal Distribution of an Affine Connection ......Page 286
9.7c. Riemann's Theorem ......Page 289
10.1a. Vector Fields Along a Surface in $M^n$ ......Page 292
10.1b. Geodesics ......Page 294
10.1c. Jacobi Fields ......Page 295
10.1d. Energy ......Page 297
10.2a. Hamilton's Principle in the Tangent Bundle ......Page 298
10.2b. Hamilton's Principle in Phase Space ......Page 300
10.2c. Jacobi's Principle of "Least" Action ......Page 301
10.2d. Closed Geodesics and Periodic Motions ......Page 304
10.3a. Gaussian Coordinates ......Page 307
10.3b. Normal Coordinates on a Surface ......Page 310
10.3c. Spiders and the Universe ......Page 311
11.1a. The Metric Potentials ......Page 314
11.1b. Einstein's Field Equations ......Page 316
11.1c. Remarks on Static Metrics ......Page 319
11.2a. Covariant Differentiation of Tensors ......Page 321
11.2b. Riemannian Connections and the Bianchi Identities ......Page 322
11.2c. Second Covariant Derivatives: The Ricci Identities ......Page 324
11.3b. Normal Coordinates, the Divergence and Laplacian ......Page 326
11.3c Hilbert's Variational Approach to General Relativity ......Page 328
11.4a. The Induced Connection and the Second Fundamental Form ......Page 332
11.4b. The Equations of Gauss and Codazzi ......Page 334
11.4c. The Interpretation of the Sectional Curvature ......Page 336
11.4d. Fixed Points of Isometries ......Page 337
11.5a. The Einstein Tensor in a (Pseudo-)Riemannian Space-Time ......Page 338
11.5b. The Relativistic Meaning of Gauss's Equation ......Page 339
11.5c. The Second Fundamental Form of a Spatial Slice ......Page 341
11.5d. The Codazzi Equations ......Page 342
11.5e. Some Remarks on the Schwarzschild Solution ......Page 343
12 Curvature and Topology: Synge's Theorem ......Page 346
12.1a. The Second Variation of Arc Length ......Page 347
12.1b. Jacobi Fields ......Page 349
12.2a. Synge's Theorem ......Page 352
12.2b. Orientability Revisited ......Page 354
13.1a. Singular Chains ......Page 356
13.1b. Some 2-Dimensional Examples ......Page 361
13.2a. Coefficient Fields ......Page 365
13.2b. Finite Simplicial Complexes ......Page 366
13.2c. Cycles, Boundaries, Homology and Betti Numbers ......Page 367
13.3a. Some Computational Tools ......Page 370
13.3b. Familiar Examples ......Page 373
13.4a. The Statement of de Rham's Theorem ......Page 378
13.4b. Two Examples ......Page 380
14.1a. The $\ast$ Operator ......Page 384
14.1b. The Codifferential Operator $\delta=d^\ast$ ......Page 387
14.1c. Maxwell's Equations in Curved Space-Time $M^4$ ......Page 389
14.1d. The Hilbert Lagrangian ......Page 390
14.2a. The Laplace Operator on Forms ......Page 391
14.2b. The Lapiacian of a 1-Form ......Page 392
14.2c. Harmonic Forms on Closed Manifolds ......Page 393
14.2d. Harmonic Forms and de Rham's Theorem ......Page 395
14.2e. Bochner's Theorem ......Page 397
14.3. Boundary Values, Relative Homology, and Morse Theory ......Page 398
14.3a. Tangential and Normal Differential Forms ......Page 399
14.3b. Hodge's Theorem for Tangential Forms ......Page 400
14.3c. Relative Homology Groups ......Page 402
14.3d. Hodge' s Theorem for Normal Forms ......Page 404
14.3e. Morse's Theory of Critical Points ......Page 405
III. Lie Groups, Bundles, and Chern Forms ......Page 412
15.1a. Lie Groups ......Page 414
