The geometry of physics: an introduction
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| Format: | Book |
| Language: | English |
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Cambridge [u.a.]
Cambridge Univ. Press
2006
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| Edition: | 2. ed., reprint. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVI, 694 S. graph. Darst. |
| ISBN: | 0521833302 0521539277 |
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| 245 | 1 | 0 | |a The geometry of physics |b an introduction |c Theodore Frankel |
| 250 | |a 2. ed., reprint. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2006 | |
| 300 | |a XXVI, 694 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Geometrie - Mathematische Physik | |
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| adam_text | Contents
Preface to the Second Edition page xix
Preface to the Revised Printing xxi
Preface to the First Edition xxiii
I Manifolds, Tensors, and Exterior Forms
1 Manifolds and Vector Fields 3
1.1. Submanifolds of Euclidean Space 3
1.1a. Submanifolds of RN 4
1.1b. The Geometry of Jacobian Matrices: The Differential 7
1.1c. The Main Theorem on Submanifolds of RN 8
l.ld. A Nontrivial Example: The Configuration Space of a
Rigid Body 9
1.2. Manifolds 11
1.2a. Some Notions from Point Set Topology 11
1.2b. The Idea of a Manifold 13
1.2c. A Rigorous Definition of a Manifold 19
1.2d. Complex Manifolds: The Riemann Sphere 21
1.3. Tangent Vectors and Mappings 22
1.3a. Tangent or Contravariant Vectors 23
1.3b. Vectors as Differentia] Operators 24
1.3c. The Tangent Space to M at a Point 25
1.3d. Mappings and Submanifolds of Manifolds 26
1.3e. Change of Coordinates 29
1.4. Vector Fields and Flows 30
1.4a. Vector Fields and Flows on R 30
1.4b. Vector Fields on Manifolds 33
1.4c. Straightening Flows 34
vii
Vlii CONTENTS
2 Tensors and Exterior Forms 37
2.1. Covectors and Riemannian Metrics 37
2.1a. Linear Functionals and the Dual Space 37
2.1b. The Differential of a Function 40
2.1c. Scalar Products in Linear Algebra 42
2.1d. Riemannian Manifolds and the Gradient Vector 45
2.1e. Curves of Steepest Ascent 46
2.2. The Tangent Bundle 48
2.2a. The Tangent Bundle 48
2.2b. The Unit Tangent Bundle 50
2.3. The Cotangent Bundle and Phase Space 52
2.3a. The Cotangent Bundle 52
2.3b. The Pull Back of a Covector 52
2.3c. The Phase Space in Mechanics 54
2.3d. The Poincare 1 Form 56
2.4. Tensors 58
2.4a. Covariant Tensors 58
2.4b. Contravariant Tensors 59
2.4c. Mixed Tensors 60
2.4d. Transformation Properties of Tensors 62
2.4e. Tensor Fields on Manifolds 63
2.5. The Grassmann or Exterior Algebra 66
2.5a. The Tensor Product of Covariant Tensors 66
2.5b. The Grassmann or Exterior Algebra 66
2.5c. The Geometric Meaning of Forms in W 70
2.5d. Special Cases of the Exterior Product 70
2.5e. Computations and Vector Analysis 71
2.6. Exterior Differentiation 73
2.6a. The Exterior Differential 73
2.6b. Examples in K3 75
2.6c. A Coordinate Expression for d 76
2.7. Pull Backs 77
2.7a. The Pull Back of a Covariant Tensor 77
2.7b. The Pull Back in Elasticity 80
2.8. Orientation and Pseudoforms 82
2.8a. Orientation of a Vector Space 82
2.8b. Orientation of a Manifold 83
2.8c. Orientability and 2 Sided Hypersurfaces 84
2.8d. Projective Spaces 85
2.8e. Pseudoforms and the Volume Form 85
2.8f. The Volume Form in a Riemannian Manifold 87
2.9. Interior Products and Vector Analysis 89
2.9a. Interior Products and Contractions 89
2.9b. Interior Product in E3 90
2.9c. Vector Analysis in K3 92
CONTENTS ix
2.10. Dictionary 94
3 Integration of Differential Forms 95
3.1. Integration over a Parameterized Subset 95
3.1a. Integration of a /? Form in M.p 95
3.1b. Integration over Parameterized Subsets 96
3.1c. Line Integrals 97
3.1d. Surface Integrals 99
3.1e. Independence of Parameterization 101
3.1f. Integrals and Pull Backs 102
3.1g. Concluding Remarks 102
3.2. Integration over Manifolds with Boundary 104
3.2a. Manifolds with Boundary 105
3.2b. Partitions of Unity 106
3.2c. Integration over a Compact Oriented Submanifold 108
3.2d. Partitions and Riemannian Metrics 109
3.3. Stokes s Theorem 110
3.3a. Orienting the Boundary 110
3.3b. Stokes s Theorem 111
3.4. Integration of Pseudoforms 114
3.4a. Integrating Pseudo n Forms on an n Manifold 115
3.4b. Submanifolds with Transverse Orientation 115
3.4c. Integration over a Submanifold with Transverse
Orientation 116
3.4d. Stokes s Theorem for Pseudoforms 117
3.5. Maxwell s Equations 118
3.5a. Charge and Current in Classical Electromagnetism 118
3.5b. The Electric and Magnetic Fields 119
3.5c. Maxwell s Equations 120
3.5d. Forms and Pseudoforms 122
4 The Lie Derivative 125
4.1. The Lie Derivative of a Vector Field 125
4.1a. The Lie Bracket 125
4.1b. Jacobi s Variational Equation 127
4.1c. The Flow Generated by [X, Y] 129
4.2. The Lie Derivative of a Form 132
4.2a. Lie Derivatives of Forms 132
4.2b. Formulas Involving the Lie Derivative 134
4.2c. Vector Analysis Again 136
4.3. Differentiation of Integrals 138
4.3a. The Autonomous (Time Independent) Case 138
4.3b. Time Dependent Fields 140
4.3c. Differentiating Integrals 142
4.4. A Problem Set on Hamiltonian Mechanics 145
