Contents
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Preliminaries and Local Theory 1
1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11
1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Differentiable Manifolds 15
2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17
2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22
2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27
2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30
2.6.1 Covering spaces and the fundamental group . . . . . . . . 30
2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36
2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43
3 The Tangent Structure 47
3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55
3.5.1 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57
3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59
4 Submanifold, Immersion and Submersion. 63
4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Regular and Critical Points and Values . . . . . . . . . . . . . . . 66
4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
iii
iv CONTENTS
4.5 Immersed Submanifolds and Initial Submanifolds . . . . . . . . . 71
4.6 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Lie Groups I 81
5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Lie Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 84
6 Fiber Bundles and Vector Bundles I 87
6.1 Transitions Maps and Structure . . . . . . . . . . . . . . . . . . . 94
6.2 Useful ways to think about vector bundles . . . . . . . . . . . . . 94
6.3 Sections of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . 97
6.4 Sheaves,Germs and Jets . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Jets and Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Vector Fields and 1-Forms 105
7.1 Definition of vector fields and 1-forms . . . . . . . . . . . . . . . 105
7.2 Pull back and push forward of functions and 1-forms . . . . . . . 106
7.3 Frame Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.6 Action by pullback and push-forward . . . . . . . . . . . . . . . . 112
7.7 Flows and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 114
7.8 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.9 Time Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . 123
8 Lie Groups II 125
8.1 Spinors and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Multilinear Bundles and Tensors Fields 137
9.1 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.1.1 Contraction of tensors . . . . . . . . . . . . . . . . . . . . 141
9.1.2 Alternating Multilinear Algebra . . . . . . . . . . . . . . . 142
9.1.3 Orientation on vector spaces . . . . . . . . . . . . . . . . 146
9.2 Multilinear Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4 Tensor Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10 Differential forms 153
10.1 Pullback of a differential form. . . . . . . . . . . . . . . . . . . . 155
10.2 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4 Lie derivative, interior product and exterior derivative. . . . . . . 161
10.5 Time Dependent Fields (Part II) . . . . . . . . . . . . . . . . . . 163
10.6 Vector valued and algebra valued forms. . . . . . . . . . . . . . . 163
CONTENTS v
10.7 Global Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.8 Orientation of manifolds with boundary . . . . . . . . . . . . . . 167
10.9 Integration of Differential Forms. . . . . . . . . . . . . . . . . . . 168
10.10Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.11Vector Bundle Valued Forms. . . . . . . . . . . . . . . . . . . . . 172
11 Distributions and Frobenius’ Theorem 175
11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.2 Integrability of Regular Distributions . . . . . . . . . . . . . . . 175
11.3 The local version Frobenius’ theorem . . . . . . . . . . . . . . . . 177
11.4 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5 The Global Frobenius Theorem . . . . . . . . . . . . . . . . . . . 183
11.6 Singular Distributions . . . . . . . . . . . . . . . . . . . . . . . . 185
12 Connections on Vector Bundles 189
12.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.2 Local Frame Fields and Connection Forms . . . . . . . . . . . . . 191
12.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13 Riemannian and semi-Riemannian Manifolds 201
13.1 The Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.2 Natural Extensions and the Star Operator . . . . . . . . . 203
13.2 Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
13.3 Riemannian and semi-Riemannian Metrics . . . . . . . . . . . . . 214
13.4 The Riemannian case (positive definite metric) . . . . . . . . . . 220
13.5 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . 221
13.6 Covariant differentiation of vector fields along maps. . . . . . . . 228
13.7 Covariant differentiation of tensor fields . . . . . . . . . . . . . . 229
13.8 Comparing the Differential Operators . . . . . . . . . . . . . . . 230
14 Formalisms for Calculation 233
14.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14.2 Covariant Exterior Calculus, Bundle-Valued Forms . . . . . . . . 234
15 Topology 235
15.1 Attaching Spaces and Quotient Topology . . . . . . . . . . . . . 235
