Adapted chart 98
Adjoint 303
Adjoint representation 304
Admissible chart 82
Almost complex manifold 284
Almost symplectic 507
Alternating 203 580
Antiholomorphic 283
Associated right action 312
Atlas 82
Atlas, submanifold 98
Base space 644
Boundary, manifold with 100 220
Bump function 125
Bundle map, local 27 638
C* algebra 531
Canonical 1-form 515
Cartesian product xv
Cauchy — Bochner integral 20 625
Causal character 381
Cell complex 454
Chain rule 18 623
Chart 82
Chirality operator 490
Clifford algebra 482
Clifford group 485
Co-distribution 329
Co-frame 124
Co-symplectic structure 526
Coadjoint action 506
Coisotropic 526
Complemented subspace 14 618
Complete vector field 168 169
COMPLEX 271
Complex manifold 284
Continuously differentiable 85
Contraction 576
Contragradient 117
Contravariant 567
coordinates 83
Cotangent bundle 116
Covariant derivative 339
Covariant, type 567
Critical value 235
CW complex 454
Darboux's theorem 510
De Rham cohomology 211
Decomposable 152 571
Derivation 109 124 306
Derivation law 125
Derivative 17 622
Diffeomorphism 18 85 622
Differentiable 84
Differentiable structure 82
Differential 109 122
Differential complex 271
Differential form, vector valued 212
Differential, Lie 260
Discrete group action 94
Effective action 316
Endomorphism algebra 578
Equivariance 309 311
Euler — Lagrange equation(s) 501
Exponential map 262
Exterior derivative 209
Exterior product 204 580
Fiber bundle 461
Fiber derivative 517
Field tensor, electromagnetic 689
Flow 166
Flow box 167
Flow, local 166
Flow, maximal 169
Foliation 330
Foliation chart 330
Form, 1-form 121
Frame field 124
Free module 148 564 570
functor 554
Fundamental vector field 311 497
G bundle 250 462
General linear group 14 618
Geodesic 391
Geodesically complete 393
Germ of function 108
Germ of section 160
Germs 161
Graded algebra 580
Grassmannian 285
Half rank 504
Hamiltonian 500
Hamiltonian action 522
Hamiltonian vector field 514
hessian 246
Hilbert space 15 618
Holomorphic 283
Holonomic 123
Homomorphism presheaf 158
Homotopy equivalence 453
Hopf bundle 251 463
Identification 119
Immersion 238
Implicit function theorem 23 632
Index of a critical point 246
Infinitesimal automorphism 335
Infinitesimal symmetry 498
Initial submanifold 242
Integrable distribution 325
Interior product 216
Involutive distribution 324
Isometry 378
Isotropy subgroup 316 678
jet 157
Jet extension 157
k-algebra 148 564
Kahler manifold 509
Killing field 498
Lagrangian subspace 504
Leaf 334
Levi Civita derivative 389
Levi decomposition 303
Lie algebra action 521
| Lie derivative 195
Lie derivative, of vector field 170
Lie derivative, on functions 124
Liebniz rule 125
Light cone 381
Lightlike curve 381
Lightlike subspace 381
Line bundle 144
Local diffeomorphism 22 627
Local immersion 25 26 634 635
Locally convex 129
Locally Hamiltonian 514
Locally splitting surjection 25 635
Lorentz manifold 375
Maxwell's equations 689
Model space 81
Module 145 561
Momentum map 522
Morphism 552
Moving frame 124
Multilinear 15 149 565 619
Musical isomophisms, symplectic 510
Musical isomorphism 371
Nice chart 220
Nonnull curve 381
Nonnull vector 381
Null-homotopic 454
One parameter subgroup 136
Orbit 315
Orient able 219 220
Orientation 219 373 583
Orthogonal group 133
Paracompact 74
Partial tangent map 119
Partition of unity 129
Pauli matrices 136
Phase space 515
Piecewise smooth 381
Plaque 330
Poisson bracket 525
Poisson structure 526
Presheaf 158
Principal bundle 465
Product group 131
Product manifold 79
Product rule 19 625
Projective plane 78
Projective space 79
Proper time 382
Pull back 165
Pull-back 165
Push-forward 165
R-algebra 578
Reflexive 150 566
Regular distribution, tangent subbundle 323
Regular Lagrangian 517
Regular point 235
Regular submanifold 98
Regular value 235
Riemann sphere 285
Riemann surface 286
Riemannian manifold 375
Rigid body 519
Sard's theorem 236
Scalar product 367 368
Scalar product bundle 474
semi-Riemannian 375
Sheaf 158
Skeleton 454
Smoothly universal 241
Spacelike 380
Spacelike curve 381
Spacelike subspace 381
Spin group 485
Splitting injection 23 633
Splitting surjection 24 634
Spray 362
Stable family 334
Star operator 473
Stationary 496
Stereographic projection 78
Stokes' theorem 228 229
Strong topology 529
Strongly Hamiltonian 522
Structure group 251 462 646
Subgroup, immersed 135
Subgroup, Lie 132
Submanifold 98
Submanifold atlas 98
Submanifold property 98
Submersion 243
Submodule 147 562
Support 227
Symplectic action 521
Symplectic form 505
Symplectic group 504
Symplectic manifold 505
Symplectic subspace 504
Symplectic vector field 514
Symplectic vector space 503
Tangent bundle 107 114
Tangent bundle of a foliation 331
Tangent map 27 113 637
Tangent space 27 108 637
Tangent vector 27 106 637
Tensor 181 574
Tensor derivation 192
Tensor product 151 567 569 572
Tensor product of tensor maps 184
Tensor, tensor map 182 566
Timelike curve 381
Timelike subspace 381
Timelike vector 381
Transitive 316
Transitive action 316
Transversality 247
Universal 151 569
Universal mapping 183 575
Vector bundle 594 644 645
Vector bundle chart 589
Vector bundle chart, type II 595 647
Vector bundle morphism 647
Weak topology 529
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