4.4a. Time Independent Hamiltonians 147
X CONTENTS
4.4b. Time Dependent Hamiltonians and Hamilton s Principle 151
4.4c. Poisson Brackets 154
5 The Poincare Lemma and Potentials 155
5.1. A More General Stokes s Theorem 155
5.2. Closed Forms and Exact Forms 156
5.3. Complex Analysis 158
5.4. The Converse to the Poincare Lemma 160
5.5. Finding Potentials 162
6 Holonomic and Nonholonomic Constraints 165
6.1. The Frobenius Integrability Condition 165
6.1a. Planes in K3 165
6.1b. Distributions and Vector Fields 167
6.1c. Distributions and 1 Forms 167
6.1d. The Frobenius Theorem 169
6.2. Integrability and Constraints 172
6.2a. Foliations and Maximal Leaves 172
6.2b. Systems of Mayer Lie 174
6.2c. Holonomic and Nonholonomic Constraints 175
6.3. Heuristic Thermodynamics via Caratheodory 178
6.3a. Introduction 178
6.3b. The First Law of Thermodynamics 179
6.3c. Some Elementary Changes of State 180
6.3d. The Second Law of Thermodynamics 181
6.3e. Entropy 183
6.3f. Increasing Entropy 185
6.3g. Chow s Theorem on Accessibility 187
II Geometry and Topology
7 R3 and Minkowski Space 191
7.1. Curvature and Special Relativity 191
7.1a. Curvature of a Space Curve in R3 191
7.1b. Minkowski Space and Special Relativity 192
7.1c. Hamiltonian Formulation 196
7.2. Electromagnetism in Minkowski Space 196
7.2a. Minkowski s Electromagnetic Field Tensor 196
7.2b. Maxwell s Equations 198
8 The Geometry of Surfaces in E3 201
8.1. The First and Second Fundamental Forms 201
8.1a. The First Fundamental Form, or Metric Tensor 201
8.1b. The Second Fundamental Form 203
8.2. Gaussian and Mean Curvatures 205
8.2a. Symmetry and Self Adjointness 205
CONTENTS Xi
8.2b. Principal Normal Curvatures 206
8.2c. Gauss and Mean Curvatures: The Gauss Normal Map 207
8.3. The Brouwer Degree of a Map: A Problem Set 210
8.3a. The Brouwer Degree 210
8.3b. Complex Analytic (Holomorphic) Maps 214
8.3c. The Gauss Normal Map Revisited: The Gauss Bonnet
Theorem 215
8.3d. The Kronecker Index of a Vector Field 215
8.3e. The Gauss Looping Integral 218
8.4. Area, Mean Curvature, and Soap Bubbles 221
8.4a. The First Variation of Area 221
8.4b. Soap Bubbles and Minimal Surfaces 226
8.5. Gauss s Theorema Egregium 228
8.5a. The Equations of Gauss and Codazzi 228
8.5b. The Theorema Egregium 230
8.6. Geodesies 232
8.6a. The First Variation of Arc Length 232
8.6b. The Intrinsic Derivative and the Geodesic Equation 234
8.7. The Parallel Displacement of Levi Civita 236
9 Covariant Differentiation and Curvature 241
9.1. Covariant Differentiation 241
9.1a. Covariant Derivative 241
9.1b. Curvature of an Affine Connection 244
9.1c. Torsion and Symmetry 245
9.2. The Riemannian Connection 246
9.3. Cartan s Exterior Covariant Differential 247
9.3a. Vector Valued Forms 247
9.3b. The Covariant Differential of a Vector Field 248
9.3c. Cartan s Structural Equations 249
9.3d. The Exterior Covariant Differential of a Vector Valued
Form 250
9.3e. The Curvature 2 Forms 251
9.4. Change of Basis and Gauge Transformations 253
9.4a. Symmetric Connections Only 253
9.4b. Change of Frame 253
9.5. The Curvature Forms in a Riemannian Manifold 255
9.5a. The Riemannian Connection 255
9.5b. Riemannian Surfaces M2 257
9.5c. An Example 257
9.6. Parallel Displacement and Curvature on a Surface 259
9.7. Riemann s Theorem and the Horizontal Distribution 263
9.7a. Flat Metrics 263
9.7b. The Horizontal Distribution of an Affine Connection 263
9.7c. Riemann s Theorem 266
Xii CONTENTS
10 Geodesies 269
10.1. Geodesies and Jacobi Fields 269
10.1a. Vector Fields Along a Surface in M 269
10.1b. Geodesies 271
10.1c. Jacobi Fields 272
lO.ld. Energy 274
10.2. Variational Principles in Mechanics 275
10.2a. Hamilton s Principle in the Tangent Bundle 275
10.2b. Hamilton s Principle in Phase Space 277
10.2c. Jacobi s Principle of Least Action 278
10.2d. Closed Geodesies and Periodic Motions 281
10.3. Geodesies, Spiders, and the Universe 284
10.3a. Gaussian Coordinates 284
10.3b. Normal Coordinates on a Surface 287
10.3c. Spiders and the Universe 288
11 Relativity, Tensors, and Curvature 291
11.1. Heuristics of Einstein s Theory 291
11.1a. The Metric Potentials 291
11.1b. Einstein s Field Equations 293
11.1c. Remarks on Static Metrics 296
11.2. Tensor Analysis 298
11.2a. Covari ant Differentiation of Tensors 298
11.2b. Riemannian Connections and the Bianchi Identities 299
11.2c. Second Covariant Derivatives: The Ricci Identities 301
11.3. Hilbert s Action Principle 303
11.3a. Geodesies in a Pseudo Riemannian Manifold 303
11.3b. Normal Coordinates, the Divergence and Laplacian 303
11.3c. Hilbert s Variational Approach to General Relativity 305
11.4. The Second Fundamental Form in the Riemannian Case 309
11.4a. The Induced Connection and the Second Fundamental
Form 309
11.4b. The Equations of Gauss and Codazzi 311
11.4c. The Interpretation of the Sectional Curvature 313
11.4d. Fixed Points of Isometries 314
11.5. The Geometry of Einstein s Equations 315
11.5a. The Einstein Tensor in a (Pseudo )Riemannian
Space Time 315
11.5b. The Relativistic Meaning of Gauss s Equation 316
11.5c. The Second Fundamental Form of a Spatial Slice 318
11.5d. The Codazzi Equations 319