15.2 Topological Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.4 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16 Algebraic Topology 245
16.1 Axioms for a Homology Theory . . . . . . . . . . . . . . . . . . . 245
16.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.3 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
vi CONTENTS
16.4 Cellular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.5 Universal Coefficient theorem . . . . . . . . . . . . . . . . . . . . 246
16.6 Axioms for a Cohomology Theory . . . . . . . . . . . . . . . . . 246
16.7 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 246
16.8 Topology of Vector Bundles . . . . . . . . . . . . . . . . . . . . . 246
16.9 de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 248
16.10The Meyer Vietoris Sequence . . . . . . . . . . . . . . . . . . . . 252
16.11Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 253
16.12Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . 253
17 Lie Groups and Lie Algebras 255
17.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
17.2 Classical complex Lie algebras . . . . . . . . . . . . . . . . . . . . 257
17.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 258
17.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . 259
17.4 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . . 261
17.5 The Adjoint Representation of a Lie group . . . . . . . . . . . . . 265
18 Group Actions and Homogenous Spaces 271
18.1 Our Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
18.1.1 Left actions . . . . . . . . . . . . . . . . . . . . . . . . . . 272
18.1.2 Right actions . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.3 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.4 The action of Diff(M) and map-related vector fields. . . 274
18.1.5 Lie derivative for equivariant bundles. . . . . . . . . . . . 274
18.2 Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 275
19 Fiber Bundles and Connections 279
19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
19.2 Principal and Associated Bundles . . . . . . . . . . . . . . . . . . 282
20 Analysis on Manifolds 285
20.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.1 Star Operator II . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.2 Divergence, Gradient, Curl . . . . . . . . . . . . . . . . . 286
20.2 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . 286
20.3 Spectral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.4 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 291
20.5.2 The Clifford group and Spinor group . . . . . . . . . . . . 296
20.6 The Structure of Clifford Algebras . . . . . . . . . . . . . . . . . 296
20.6.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . 297
20.7 Clifford Algebra Structure and Representation . . . . . . . . . . 298
20.7.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . 298
20.7.2 Hyperbolic Spaces And Witt Decomposition . . . . . . . . 299
CONTENTS vii
20.7.3 Witt’s Decomposition and Clifford Algebras . . . . . . . . 300
20.7.4 The Chirality operator . . . . . . . . . . . . . . . . . . . 301
20.7.5 Spin Bundles and Spin-c Bundles . . . . . . . . . . . . . . 302
20.7.6 Harmonic Spinors . . . . . . . . . . . . . . . . . . . . . . 302
21 Complex Manifolds 303
21.1 Some complex linear algebra . . . . . . . . . . . . . . . . . . . . 303
21.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . 306
21.3 Complex Tangent Structures . . . . . . . . . . . . . . . . . . . . 309
21.4 The holomorphic tangent map. . . . . . . . . . . . . . . . . . . . 310
21.5 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
21.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
21.7 The holomorphic inverse and implicit functions theorems. . . . . 312
22 Classical Mechanics 315
22.1 Particle motion and Lagrangian Systems . . . . . . . . . . . . . . 315
22.1.1 Basic Variational Formalism for a Lagrangian . . . . . . . 316
22.1.2 Two examples of a Lagrangian . . . . . . . . . . . . . . . 319
22.2 Symmetry, Conservation and Noether’s Theorem . . . . . . . . . 319
22.2.1 Lagrangians with symmetries. . . . . . . . . . . . . . . . . 321
22.2.2 Lie Groups and Left Invariants Lagrangians . . . . . . . . 322
22.3 The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 322
23 Symplectic Geometry 325
23.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 325
23.2 Canonical Form (Linear case) . . . . . . . . . . . . . . . . . . . . 327
23.3 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 327
23.4 Complex Structure and K¨ahler Manifolds . . . . . . . . . . . . . 329
23.5 Symplectic musical isomorphisms . . . . . . . . . . . . . . . . . 332
23.6 Darboux’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 332
23.7 Poisson Brackets and Hamiltonian vector fields . . . . . . . . . . 334
23.8 Configuration space and Phase space . . . . . . . . . . . . . . . 337
23.9 Transfer of symplectic structure to the Tangent bundle . . . . . . 338
23.10Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
23.11The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
23.11.1 The configuration in R3N . . . . . . . . . . . . . . . . . . 342
23.11.2Modelling the rigid body on SO(3) . . . . . . . . . . . . . 342