15.1b. Invariant Vector Fields and Forms ......Page 418
15.2. One Parameter Subgroups ......Page 421
15.3a. The Lie Algebra ......Page 425
15.3b. The Exponential Map ......Page 426
15.3c. Examples of Lie Algebras ......Page 427
15.3d. Do the 1-Parameter Subgroups Cover $G$? ......Page 428
15.4a. Left Invariant Fields Generate Right Translations ......Page 430
15.4b. Commutators of Matrices ......Page 431
15.4c. Right Invariant Fields ......Page 432
15.4d. Subgroups and Subalgebras ......Page 433
16.1a. Motivation by Two Examples ......Page 436
16.1b. Vector Bundles ......Page 438
16.1c. Local Trivializations ......Page 440
16.1d. The Normal Bundle to a Submanifold ......Page 442
16.2. Poincare's Theorem and the Euler Characteristic ......Page 444
16.2a. Poincare's Theorem ......Page 445
16.2b. The Stiefel Vector Field and Euler's Theorem ......Page 449
16.3a. Connection in a Vector Bundle ......Page 451
16.3b. Complex Vector Spaces ......Page 454
16.3d. Complex Line Bundles ......Page 456
16.4a. Lagrange's Equations Without Electromagnetism ......Page 458
16.4b. The Modified Lagrangian and Hamiltonian ......Page 459
16.4c. Schrodinger's Equation in an Electromagnetic Field ......Page 462
16.4d. Global Potentials ......Page 466
16.4e. The Dirac Monopole ......Page 467
16.4f. The Aharonov-Bohm Effect ......Page 469
17.1a. Fiber Bundles ......Page 474
17.1b. Principal Bundles and Frame Bundles ......Page 476
17.1c. Action of the Structure Group on a Principal Bundle ......Page 477
17.2a. Cosets ......Page 479
17.2b. Grassmann Manifolds ......Page 482
17.3a. A Connection in the Frame Bundle of a Surface ......Page 483
17.3b. The Gauss-Bonnet-Poincare Theorem ......Page 485
17.4a. A Generalization of Gauss-Bonnet ......Page 488
17.4b. Berry Phase ......Page 491
17.4c. Monopoles and the Hopf Bundle ......Page 496
18.1a. The Maurer-Cartan Form ......Page 498
18.1b. $\mathscr{g}$-Valued $p$-Forms on a Manifold ......Page 500
18.1c. Connections in a Principal Bundle ......Page 502
18.2a. Associated Bundles ......Page 504
18.2b. Connections in Associated Bundles ......Page 506
18.2c. The Associated $Ad$ Bundle ......Page 508
18.3a. $r$-Form sections of $E$ ......Page 511
18.3b. Curvature and the $Ad$ Bundle ......Page 512
19.1. The Groups $S0$(3) and $SU$(2) ......Page 514
19.1a. The Rotation Group $SO$(3) of $\mathbb{R}^3$ ......Page 515
19.1b. $SU$(2): The Lie algebra $\mathscr{su}$(2) ......Page 516
19.1c. $SU$(2) is Topologically the 3-Sphere ......Page 518
19.1d. $Ad$ : $SU$(2)$\to S0$(3) in More Detail ......Page 519
19.2a. Spinors and Rotations of $\mathbb{R}^3$ ......Page 520
19.2b. Hamilton on Composing Two Rotations ......Page 522
19.2c. Clifford Algebras ......Page 523
19.2d. The Dirac Program: The Square Root of the d'Alembertian ......Page 525
19.3a. The Lorentz Group ......Page 527
19.3b. The Dirac Algebra ......Page 532
19.4a. Dirac Spinors ......Page 534
19.4b. The Dirac Operator ......Page 536
19.5a. The Spinor Bundle ......Page 538
19.5b. The Spin Connection in $\mathcal{S}M$ ......Page 541
20.1a. The Tensorial Nature of Lagrange's Equations ......Page 546
20.1b. Boundary Conditions ......Page 549
20.1c. Noether's Theorem for Internal Symmetries ......Page 550
20.1d. Noether's Principle ......Page 551