11.5e. Some Remarks on the Schwarzschild Solution 320
12 Curvature and Topology: Synge s Theorem 323
12.1. Synge s Formula for Second Variation 324
12.1a. The Second Variation of Arc Length 324
12.1b. Jacobi Fields 326
CONTENTS Xiii
12.2. Curvature and Simple Connectivity 329
12.2a. Synge s Theorem 329
12.2b. Orientability Revisited 331
13 Betti Numbers and De Rham s Theorem 333
13.1. Singular Chains and Their Boundaries 333
13.1a. Singular Chains 333
13.1b. Some 2 DimensionaI Examples 338
13.2. The Singular Homology Groups 342
13.2a. Coefficient Fields 342
13.2b. Finite Simplicial Complexes 343
13.2c. Cycles, Boundaries, Homology, and Betti Numbers 344
13.3. Homology Groups of Familiar Manifolds 347
13.3a. Some Computational Tools 347
13.3b. Familiar Examples 350
13.4. De Rham s Theorem 355
13.4a. The Statement of De Rham s Theorem 355
13.4b. Two Examples 357
14 Harmonic Forms 361
14.1. The Hodge Operators 361
14.1a. The* Operator 361
14.1b. The Codifferential Operator S d* 364
14.1c. Maxwell s Equations in Curved Space Time M4 366
14.1d. The Hilbert Lagrangian 367
14.2. Harmonic Forms 368
14.2a. The Laplace Operator on Forms 368
14.2b. The Laplacian of a 1 Form 369
14.2c. Harmonic Forms on Closed Manifolds 370
14.2d. Harmonic Forms and De Rham s Theorem 372
14.2e. Bochner s Theorem 374
14.3. Boundary Values, Relative Homology, and Morse Theory 375
14.3a. Tangential and Normal Differential Forms 376
14.3b. Hodge s Theorem for Tangential Forms 377
14.3c. Relative Homology Groups 379
14.3d. Hodge s Theorem for Normal Forms 381
14.3e. Morse s Theory of Critical Points 382
III Lie Groups, Bundles, and Chern Forms
15 Lie Groups 391
15.1. Lie Groups, Invariant Vector Fields, and Forms 391
15.1a. Lie Groups 391
15.1b. Invariant Vector Fields and Forms 395
15.2. One Parameter Subgroups 398
15.3. The Lie Algebra of a Lie Group 402
15.3a. The Lie Algebra 402
Xiv CONTENTS
15.3b. The Exponential Map 403
15.3c. Examples of Lie Algebras 404
15.3d. Do the 1 Parameter Subgroups Cover G? 405
15.4. Subgroups and Subalgebras 407
15.4a. Left Invariant Fields Generate Right Translations 407
15.4b. Commutators of Matrices 408
15.4c. Right Invariant Fields 409
15.4d. Subgroups and Subalgebras 410
16 Vector Bundles in Geometry and Physics 413
16.1. Vector Bundles 413
16.1a. Motivation by Two Examples 413
16.1b. Vector Bundles 415
16.1c. Local Trivializations 417
16.1d. The Normal Bundle to a Submanifold 419
16.2. Poincare s Theorem and the Euler Characteristic 421
16.2a. Poincare s Theorem 422
16.2b. The Stiefel Vector Field and Euler s Theorem 426
16.3. Connections in a Vector Bundle 428
16.3a. Connection in a Vector Bundle 428
16.3b. Complex Vector Spaces 431
16.3c. The Structure Group of a Bundle 433
16.3d. Complex Line Bundles 433
16.4. The Electromagnetic Connection 435
16.4a. Lagrange s Equations without Electromagnetism 435
16.4b. The Modified Lagrangian and Hamiltonian 436
16.4c. Schrodinger s Equation in an Electromagnetic Field 439
16.4d. Global Potentials 443
16.4e. The Dirac Monopole 444
16.4f. The Aharonov Bohm Effect 446
17 Fiber Bundles, Gauss Bonnet, and Topological Quantization 451
17.1. Fiber Bundles and Principal Bundles 451
17.1a. Fiber Bundles 451
17.1b. Principal Bundles and Frame Bundles 453
17.1c. Action of the Structure Group on a Principal Bundle 454
17.2. Coset Spaces 456
17.2a. Cosets 456
17.2b. Grassmann Manifolds 459
17.3. Chern s Proof of the Gauss Bonnet Poincare Theorem 460
17.3a. A Connection in the Frame Bundle of a Surface 460
17.3b. The Gauss Bonnet Poincare Theorem 462
17.3c. Gauss Bonnet as an Index Theorem 465
17.4. Line Bundles, Topological Quantization, and Berry Phase 465
17.4a. A Generalization of Gauss Bonnet 465
17.4b. Berry Phase 468
17.4c. Monopoles and the Hopf Bundle 473
CONTENTS XV
18 Connections and Associated Bundles 475
18.1. Forms with Values in a Lie Algebra 475
18.1a. The Maurer Cartan Form 475
18.1b. op Valued p Forms on a Manifold 477
18.1c. Connections in a Principal Bundle 479
18.2. Associated Bundles and Connections 481
18.2a. Associated Bundles 481
18.2b. Connections in Associated Bundles 483
18.2c. The Associated Ad Bundle 485
18.3. r Form Sections of a Vector Bundle: Curvature 488
18.3a. / Form Sections of E 488
18.3b. Curvature and the Ad Bundle 489
19 The Dirac Equation 491
19.1. The Groups SO(3) and SU(2) 491
19.1a. The Rotation Group S O (3) of R3 492
19.1b. 51/(2): The Lie Algebra a*i. 2) 493
19.1c. 5(7(2) Is Topologically the 3 Sphere 495
19.1d. Ad : SU(2) » 50(3) in More Detail 496
19.2. Hamilton, Clifford, and Dirac 497
19.2a. Spinors and Rotations of M3 497
19.2b. Hamilton on Composing Two Rotations 499
19.2c. Clifford Algebras 500
19.2d. The Dirac Program: The Square Root of the
d Alembertian 502
19.3. The Dirac Algebra 504
19.3a. The Lorentz Group 504
19.3b. The Dirac Algebra 509
19.4. The Dirac Operator jl in Minkowski Space 511
19.4a. Dirac Spinors 511
19.4b. The Dirac Operator 513
19.5. The Dirac Operator in Curved Space Time 515
19.5a. The Spinor Bundle 515
19.5b. The Spin Connection in M 518
20 Yang Mills Fields 523
20.1. Noether s Theorem for Internal Symmetries 523
20.1a. The Tensorial Nature of Lagrange s Equations 523
