Adjoint representation 227
Adjoint system 28
Algebra of contravariant tensors 10
Algebraic preliminaries 1
Almost continuous 113
Ambrose — Singer Theorem 305
Antiderivation 21 102
Antisymmetric tensors 14
Approximation theorems 55
ARC 207
Associated bundle 307
Associated system 24
Barycentric coordinates 104 105
Bases 8
Betti numbers 289
Bi-invariant Riemann metric 232
Bianchi — Cartan identities 330
Bianchj's identities 360
Bilinear map 2
binomial 253
Bonnet's theorem 292
Buler vector field 159
Bundle of frames 80
Bundle of orthonormai frames 247
Calculus of variations 148
Cartan's Lemma 18
Cellulated surface 288
Characteristic system 135
Classical mechanics 141
Closed differential form 108
Closed subgroups 228
Closed submanifold 43
Cohomology group 108
Complete Riemannian manifold 173
Complete spray 199
Completely integrable 130
Completeness 207
Conjugate points 182
Connection form 298
Connection on a principal bundle 298
Connection on an associated bundle 308
Connections 294
Conservation laws 170
Conservation of energy 146
Content 113
Content zero 113
Contraction of tensors 6
Contravariant and symmetric algebras 10
Contravariant degree 6
Contravariant vector field 79
Coordinate chart 35
Coordinate map 34
Coordinate neighborhood 34
Coordinates of die chart 36
Cotangent bundle 79
Covariant degree 6
Covariant derivative, with respect to a linear connection 352
Covariant derivative, with respect to a parallelism 341
Covariant differential 355
Covariant vector field 79
Critical points 47
Critical value 47
Cross section 295
Curvature of a connection 300
Curvature of a ribbon 253
Curves in Euclidean space 252
Darboux's theorem 137 141
Decomposable p-vectors 19
Degree of a map 120
densities 111 112
Derivation 102
Diffeomorphism 43
Differentiable function 45
Differentiable manifolds 31 35
Differentiable p-chains 106
Differential form of degree p 82
Differential system 130
Energy of a Lagrangian 152
Equations of structure of Euclidean space 237 238
Equations of structure, of a Riemann manifold 246 249
Equations of structure, of a submanifold 245
Equations of structure, of a submanifold of Euclidean space 251
Euler — PoincarЈ characteristic 290
Euler's differential equation 159 166
Exact differential form 108
Exponential map 222
Exterior 2-fornis 23
Exterior algebra 14. 15
Exterior differential form of degree p 82
Exterior differentiation 100
Exterior equations 23
Exterior forms 20
Exterior multiplication 20
Extremal 167
F-related 95
Field of quantities 78
Flat G-structure 315
Focal point 18
frames 8
Frenet equations 253
Frenet ribbon 253
Frenet trihedral 253
Frobenius' Theorem 130
Fundamental form 143
G-connection 352
G-structures 293 310
Gauss' lemma 201
Gaussian curvature 266
General equivalence problem 314
Geodesic flow on T(M) 199
Geodesic spray 200
Geodesies of a Riemann metric 167
Hamilton — Jacobi equation 179
Hamiltonian manifold 141
Hamiltonian structures 141
Hamiltonian vector field 143
Higher order structures 337
| Holonomy group 300 301
Homogeneous ideal 26
Homomorphism of principal bundle 294
Horizontal vector field 299
Imbedding 42
Immersed submanifold 43
Immersion 42
Indefinite metric 13
Inner product 22
Integral calculus on manifolds 97
Integral manifold 130
Interior multiplication 20
Invariant metrics 231
Isometries 212
Isomorphism of principal bundles 294
Jacobi equations 186
Jacobi vector field 185
Jacobi's condition 182
Jacobi's identity 94 222
kinetic energy 149
Knbayashi's theorem 347
Left invariant forms 217
Left invariant spray 222
Left invariant vector fields 218
Legendre transformations 150
Lensor algebra of a vector space 5
Levi — Civita's theorem 334
Lie algebra 94 217 218
Lie bracket 94
Lie derivatives 89 91
Lie groups 214
Lie homomorphism 216
Lie subgroups 216
Linear connection 352
Linear differential form 79
Linear map 2
Linear quantity 10
Liouville form 171
Locally bounded 112
Lorentzian metric 14
Maurer — Cartan equation 218
Maximal integral manifold 136
Moving frames 244
Multilinear map 3
Necessary conditions for a minimum 153
Non-singular metric 13
Normal coordinates 222
One parameter group 89
Operator D 99
Orientable 115
Orientation on a manifold 116
Oriented manifold 116
Orthogonal frames 240
Partitions of unity 55 57
Path 207
Poincare's lemma 121
Poisson bracket 142
Positive definite Lagrangian 176
Principal bundles 73 294
Principal normal 253
Prolongations of  331
Prolongations of a G-structure 331
Proper imbedding 43
Proper map 43
quantities 73 76
Rank of a collection of functions 341
Ray 208
Regular family of functions 342
Regular Lagrangian 152
Restricted equivalence problem 339
ribbon 252
Right interior multiplication 22
Sard's theorem 45
Scalar densities 10
Scalar product 22
Second fundamental form 264
Second order differential equation 166
Second order structures 337
Segment 207
Singulars-simplex 106
Space of exterior p-forms over V 20
Space of m-frames of V 36
Space of symmetric tensors 12
Spherical map 264
Spray 199
Spray of a linear connection 361
Stokes' theorem 109 119
Structure function of a G-structure 318
Structure group of a bundle 294
Sturm's theorem 291
Submanifold 43
Sufficient conditions in the calculus of variations 174
Support of a density 112
Support of a form 109
Surfaces 279
Symmetric bilinear forms 13
Symmetric tensors 14
Tangent space 69 70
Tangent vector 69
Tensor bundles 84
Tensor field 80
Tensor products of vector spaces 2 4 5
Theorem Egregium 267
Thorn's transversality theorem 65
Topological group 231
Torsion of a curve 253
Torsion of a linear connection 356
Torsion of a ribbon 253
Total angular momenta 147
Total linear momenta 147
Transitive G-structures 321
Trivial bundle 295
Twist of a ribbon 253
Universal property 1C
Vector fields 89
Vector product 23
Vector space 2
Vector valued forms 234
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