20.2a. The Dirac Lagrangian ......Page 554
20.2b. Weyl's Gauge Invariance Revisited ......Page 556
20.2c. The Electromagnetic Lagrangian ......Page 557
20.2d. Quantization of the A Field: Photons ......Page 559
20.3a. The Heisenberg Nucleon ......Page 560
20.3b. The Yang-Mills Nucleon ......Page 561
20.3c. A Remark on Terminology ......Page 563
20.4b. Averaging over a Compact Group ......Page 564
20.4c. Compact Matrix Groups Are Subgroups of Unitary Groups ......Page 565
20.4d. $Ad$ Invariant Scalar Products in the Lie Algebra of a Compact Group ......Page 566
20.4e. The Yang-Mills Action ......Page 567
20.5a. The Exterior Covariant Divergence $\nabla^\ast$ ......Page 568
20.5b. The Yang-Mills Analogy with Electromagnetism ......Page 570
20.5c. Further Remarks on the Yang-Mills Equations ......Page 571
20.6a. Instantons ......Page 573
20.6b. Chern's Proof Revisited ......Page 576
20.6c. Instantons and the Vacuum ......Page 580
21.1a. Bi-invariant $p$-Forms ......Page 584
21.1b. The Cartan $p$-Forms ......Page 585
21.1c. Bi-invariant Riemannian Metrics ......Page 586
21.1d. Harmonic Forms in the Bi-invariant Metric ......Page 587
21.1e. Weyl and Cartan on the Betti Numbers of $G$ ......Page 588
21.2a. Poincare's Fundamental Group $\pi_1(M)$ ......Page 590
21.2b. The Concept of a Covering Space ......Page 592
21.2c. The Universal Covering ......Page 593
21.2d. The Orientable Covering ......Page 596
21.2e. Lifting Paths ......Page 597
21.2g. The Universal Covering Group ......Page 598
21.3. The Theorem of S. B. Myers: A Problem Set ......Page 599
21.4a. The Connection of a Bi-invariant Metric ......Page 603
21.4b. The Flat Connections ......Page 604
22.1a. The Yang-Mills "Winding Number" ......Page 606
22.1b. Winding Number in Terms of Field Strength ......Page 608
22.1c. The Chern Forms for $U(n)$ Bundle ......Page 610
22.2a. Homotopy ......Page 614
22.2b. Covering Homotopy ......Page 615
22.2c. Some Topology of $SU(n)$ ......Page 617
22.3a. $\pi_k(M)$ ......Page 619
22.3b. Homotopy Groups of Spheres ......Page 620
22.3c. Exact Sequences of Groups ......Page 621
22.3d. The Homotopy Sequence of a Bundle ......Page 623
22.3e. The Relation Between Homotopy and Homology Groups ......Page 626
22.4a. Lifting Spheres from $M$ into the Bundle $P$ ......Page 628
22.4c. The Hopf Map and Fibering ......Page 629
22.5a. The Chern Forms $c_r$ for an $SU(n)$ Bundle Revisited ......Page 631
22.5b. $c_2$ as an "Obstruction Cocycle" ......Page 632
22.5d. Chern's Integral ......Page 635
22.5e. Concluding Remarks ......Page 638
A.a. The Classical Cauchy Stress Tensor and Equations of Motion ......Page 640
A.b. Stresses in Terms of Exterior Forms ......Page 641
A.c. Symmetry of Cauchy's Stress Tensor in $\mathbb{R}^n$ ......Page 643
A.d. The Piola-Kirchhoff Stress Tensors ......Page 645
A.e. Stored Energy of Deformation ......Page 646
A.f. Hamilton's Principle in Elasticity ......Page 649
A.g. Some Typical Computations Using Forms ......Page 652
A.h. Concluding Remarks ......Page 658
B.a. Chain Complexes ......Page 659
B.b. Cochains and Cohomology ......Page 661
B.c. Transpose and Adjoint ......Page 662
B.d. Laplacians and Harmonic Cochains ......Page 664
B.e. Kirchhoff's Circuit Laws ......Page 666
References ......Page 674
Index ......Page 678