20.1b. Boundary Conditions 526
20.1c. Noether s Theorem for Internal Symmetries 527
20.1d. Noether s Principle 528
20.2. Weyl s Gauge Invariance Revisited 531
20.2a. The Dirac Lagrangian 531
20.2b. Weyl s Gauge Invariance Revisited 533
20.2c. The Electromagnetic Lagrangian 534
20.2d. Quantization of the A Field: Photons 536
XVi CONTENTS
20.3. The Yang Mills Nucleon 537
20.3a. The Heisenberg Nucleon 537
20.3b. The Yang Mills Nucleon 538
20.3c. A Remark on Terminology 540
20.4. Compact Groups and Yang Mills Action 541
20.4a. The Unitary Group Is Compact 541
20.4b. Averaging over a Compact Group 541
20.4c. Compact Matrix Groups Are Subgroups of Unitary
Groups 542
20.4d. Ad Invariant Scalar Products in the Lie Algebra of a
Compact Group 543
20.4e. The Yang Mills Action 544
20.5. The Yang Mills Equation 545
20.5a. The Exterior Covariant Divergence V* 545
20.5b. The Yang Mills Analogy with Electromagnetism 547
20.5c. Further Remarks on the Yang Mills Equations 548
20.6. Yang Mills Instantons 550
20.6a. Instantons 550
20.6b. Chern s Proof Revisited 553
20.6c. Instantons and the Vacuum 557
21 Betti Numbers and Covering Spaces 561
21.1. Bi invariant Forms on Compact Groups 561
21.1a. Bi invariant p Forms 561
21.1b. The Cartan /j Forms 562
21.1c. Bi invariant Riemannian Metrics 563
21.Id. Harmonic Forms in the Bi invariant Metric 564
21.1e. Weyl and Cartan on the Betti Numbers of G 565
21.2. The Fundamental Group and Covering Spaces 567
21.2a. Poincare s Fundamental Group Tt (M) 567
21.2b. The Concept of a Covering Space 569
21.2c. The Universal Covering 570
21.2d. The Orientable Covering 573
21.2e. Lifting Paths 574
21.2f. Subgroups of tt,(M) 575
21.2g. The Universal Covering Group 575
21.3. The Theorem of S. B. Myers: A Problem Set 576
21.4. The Geometry of a Lie Group 580
21.4a. The Connection of a Bi invariant Metric 580
21.4b. The Flat Connections 581
22 Chern Forms and Homotopy Groups 583
22.1. Chern Forms and Winding Numbers 583
22.1a. The Yang Mills Winding Number 583
22.1b. Winding Number in Terms of Field Strength 585
22.1c. The Chern Forms for a U(n) Bundle 587
CONTENTS XVii
22.2. Homotopies and Extensions 591
22.2a. Homotopy 591
22.2b. Covering Homotopy 592
22.2c. Some Topology of SU (n) 594
22.3. The Higher Homotopy Groups jr* (M) 596
22.3a. nk{M) 596
22.3b. Homotopy Groups of Spheres 597
22.3c. Exact Sequences of Groups 598
22.3d. The Homotopy Sequence of a Bundle 600
22.3e. The Relation between Homotopy and Homology
Groups 603
22.4. Some Computations of Homotopy Groups 605
22.4a. Lifting Spheres from M into the Bundle P 605
22.4b. SU(n) Again 606
22.4c. TheHopf Map and Fibering 606
22.5. Chern Forms as Obstructions 608
22.5a. The Chern Forms cr for an SU(n) Bundle Revisited 608
22.5b. c% as an Obstruction Cocycle 609
22.5c. The Meaning of the Integer j (A4) 612
22.5d. Chern s Integral 612
22.5e. Concluding Remarks 615
Appendix A. Forms in Continuum Mechanics 617
A.a. The Classical Cauchy Stress Tensor and Equations of Motion 617
A.b. Stresses in Terms of Exterior Forms 618
A.c. Symmetry of Cauchy s Stress Tensor in K 620
A.d. The Piola Kirchhoff Stress Tensors 622
A.e. Stored Energy of Deformation 623
A.f. Hamilton s Principle in Elasticity 626
A.g. Some Typical Computations Using Forms 629
A.h. Concluding Remarks 635
Appendix B. Harmonic Chains and Kirchhoff s Circuit Laws 636
B.a. Chain Complexes 636
B.b. Cochains and Cohomology 638
B.C. Transpose and Adjoint 639
B.d. Laplacians and Harmonic Cochains 641
B.e. Kirchhoff s Circuit Laws 643
Appendix C. Symmetries, Quarks, and Meson Masses 648
C.a. Flavored Quarks 648
C.b. Interactions of Quarks and Antiquarks 650
C.c. The Lie Algebra of SU(3) 652
C.d. Pions, Kaons, and Etas 653
C.e. A Reduced Symmetry Group 656
C.f. Meson Masses 658
xviii contents
Appendix D. Representations and Hyperelastic Bodies 660
D.a. Hyperelastic Bodies 660
D.b. Isotropic Bodies 661
D.c. Application of Schur s Lemma 662
D.d. Frobenius Schur Relations 664
D.e. The Symmetric Traceless 3x3 Matrices Are Irreducible 666
Appendix E. Orbits and Morse Bott Theory in Compact Lie Groups 670
E.a. The Topology of Conjugacy Orbits 670
E.b. Application of Bott s Extension of Morse Theory 673
References 679
Index 683
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| adam_txt |
Contents
Preface to the Second Edition page xix
Preface to the Revised Printing xxi
Preface to the First Edition xxiii
I Manifolds, Tensors, and Exterior Forms
1 Manifolds and Vector Fields 3
1.1. Submanifolds of Euclidean Space 3
1.1a. Submanifolds of RN 4
1.1b. The Geometry of Jacobian Matrices: The "Differential" 7
1.1c. The Main Theorem on Submanifolds of RN 8
l.ld. A Nontrivial Example: The Configuration Space of a
Rigid Body 9
1.2. Manifolds 11
1.2a. Some Notions from Point Set Topology 11
1.2b. The Idea of a Manifold 13
1.2c. A Rigorous Definition of a Manifold 19
1.2d. Complex Manifolds: The Riemann Sphere 21
1.3. Tangent Vectors and Mappings 22
1.3a. Tangent or "Contravariant" Vectors 23
1.3b. Vectors as Differentia] Operators 24
1.3c. The Tangent Space to M" at a Point 25
1.3d. Mappings and Submanifolds of Manifolds 26