23.11.3 The trivial bundle picture . . . . . . . . . . . . . . . . . . 343
23.12The momentum map and Hamiltonian actions . . . . . . . . . . . 343
24 Poisson Geometry 347
24.1 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 347
viii CONTENTS
25 Quantization 351
25.1 Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . 351
25.2 C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
25.2.1 Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . . 354
25.3 Jordan-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 354
26 Appendices 357
26.1 A. Primer for Manifold Theory . . . . . . . . . . . . . . . . . . . 357
26.1.1 Fixing a problem . . . . . . . . . . . . . . . . . . . . . . . 360
26.2 B. Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 361
26.2.1 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 363
26.2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 364
26.3 C. Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . 365
26.3.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 367
26.3.2 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . 368
26.4 D. Overview of Classical Physics . . . . . . . . . . . . . . . . . . 368
26.4.1 Units of measurement . . . . . . . . . . . . . . . . . . . . 368
26.4.2 Newton’s equations . . . . . . . . . . . . . . . . . . . . . . 369
26.4.3 Classical particle motion in a conservative field . . . . . . 370
26.4.4 Some simple mechanical systems . . . . . . . . . . . . . . 375
26.4.5 The Basic Ideas of Relativity . . . . . . . . . . . . . . . . 380
26.4.6 Variational Analysis of Classical Field Theory . . . . . . . 385
26.4.7 Symmetry and Noether’s theorem for field theory . . . . . 386
26.4.8 Electricity and Magnetism . . . . . . . . . . . . . . . . . . 388
26.4.9 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 390
26.5 E. Calculus on Banach Spaces . . . . . . . . . . . . . . . . . . . . 390
26.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
26.7 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
26.8 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 400
26.9 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 405
26.9.1 Linear case. . . . . . . . . . . . . . . . . . . . . . . . . . 411
26.9.2 Local (nonlinear) case. . . . . . . . . . . . . . . . . . 412
26.10The Tangent Bundle of an Open Subset of a Banach Space . . . 413
26.11Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
26.11.1 Existence and uniqueness for differential equations . . . . 417
26.11.2 Differential equations depending on a parameter. . . . . . 418
26.12Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 418
26.12.1Smooth Banach Vector Bundles . . . . . . . . . . . . . . . 435
26.12.2Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
26.13Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.14Group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.15Notation and font usage guide . . . . . . . . . . . . . . . . . . . . 445
27 Bibliography 453
0.1. PREFACE ix
0.1 Preface
In this book I present differential geometry and related mathematical topics with
the help of examples from physics. It is well known that there is something
strikingly mathematical about the physical universe as it is conceived of in
the physical sciences. The convergence of physics with mathematics, especially
differential geometry, topology and global analysis is even more pronounced in
the newer quantum theories such as gauge field theory and string theory. The
amount of mathematical sophistication required for a good understanding of
modern physics is astounding. On the other hand, the philosophy of this book
is that mathematics itself is illuminated by physics and physical thinking.
The ideal of a truth that transcends all interpretation is perhaps unattain-
able. Even the two most impressively objective realities, the physical and the
mathematical, are still only approachable through, and are ultimately insepa-
rable from, our normative and linguistic background. And yet it is exactly the
tendency of these two sciences to point beyond themselves to something tran-
scendentally real that so inspires us. Whenever we interpret something real,
whether physical or mathematical, there will be those aspects which arise as
mere artifacts of our current descriptive scheme and those aspects that seem to
be objective realities which are revealed equally well through any of a multitude
of equivalent descriptive schemes-“cognitive inertial frames” as it were. This
theme is played out even within geometry itself where a viewpoint or interpre-
tive scheme translates to the notion of a coordinate system on a differentiable
manifold.
A physicist has no trouble believing that a vector field is something beyond
its representation in any particular coordinate system since the vector field it-
self is something physical. It is the way that the various coordinate descriptions
relate to each other (covariance) that manifests to the understanding the pres-
ence of an invariant physical reality. This seems to be very much how human
perception works and it is interesting that the language of tensors has shown up
in the cognitive science literature.