1.3e. Change of Coordinates 29
1.4. Vector Fields and Flows 30
1.4a. Vector Fields and Flows on R" 30
1.4b. Vector Fields on Manifolds 33
1.4c. Straightening Flows 34
vii
Vlii CONTENTS
2 Tensors and Exterior Forms 37
2.1. Covectors and Riemannian Metrics 37
2.1a. Linear Functionals and the Dual Space 37
2.1b. The Differential of a Function 40
2.1c. Scalar Products in Linear Algebra 42
2.1d. Riemannian Manifolds and the Gradient Vector 45
2.1e. Curves of Steepest Ascent 46
2.2. The Tangent Bundle 48
2.2a. The Tangent Bundle 48
2.2b. The Unit Tangent Bundle 50
2.3. The Cotangent Bundle and Phase Space 52
2.3a. The Cotangent Bundle 52
2.3b. The Pull Back of a Covector 52
2.3c. The Phase Space in Mechanics 54
2.3d. The Poincare 1 Form 56
2.4. Tensors 58
2.4a. Covariant Tensors 58
2.4b. Contravariant Tensors 59
2.4c. Mixed Tensors 60
2.4d. Transformation Properties of Tensors 62
2.4e. Tensor Fields on Manifolds 63
2.5. The Grassmann or Exterior Algebra 66
2.5a. The Tensor Product of Covariant Tensors 66
2.5b. The Grassmann or Exterior Algebra 66
2.5c. The Geometric Meaning of Forms in W 70
2.5d. Special Cases of the Exterior Product 70
2.5e. Computations and Vector Analysis 71
2.6. Exterior Differentiation 73
2.6a. The Exterior Differential 73
2.6b. Examples in K3 75
2.6c. A Coordinate Expression for d 76
2.7. Pull Backs 77
2.7a. The Pull Back of a Covariant Tensor 77
2.7b. The Pull Back in Elasticity 80
2.8. Orientation and Pseudoforms 82
2.8a. Orientation of a Vector Space 82
2.8b. Orientation of a Manifold 83
2.8c. Orientability and 2 Sided Hypersurfaces 84
2.8d. Projective Spaces 85
2.8e. Pseudoforms and the Volume Form 85
2.8f. The Volume Form in a Riemannian Manifold 87
2.9. Interior Products and Vector Analysis 89
2.9a. Interior Products and Contractions 89
2.9b. Interior Product in E3 90
2.9c. Vector Analysis in K3 92
CONTENTS ix
2.10. Dictionary 94
3 Integration of Differential Forms 95
3.1. Integration over a Parameterized Subset 95
3.1a. Integration of a /? Form in M.p 95
3.1b. Integration over Parameterized Subsets 96
3.1c. Line Integrals 97
3.1d. Surface Integrals 99
3.1e. Independence of Parameterization 101
3.1f. Integrals and Pull Backs 102
3.1g. Concluding Remarks 102
3.2. Integration over Manifolds with Boundary 104
3.2a. Manifolds with Boundary 105
3.2b. Partitions of Unity 106
3.2c. Integration over a Compact Oriented Submanifold 108
3.2d. Partitions and Riemannian Metrics 109
3.3. Stokes's Theorem 110
3.3a. Orienting the Boundary 110
3.3b. Stokes's Theorem 111
3.4. Integration of Pseudoforms 114
3.4a. Integrating Pseudo n Forms on an n Manifold 115
3.4b. Submanifolds with Transverse Orientation 115
3.4c. Integration over a Submanifold with Transverse
Orientation 116
3.4d. Stokes's Theorem for Pseudoforms 117
3.5. Maxwell's Equations 118
3.5a. Charge and Current in Classical Electromagnetism 118
3.5b. The Electric and Magnetic Fields 119
3.5c. Maxwell's Equations 120
3.5d. Forms and Pseudoforms 122
4 The Lie Derivative 125
4.1. The Lie Derivative of a Vector Field 125
4.1a. The Lie Bracket 125
4.1b. Jacobi's Variational Equation 127
4.1c. The Flow Generated by [X, Y] 129
4.2. The Lie Derivative of a Form 132
4.2a. Lie Derivatives of Forms 132
4.2b. Formulas Involving the Lie Derivative 134
4.2c. Vector Analysis Again 136
4.3. Differentiation of Integrals 138
4.3a. The Autonomous (Time Independent) Case 138
4.3b. Time Dependent Fields 140
4.3c. Differentiating Integrals 142
4.4. A Problem Set on Hamiltonian Mechanics 145
4.4a. Time Independent Hamiltonians 147
X CONTENTS
4.4b. Time Dependent Hamiltonians and Hamilton's Principle 151
4.4c. Poisson Brackets 154
5 The Poincare Lemma and Potentials 155
5.1. A More General Stokes's Theorem 155
5.2. Closed Forms and Exact Forms 156
5.3. Complex Analysis 158
5.4. The Converse to the Poincare Lemma 160
5.5. Finding Potentials 162
6 Holonomic and Nonholonomic Constraints 165
6.1. The Frobenius Integrability Condition 165
6.1a. Planes in K3 165
6.1b. Distributions and Vector Fields 167
6.1c. Distributions and 1 Forms 167
6.1d. The Frobenius Theorem 169
6.2. Integrability and Constraints 172
6.2a. Foliations and Maximal Leaves 172
6.2b. Systems of Mayer Lie 174
6.2c. Holonomic and Nonholonomic Constraints 175
6.3. Heuristic Thermodynamics via Caratheodory 178
6.3a. Introduction 178
6.3b. The First Law of Thermodynamics 179
6.3c. Some Elementary Changes of State 180
6.3d. The Second Law of Thermodynamics 181
6.3e. Entropy 183
6.3f. Increasing Entropy 185
6.3g. Chow's Theorem on Accessibility 187
II Geometry and Topology
7 R3 and Minkowski Space 191
7.1. Curvature and Special Relativity 191
7.1a. Curvature of a Space Curve in R3 191
7.1b. Minkowski Space and Special Relativity 192
7.1c. Hamiltonian Formulation 196
7.2. Electromagnetism in Minkowski Space 196
7.2a. Minkowski's Electromagnetic Field Tensor 196
7.2b. Maxwell's Equations 198
8 The Geometry of Surfaces in E3 201
8.1. The First and Second Fundamental Forms 201
8.1a. The First Fundamental Form, or Metric Tensor 201
8.1b. The Second Fundamental Form 203
8.2. Gaussian and Mean Curvatures 205
8.2a. Symmetry and Self Adjointness 205
CONTENTS Xi
8.2b. Principal Normal Curvatures 206
8.2c. Gauss and Mean Curvatures: The Gauss Normal Map 207
8.3. The Brouwer Degree of a Map: A Problem Set 210
8.3a. The Brouwer Degree 210
8.3b. Complex Analytic (Holomorphic) Maps 214
8.3c. The Gauss Normal Map Revisited: The Gauss Bonnet
Theorem 215
8.3d. The Kronecker Index of a Vector Field 215
8.3e. The Gauss Looping Integral 218
8.4. Area, Mean Curvature, and Soap Bubbles 221
8.4a. The First Variation of Area 221
8.4b. Soap Bubbles and Minimal Surfaces 226
8.5. Gauss's Theorema Egregium 228
8.5a. The Equations of Gauss and Codazzi 228
8.5b. The Theorema Egregium 230
8.6. Geodesies 232
8.6a. The First Variation of Arc Length 232
8.6b. The Intrinsic Derivative and the Geodesic Equation 234
8.7. The Parallel Displacement of Levi Civita 236
9 Covariant Differentiation and Curvature 241
9.1. Covariant Differentiation 241
9.1a. Covariant Derivative 241
9.1b. Curvature of an Affine Connection 244
9.1c. Torsion and Symmetry 245
9.2. The Riemannian Connection 246
9.3. Cartan's Exterior Covariant Differential 247
9.3a. Vector Valued Forms 247
9.3b. The Covariant Differential of a Vector Field 248
9.3c. Cartan's Structural Equations 249
9.3d. The Exterior Covariant Differential of a Vector Valued
Form 250
9.3e. The Curvature 2 Forms 251
9.4. Change of Basis and Gauge Transformations 253
9.4a. Symmetric Connections Only 253
9.4b. Change of Frame 253
9.5. The Curvature Forms in a Riemannian Manifold 255
9.5a. The Riemannian Connection 255
9.5b. Riemannian Surfaces M2 257
9.5c. An Example 257
9.6. Parallel Displacement and Curvature on a Surface 259
9.7. Riemann's Theorem and the Horizontal Distribution 263
9.7a. Flat Metrics 263
9.7b. The Horizontal Distribution of an Affine Connection 263
9.7c. Riemann's Theorem 266
Xii CONTENTS
10 Geodesies 269
10.1. Geodesies and Jacobi Fields 269
10.1a. Vector Fields Along a Surface in M" 269
10.1b. Geodesies 271
10.1c. Jacobi Fields 272
lO.ld. Energy 274
10.2. Variational Principles in Mechanics 275
10.2a. Hamilton's Principle in the Tangent Bundle 275
10.2b. Hamilton's Principle in Phase Space 277
10.2c. Jacobi's Principle of "Least" Action 278
10.2d. Closed Geodesies and Periodic Motions 281
10.3. Geodesies, Spiders, and the Universe 284
10.3a. Gaussian Coordinates 284
10.3b. Normal Coordinates on a Surface 287
10.3c. Spiders and the Universe 288
11 Relativity, Tensors, and Curvature 291
11.1. Heuristics of Einstein's Theory 291
11.1a. The Metric Potentials 291
11.1b. Einstein's Field Equations 293
11.1c. Remarks on Static Metrics 296
11.2. Tensor Analysis 298
11.2a. Covari ant Differentiation of Tensors 298
11.2b. Riemannian Connections and the Bianchi Identities 299
11.2c. Second Covariant Derivatives: The Ricci Identities 301
11.3. Hilbert's Action Principle 303
11.3a. Geodesies in a Pseudo Riemannian Manifold 303
11.3b. Normal Coordinates, the Divergence and Laplacian 303
11.3c. Hilbert's Variational Approach to General Relativity 305
11.4. The Second Fundamental Form in the Riemannian Case 309
11.4a. The Induced Connection and the Second Fundamental
Form 309
11.4b. The Equations of Gauss and Codazzi 311
11.4c. The Interpretation of the Sectional Curvature 313
11.4d. Fixed Points of Isometries 314
11.5. The Geometry of Einstein's Equations 315
11.5a. The Einstein Tensor in a (Pseudo )Riemannian
Space Time 315
11.5b. The Relativistic Meaning of Gauss's Equation 316
11.5c. The Second Fundamental Form of a Spatial Slice 318
11.5d. The Codazzi Equations 319
11.5e. Some Remarks on the Schwarzschild Solution 320
12 Curvature and Topology: Synge's Theorem 323
12.1. Synge's Formula for Second Variation 324
12.1a. The Second Variation of Arc Length 324
12.1b. Jacobi Fields 326
CONTENTS Xiii
12.2. Curvature and Simple Connectivity 329
12.2a. Synge's Theorem 329
12.2b. Orientability Revisited 331
13 Betti Numbers and De Rham's Theorem 333
13.1. Singular Chains and Their Boundaries 333
13.1a. Singular Chains 333
13.1b. Some 2 DimensionaI Examples 338
13.2. The Singular Homology Groups 342
13.2a. Coefficient Fields 342
13.2b. Finite Simplicial Complexes 343
13.2c. Cycles, Boundaries, Homology, and Betti Numbers 344
13.3. Homology Groups of Familiar Manifolds 347
13.3a. Some Computational Tools 347
13.3b. Familiar Examples 350
13.4. De Rham's Theorem 355
13.4a. The Statement of De Rham's Theorem 355
13.4b. Two Examples 357
14 Harmonic Forms 361
14.1. The Hodge Operators 361
14.1a. The* Operator 361
14.1b. The Codifferential Operator S d* 364
14.1c. Maxwell's Equations in Curved Space Time M4 366
14.1d. The Hilbert Lagrangian 367
14.2. Harmonic Forms 368
14.2a. The Laplace Operator on Forms 368
14.2b. The Laplacian of a 1 Form 369
14.2c. Harmonic Forms on Closed Manifolds 370
14.2d. Harmonic Forms and De Rham's Theorem 372
14.2e. Bochner's Theorem 374
14.3. Boundary Values, Relative Homology, and Morse Theory 375
14.3a. Tangential and Normal Differential Forms 376
14.3b. Hodge's Theorem for Tangential Forms 377
14.3c. Relative Homology Groups 379
14.3d. Hodge's Theorem for Normal Forms 381
14.3e. Morse's Theory of Critical Points 382
III Lie Groups, Bundles, and Chern Forms
15 Lie Groups 391