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Preliminaries and Local Theory 1
1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11
1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Differentiable Manifolds 15
2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17
2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22
2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27
2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30
2.6.1 Covering spaces and the fundamental group . . . . . . . . 30
2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36
2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43
3 The Tangent Structure 47
3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55
3.5.1 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57
3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59
4 Submanifold, Immersion and Submersion. 63
4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Regular and Critical Points and Values . . . . . . . . . . . . . . . 66
4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
iii
iv CONTENTS
4.5 Immersed Submanifolds and Initial Submanifolds . . . . . . . . . 71
4.6 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Lie Groups I 81
5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Lie Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 84
6 Fiber Bundles and Vector Bundles I 87
6.1 Transitions Maps and Structure . . . . . . . . . . . . . . . . . . . 94
6.2 Useful ways to think about vector bundles . . . . . . . . . . . . . 94
6.3 Sections of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . 97
6.4 Sheaves,Germs and Jets . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Jets and Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . 102
7 Vector Fields and 1-Forms 105
7.1 Definition of vector fields and 1-forms . . . . . . . . . . . . . . . 105
7.2 Pull back and push forward of functions and 1-forms . . . . . . . 106
7.3 Frame Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.5 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.6 Action by pullback and push-forward . . . . . . . . . . . . . . . . 112
7.7 Flows and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 114
7.8 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.9 Time Dependent Fields . . . . . . . . . . . . . . . . . . . . . . . 123
8 Lie Groups II 125
8.1 Spinors and rotation . . . . . . . . . . . . . . . . . . . . . . . . . 133
9 Multilinear Bundles and Tensors Fields 137
9.1 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.1.1 Contraction of tensors . . . . . . . . . . . . . . . . . . . . 141
9.1.2 Alternating Multilinear Algebra . . . . . . . . . . . . . . . 142
9.1.3 Orientation on vector spaces . . . . . . . . . . . . . . . . 146
9.2 Multilinear Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4 Tensor Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 149
10 Differential forms 153
10.1 Pullback of a differential form. . . . . . . . . . . . . . . . . . . . 155
10.2 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.3 Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . . . . 159
10.4 Lie derivative, interior product and exterior derivative. . . . . . . 161
10.5 Time Dependent Fields (Part II) . . . . . . . . . . . . . . . . . . 163
10.6 Vector valued and algebra valued forms. . . . . . . . . . . . . . . 163
CONTENTS v
10.7 Global Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.8 Orientation of manifolds with boundary . . . . . . . . . . . . . . 167
10.9 Integration of Differential Forms. . . . . . . . . . . . . . . . . . . 168
10.10Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.11Vector Bundle Valued Forms. . . . . . . . . . . . . . . . . . . . . 172
11 Distributions and Frobenius’ Theorem 175
11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.2 Integrability of Regular Distributions . . . . . . . . . . . . . . . 175
11.3 The local version Frobenius’ theorem . . . . . . . . . . . . . . . . 177
11.4 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.5 The Global Frobenius Theorem . . . . . . . . . . . . . . . . . . . 183
11.6 Singular Distributions . . . . . . . . . . . . . . . . . . . . . . . . 185
12 Connections on Vector Bundles 189
12.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
12.2 Local Frame Fields and Connection Forms . . . . . . . . . . . . . 191
12.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13 Riemannian and semi-Riemannian Manifolds 201
13.1 The Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . 201
13.1.2 Natural Extensions and the Star Operator . . . . . . . . . 203
13.2 Surface Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
13.3 Riemannian and semi-Riemannian Metrics . . . . . . . . . . . . . 214
13.4 The Riemannian case (positive definite metric) . . . . . . . . . . 220
13.5 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . 221
13.6 Covariant differentiation of vector fields along maps. . . . . . . . 228
13.7 Covariant differentiation of tensor fields . . . . . . . . . . . . . . 229
13.8 Comparing the Differential Operators . . . . . . . . . . . . . . . 230
14 Formalisms for Calculation 233
14.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14.2 Covariant Exterior Calculus, Bundle-Valued Forms . . . . . . . . 234
15 Topology 235