15.1. Lie Groups, Invariant Vector Fields, and Forms 391
15.1a. Lie Groups 391
15.1b. Invariant Vector Fields and Forms 395
15.2. One Parameter Subgroups 398
15.3. The Lie Algebra of a Lie Group 402
15.3a. The Lie Algebra 402
Xiv CONTENTS
15.3b. The Exponential Map 403
15.3c. Examples of Lie Algebras 404
15.3d. Do the 1 Parameter Subgroups Cover G? 405
15.4. Subgroups and Subalgebras 407
15.4a. Left Invariant Fields Generate Right Translations 407
15.4b. Commutators of Matrices 408
15.4c. Right Invariant Fields 409
15.4d. Subgroups and Subalgebras 410
16 Vector Bundles in Geometry and Physics 413
16.1. Vector Bundles 413
16.1a. Motivation by Two Examples 413
16.1b. Vector Bundles 415
16.1c. Local Trivializations 417
16.1d. The Normal Bundle to a Submanifold 419
16.2. Poincare's Theorem and the Euler Characteristic 421
16.2a. Poincare's Theorem 422
16.2b. The Stiefel Vector Field and Euler's Theorem 426
16.3. Connections in a Vector Bundle 428
16.3a. Connection in a Vector Bundle 428
16.3b. Complex Vector Spaces 431
16.3c. The Structure Group of a Bundle 433
16.3d. Complex Line Bundles 433
16.4. The Electromagnetic Connection 435
16.4a. Lagrange's Equations without Electromagnetism 435
16.4b. The Modified Lagrangian and Hamiltonian 436
16.4c. Schrodinger's Equation in an Electromagnetic Field 439
16.4d. Global Potentials 443
16.4e. The Dirac Monopole 444
16.4f. The Aharonov Bohm Effect 446
17 Fiber Bundles, Gauss Bonnet, and Topological Quantization 451
17.1. Fiber Bundles and Principal Bundles 451
17.1a. Fiber Bundles 451
17.1b. Principal Bundles and Frame Bundles 453
17.1c. Action of the Structure Group on a Principal Bundle 454
17.2. Coset Spaces 456
17.2a. Cosets 456
17.2b. Grassmann Manifolds 459
17.3. Chern's Proof of the Gauss Bonnet Poincare Theorem 460
17.3a. A Connection in the Frame Bundle of a Surface 460
17.3b. The Gauss Bonnet Poincare Theorem 462
17.3c. Gauss Bonnet as an Index Theorem 465
17.4. Line Bundles, Topological Quantization, and Berry Phase 465
17.4a. A Generalization of Gauss Bonnet 465
17.4b. Berry Phase 468
17.4c. Monopoles and the Hopf Bundle 473
CONTENTS XV
18 Connections and Associated Bundles 475
18.1. Forms with Values in a Lie Algebra 475
18.1a. The Maurer Cartan Form 475
18.1b. op Valued p Forms on a Manifold 477
18.1c. Connections in a Principal Bundle 479
18.2. Associated Bundles and Connections 481
18.2a. Associated Bundles 481
18.2b. Connections in Associated Bundles 483
18.2c. The Associated Ad Bundle 485
18.3. r Form Sections of a Vector Bundle: Curvature 488
18.3a. / Form Sections of E 488
18.3b. Curvature and the Ad Bundle 489
19 The Dirac Equation 491
19.1. The Groups SO(3) and SU(2) 491
19.1a. The Rotation Group S O (3) of R3 492
19.1b. 51/(2): The Lie Algebra a*i. 2) 493
19.1c. 5(7(2) Is Topologically the 3 Sphere 495
19.1d. Ad : SU(2) » 50(3) in More Detail 496
19.2. Hamilton, Clifford, and Dirac 497
19.2a. Spinors and Rotations of M3 497
19.2b. Hamilton on Composing Two Rotations 499
19.2c. Clifford Algebras 500
19.2d. The Dirac Program: The Square Root of the
d'Alembertian 502
19.3. The Dirac Algebra 504
19.3a. The Lorentz Group 504
19.3b. The Dirac Algebra 509
19.4. The Dirac Operator jl in Minkowski Space 511
19.4a. Dirac Spinors 511
19.4b. The Dirac Operator 513
19.5. The Dirac Operator in Curved Space Time 515
19.5a. The Spinor Bundle 515
19.5b. The Spin Connection in M 518
20 Yang Mills Fields 523
20.1. Noether's Theorem for Internal Symmetries 523
20.1a. The Tensorial Nature of Lagrange's Equations 523
20.1b. Boundary Conditions 526
20.1c. Noether's Theorem for Internal Symmetries 527
20.1d. Noether's Principle 528
20.2. Weyl's Gauge Invariance Revisited 531
20.2a. The Dirac Lagrangian 531
20.2b. Weyl's Gauge Invariance Revisited 533
20.2c. The Electromagnetic Lagrangian 534
20.2d. Quantization of the A Field: Photons 536
XVi CONTENTS
20.3. The Yang Mills Nucleon 537
20.3a. The Heisenberg Nucleon 537
20.3b. The Yang Mills Nucleon 538
20.3c. A Remark on Terminology 540
20.4. Compact Groups and Yang Mills Action 541
20.4a. The Unitary Group Is Compact 541
20.4b. Averaging over a Compact Group 541
20.4c. Compact Matrix Groups Are Subgroups of Unitary
Groups 542
20.4d. Ad Invariant Scalar Products in the Lie Algebra of a
Compact Group 543
20.4e. The Yang Mills Action 544
20.5. The Yang Mills Equation 545
20.5a. The Exterior Covariant Divergence V* 545
20.5b. The Yang Mills Analogy with Electromagnetism 547
20.5c. Further Remarks on the Yang Mills Equations 548
20.6. Yang Mills Instantons 550
20.6a. Instantons 550
20.6b. Chern's Proof Revisited 553
20.6c. Instantons and the Vacuum 557
21 Betti Numbers and Covering Spaces 561
21.1. Bi invariant Forms on Compact Groups 561
21.1a. Bi invariant p Forms 561
21.1b. The Cartan /j Forms 562
21.1c. Bi invariant Riemannian Metrics 563
21.Id. Harmonic Forms in the Bi invariant Metric 564
21.1e. Weyl and Cartan on the Betti Numbers of G 565
21.2. The Fundamental Group and Covering Spaces 567
21.2a. Poincare's Fundamental Group Tt\ (M) 567
21.2b. The Concept of a Covering Space 569
21.2c. The Universal Covering 570
21.2d. The Orientable Covering 573
21.2e. Lifting Paths 574
21.2f. Subgroups of tt,(M) 575
21.2g. The Universal Covering Group 575
21.3. The Theorem of S. B. Myers: A Problem Set 576
21.4. The Geometry of a Lie Group 580
21.4a. The Connection of a Bi invariant Metric 580
21.4b. The Flat Connections 581
22 Chern Forms and Homotopy Groups 583
22.1. Chern Forms and Winding Numbers 583
22.1a. The Yang Mills "Winding Number" 583
22.1b. Winding Number in Terms of Field Strength 585
22.1c. The Chern Forms for a U(n) Bundle 587
CONTENTS XVii
22.2. Homotopies and Extensions 591
22.2a. Homotopy 591
22.2b. Covering Homotopy 592
22.2c. Some Topology of SU (n) 594
22.3. The Higher Homotopy Groups jr* (M) 596
22.3a. nk{M) 596
22.3b. Homotopy Groups of Spheres 597
22.3c. Exact Sequences of Groups 598
22.3d. The Homotopy Sequence of a Bundle 600
22.3e. The Relation between Homotopy and Homology
Groups 603
22.4. Some Computations of Homotopy Groups 605
22.4a. Lifting Spheres from M into the Bundle P 605
22.4b. SU(n) Again 606
22.4c. TheHopf Map and Fibering 606
22.5. Chern Forms as Obstructions 608
22.5a. The Chern Forms cr for an SU(n) Bundle Revisited 608
22.5b. c% as an "Obstruction Cocycle" 609
22.5c. The Meaning of the Integer j (A4) 612
22.5d. Chern's Integral 612
22.5e. Concluding Remarks 615
Appendix A. Forms in Continuum Mechanics 617
A.a. The Classical Cauchy Stress Tensor and Equations of Motion 617
A.b. Stresses in Terms of Exterior Forms 618
A.c. Symmetry of Cauchy's Stress Tensor in K" 620
A.d. The Piola Kirchhoff Stress Tensors 622
A.e. Stored Energy of Deformation 623
A.f. Hamilton's Principle in Elasticity 626
A.g. Some Typical Computations Using Forms 629
A.h. Concluding Remarks 635
Appendix B. Harmonic Chains and Kirchhoff 's Circuit Laws 636
B.a. Chain Complexes 636
B.b. Cochains and Cohomology 638
B.C. Transpose and Adjoint 639
B.d. Laplacians and Harmonic Cochains 641
B.e. Kirchhoff s Circuit Laws 643
Appendix C. Symmetries, Quarks, and Meson Masses 648
C.a. Flavored Quarks 648
C.b. Interactions of Quarks and Antiquarks 650
C.c. The Lie Algebra of SU(3) 652
C.d. Pions, Kaons, and Etas 653
C.e. A Reduced Symmetry Group 656
C.f. Meson Masses 658
xviii contents
Appendix D. Representations and Hyperelastic Bodies 660
D.a. Hyperelastic Bodies 660
D.b. Isotropic Bodies 661
D.c. Application of Schur's Lemma 662
D.d. Frobenius Schur Relations 664
D.e. The Symmetric Traceless 3x3 Matrices Are Irreducible 666
Appendix E. Orbits and Morse Bott Theory in Compact Lie Groups 670
E.a. The Topology of Conjugacy Orbits 670
E.b. Application of Bott's Extension of Morse Theory 673
References 679
Index 683 |
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| author | Frankel, Theodore 1929- |
| author_GND | (DE-588)134187318 |
| author_facet | Frankel, Theodore 1929- |
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| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 2. ed., reprint. |
| format | Book |
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| genre | Einführung - Differentialgeometrie |
| genre_facet | Einführung - Differentialgeometrie |
| id | DE-604.BV021757247 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:33:53Z |
| indexdate | 2024-07-09T20:43:21Z |
| institution | BVB |
| isbn | 0521833302 0521539277 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014970392 |
| oclc_num | 253986863 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XXVI, 694 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Frankel, Theodore 1929- Verfasser (DE-588)134187318 aut The geometry of physics an introduction Theodore Frankel 2. ed., reprint. Cambridge [u.a.] Cambridge Univ. Press 2006 XXVI, 694 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometrie - Mathematische Physik Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Einführung - Differentialgeometrie Mathematische Physik (DE-588)4037952-8 s Geometrie (DE-588)4020236-7 s DE-604 Differentialgeometrie (DE-588)4012248-7 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014970392&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Frankel, Theodore 1929- The geometry of physics an introduction Geometrie - Mathematische Physik Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4012248-7 (DE-588)4020236-7 |
| title | The geometry of physics an introduction |
| title_auth | The geometry of physics an introduction |
| title_exact_search | The geometry of physics an introduction |
| title_exact_search_txtP | The geometry of physics an introduction |
| title_full | The geometry of physics an introduction Theodore Frankel |
| title_fullStr | The geometry of physics an introduction Theodore Frankel |
| title_full_unstemmed | The geometry of physics an introduction Theodore Frankel |
| title_short | The geometry of physics |
| title_sort | the geometry of physics an introduction |
| title_sub | an introduction |
| topic | Geometrie - Mathematische Physik Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Geometrie - Mathematische Physik Mathematische Physik Differentialgeometrie Geometrie Einführung - Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014970392&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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