15.1 Attaching Spaces and Quotient Topology . . . . . . . . . . . . . 235
15.2 Topological Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
15.4 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16 Algebraic Topology 245
16.1 Axioms for a Homology Theory . . . . . . . . . . . . . . . . . . . 245
16.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.3 Singular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
vi CONTENTS
16.4 Cellular Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.5 Universal Coefficient theorem . . . . . . . . . . . . . . . . . . . . 246
16.6 Axioms for a Cohomology Theory . . . . . . . . . . . . . . . . . 246
16.7 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 246
16.8 Topology of Vector Bundles . . . . . . . . . . . . . . . . . . . . . 246
16.9 de Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 248
16.10The Meyer Vietoris Sequence . . . . . . . . . . . . . . . . . . . . 252
16.11Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 253
16.12Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . 253
17 Lie Groups and Lie Algebras 255
17.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
17.2 Classical complex Lie algebras . . . . . . . . . . . . . . . . . . . . 257
17.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 258
17.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . 259
17.4 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . . 261
17.5 The Adjoint Representation of a Lie group . . . . . . . . . . . . . 265
18 Group Actions and Homogenous Spaces 271
18.1 Our Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
18.1.1 Left actions . . . . . . . . . . . . . . . . . . . . . . . . . . 272
18.1.2 Right actions . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.3 Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.1.4 The action of Diff(M) and map-related vector fields. . . 274
18.1.5 Lie derivative for equivariant bundles. . . . . . . . . . . . 274
18.2 Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . 275
19 Fiber Bundles and Connections 279
19.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
19.2 Principal and Associated Bundles . . . . . . . . . . . . . . . . . . 282
20 Analysis on Manifolds 285
20.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.1 Star Operator II . . . . . . . . . . . . . . . . . . . . . . . 285
20.1.2 Divergence, Gradient, Curl . . . . . . . . . . . . . . . . . 286
20.2 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . 286
20.3 Spectral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.4 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5 Dirac Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
20.5.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 291
20.5.2 The Clifford group and Spinor group . . . . . . . . . . . . 296
20.6 The Structure of Clifford Algebras . . . . . . . . . . . . . . . . . 296
20.6.1 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . 297
20.7 Clifford Algebra Structure and Representation . . . . . . . . . . 298
20.7.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . 298
20.7.2 Hyperbolic Spaces And Witt Decomposition . . . . . . . . 299
CONTENTS vii
20.7.3 Witt’s Decomposition and Clifford Algebras . . . . . . . . 300
20.7.4 The Chirality operator . . . . . . . . . . . . . . . . . . . 301
20.7.5 Spin Bundles and Spin-c Bundles . . . . . . . . . . . . . . 302
20.7.6 Harmonic Spinors . . . . . . . . . . . . . . . . . . . . . . 302
21 Complex Manifolds 303
21.1 Some complex linear algebra . . . . . . . . . . . . . . . . . . . . 303
21.2 Complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . 306
21.3 Complex Tangent Structures . . . . . . . . . . . . . . . . . . . . 309
21.4 The holomorphic tangent map. . . . . . . . . . . . . . . . . . . . 310
21.5 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
21.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
21.7 The holomorphic inverse and implicit functions theorems. . . . . 312
22 Classical Mechanics 315
22.1 Particle motion and Lagrangian Systems . . . . . . . . . . . . . . 315
22.1.1 Basic Variational Formalism for a Lagrangian . . . . . . . 316
22.1.2 Two examples of a Lagrangian . . . . . . . . . . . . . . . 319
22.2 Symmetry, Conservation and Noether’s Theorem . . . . . . . . . 319
22.2.1 Lagrangians with symmetries. . . . . . . . . . . . . . . . . 321
22.2.2 Lie Groups and Left Invariants Lagrangians . . . . . . . . 322
22.3 The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 322
23 Symplectic Geometry 325
23.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 325
23.2 Canonical Form (Linear case) . . . . . . . . . . . . . . . . . . . . 327
23.3 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 327
23.4 Complex Structure and K¨ahler Manifolds . . . . . . . . . . . . . 329
23.5 Symplectic musical isomorphisms . . . . . . . . . . . . . . . . . 332
23.6 Darboux’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 332
23.7 Poisson Brackets and Hamiltonian vector fields . . . . . . . . . . 334
23.8 Configuration space and Phase space . . . . . . . . . . . . . . . 337
23.9 Transfer of symplectic structure to the Tangent bundle . . . . . . 338
23.10Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
23.11The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
23.11.1 The configuration in R3N . . . . . . . . . . . . . . . . . . 342
23.11.2Modelling the rigid body on SO(3) . . . . . . . . . . . . . 342
23.11.3 The trivial bundle picture . . . . . . . . . . . . . . . . . . 343
23.12The momentum map and Hamiltonian actions . . . . . . . . . . . 343
24 Poisson Geometry 347
24.1 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 347
viii CONTENTS
25 Quantization 351
25.1 Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . 351
25.2 C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
25.2.1 Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . . 354
25.3 Jordan-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 354
26 Appendices 357
26.1 A. Primer for Manifold Theory . . . . . . . . . . . . . . . . . . . 357
26.1.1 Fixing a problem . . . . . . . . . . . . . . . . . . . . . . . 360
26.2 B. Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 361
26.2.1 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . 363
26.2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 364
26.3 C. Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . 365
26.3.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 367
26.3.2 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . 368
26.4 D. Overview of Classical Physics . . . . . . . . . . . . . . . . . . 368
26.4.1 Units of measurement . . . . . . . . . . . . . . . . . . . . 368
26.4.2 Newton’s equations . . . . . . . . . . . . . . . . . . . . . . 369
26.4.3 Classical particle motion in a conservative field . . . . . . 370
26.4.4 Some simple mechanical systems . . . . . . . . . . . . . . 375
26.4.5 The Basic Ideas of Relativity . . . . . . . . . . . . . . . . 380
26.4.6 Variational Analysis of Classical Field Theory . . . . . . . 385
26.4.7 Symmetry and Noether’s theorem for field theory . . . . . 386
26.4.8 Electricity and Magnetism . . . . . . . . . . . . . . . . . . 388
26.4.9 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 390
26.5 E. Calculus on Banach Spaces . . . . . . . . . . . . . . . . . . . . 390
26.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
26.7 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
26.8 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 400
26.9 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 405
26.9.1 Linear case. . . . . . . . . . . . . . . . . . . . . . . . . . 411
26.9.2 Local (nonlinear) case. . . . . . . . . . . . . . . . . . 412
26.10The Tangent Bundle of an Open Subset of a Banach Space . . . 413
26.11Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
26.11.1 Existence and uniqueness for differential equations . . . . 417
26.11.2 Differential equations depending on a parameter. . . . . . 418
26.12Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 418
26.12.1Smooth Banach Vector Bundles . . . . . . . . . . . . . . . 435
26.12.2Formulary . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
26.13Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.14Group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
26.15Notation and font usage guide . . . . . . . . . . . . . . . . . . . . 445
27 Bibliography 453
0.1. PREFACE ix
0.1 Preface
In this book I present differential geometry and related mathematical topics with
the help of examples from physics. It is well known that there is something
strikingly mathematical about the physical universe as it is conceived of in
the physical sciences. The convergence of physics with mathematics, especially
differential geometry, topology and global analysis is even more pronounced in
the newer quantum theories such as gauge field theory and string theory. The
amount of mathematical sophistication required for a good understanding of
modern physics is astounding. On the other hand, the philosophy of this book
is that mathematics itself is illuminated by physics and physical thinking.
The ideal of a truth that transcends all interpretation is perhaps unattain-
able. Even the two most impressively objective realities, the physical and the
mathematical, are still only approachable through, and are ultimately insepa-
rable from, our normative and linguistic background. And yet it is exactly the
tendency of these two sciences to point beyond themselves to something tran-
scendentally real that so inspires us. Whenever we interpret something real,
whether physical or mathematical, there will be those aspects which arise as
mere artifacts of our current descriptive scheme and those aspects that seem to
be objective realities which are revealed equally well through any of a multitude
of equivalent descriptive schemes-“cognitive inertial frames” as it were. This
theme is played out even within geometry itself where a viewpoint or interpre-
tive scheme translates to the notion of a coordinate system on a differentiable
manifold.
A physicist has no trouble believing that a vector field is something beyond
its representation in any particular coordinate system since the vector field it-
self is something physical. It is the way that the various coordinate descriptions
relate to each other (covariance) that manifests to the understanding the pres-
ence of an invariant physical reality. This seems to be very much how human
perception works and it is interesting that the language of tensors has shown up
in the cognitive science literature.
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