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| Kobayashi S., Nomizu K. — Foundations of Differential Geometry, Volume 1 |
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| Предметный указатель |
1-parameter, group of transformations 12
1-parameter, subgroup 39
Absolute parallclism 122
Adjoint representation 40
Affine connection 129
Affine connection, generalized 127
Affine frame 126
Affine holonomy group 130
Affine mapping 225
Affine parallel displacement 130
Affine parameter 138
Affine space 125
Affine space, tangent 125
Affine transformation 125 226
Affine transformation, infinitesimal 230
Alternation 28
Analytic continuation 254
Arc-length 157
Atlas 2
Atlas, complete 2
Automorphism of a connection 81
Automorphism of a Lie algebra 40
Automorphism of a Lie group 40
Bianchi's identities 78 121 135
Bundle of affine frames 126
Bundle of linear frames 56
Bundle of orthonormal frames 60
Bundle, associated 55
Bundle, holonomy 85
Bundle, homomorphism of 53
Bundle, induced 60
Bundle, principal fibre 50
Bundle, reduced 53
Bundle, sub- 53
Bundle, tangent 56
Bundle, tensor 56
Bundle, trivial 50
Bundle, vector 113
Canonical 1-form on a group 41
Canonical decomposition (= de Rham decomposition) 185 192
Canonical flat connection 92
Canonical form on L(M) 118
Canonical invariant connection 110 301
Canonical invariant Riemannian metric 155
Canonical linear connection 302
Canonical metric 155
Canonical parameter of a geodesic 162
Chart 2
Christoffel’s symbols ( ) 141
Compact-open topology 46
Complete linear connection 139
Complete Riemannian manifold 172
Complete Riemannian metric 172
Complete vector field 13
Components of a 1-form 6
Components of a linear connection 141
Components of a tensor (field) 21 26
Components of a vector (field) 5
Conformal transformation 309
Conformal transformation, infinitesimal 309
Connection 63
Connection, affine 129
Connection, canonical 110 301
Connection, canonical flat 92
Connection, canonical linear 302
Connection, flat 92
Connection, flat affine 209
Connection, form 64
Connection, generalized affine 127
Connection, induced 82
Connection, invariant 81 103
Connection, invariant, by parallelism 262
Connection, Levi-Civita 158
Connection, linear 119
Connection, metric 117 158
Connection, Riemannian 158
Connection, universal 290
Constant curvature 202
Constant curvature, space of 202 204
Contraction 22
Contravariant tensor (space) 20
Convex neighborhood 149 166
Coordinate neighborhood 3
Covariant derivative 114 115 122
Covariant differential 124
Covariant differentiation 115 116 123
Covariant tensor (space) 20
Covector 6
Covering space 61
Cross section 57
Cross section adapted to a normal coordinate system 257
Cubic neighborhood 3
Curvature 132
Curvature, constant 202
Curvature, form 77
Curvature, recurrent 305
Curvature, scalar 294
Curvature, sectional 202
Curvature, tensor (field) 132 145
Curvature, tensor (field), Riemannian 201
Curvature, transformation 133
Cylinder 223
Cylinder, Euclidean 210
Cylinder, twisted 223
de Rham decomposition 185 192
Derivation of  33
Derivation of  30
Derivation of the tensor algebra 25
Development 131
Diffeomorphism 9
Differential covariant 124
Differential of a function 8
Differential of a mapping 8
Discontinuous group 44
Discontinuous group, properly 43
Distance function 157
distribution 10
Distribution, involutive 10
Divergence 281
Effective action of a group 42
Einstein manifold 294
Elliptic 209
Euclidean cylinder 210
Euclidean metric 154
Euclidean motion 215
Euclidean subspace 218
Euclidean tangent space 193
Euclidean torus 210
Euclidean, locally 197 209 210
Exponential mapping 39 140 147
Exterior, covariant derivative 77
Exterior, covariant differentiation 77
Exterior, derivative 7 36
Exterior, differentiation 7 36
Fibre 55
Fibre, bundle, principal 50
Fibre, metric 116
Fibre, transitive 106
Flat, affine connection 209
Flat, connection 92
Flat, connection, canonical 92
Flat, linear connection 210
Flat, Riemannian manifold 209 210
Form, 1-form 6
Form, curvature 77
Form, r-form 7
Form, tensorial 75
Form, tensorial, pseudo- 75
Form, torsion 120
Frame, affine 126
| Frame, linear 55
Frame, orthonormal 60
Free action of a group 42
Frobenius, theorem of 10
Fundamental vector field 51
G-structure 288
Geodesic 138 146
Geodesic, minimizing 166
Geodesic, totally 180
Green’s Theorem 281
Holomorphic 2
Holonomy bundle 85
Holonomy group 71 72
Holonomy group, affine 130
Holonomy group, homogeneous 130
Holonomy group, infinitesimal 96 151
Holonomy group, linear 130
Holonomy group, local 94 151
Holonomy group, restricted 71 72
Holonomy theorem 89
Homogeneous Riemannian manifold 155 176
Homogeneous space (quotient space) 43
Homogeneous space (quotient space), symmetric 301
Homomorphism of fibre bundles 53
Homothetic transformation 242 309
Homothetic transformation, infinitesimal 309
Homotopic 284
Homotopic,  284
Horizontal component 63
Horizontal curve 68 88
Horizontal lift 64 68 88
Horizontal subspace 63 87
Horizontal vector 63
Hyperbolic 209
Hypersurface 9
Imbedding 9 53
Imbedding, isometric 161
Immersion 9
Immersion, isometric 161
Indefinite Riemannian metric 155
Induced bundle 60
Induced connection 82
Induced Riemannian metric 154
Inner product 24
Integral curve 12
Integral manifold 10
Interior product 35
Invariant, by parallelism 262
Invariant, connection 81 103
Invariant, Riemannian metric 154
Involutive distribution 10
Irreducible, group of Euclidean motions 218
Irreducible, Riemannian manifold 179
Isometric 161
Isometric imbedding 161
Isometric immersion 161
Isometry 46 161 236
Isometry, infinitesimal 237
Isotropy, group, linear 154
Isotropy, subgroup 49
Killing vector field 237
Killing — Cartan form 155
Klein bottle 223
Lasso 73 184 284
Leibniz’s formula 11
Levi-Civita connection 158
Lie algebra 38
Lie derivative 29
Lie differentiation 29
Lie group 38
Lie subgroup 39
Lie transformation group 41
Lift 64 68 88
Lift, horizontal 64 68 88
Lift, natural 230
Linear connection 119
Linear frame 55
Linear holonomy group 130
Linear isotropy group 154
Local basis of a distribution 10
Local coordinate system 3
Locally affine 210
Locally Euclidean 197 209 210
Locally symmetric 303
Lorentz manifold (metric) 292 297
Manifold 2 3
Manifold, complex analytic 3
Manifold, differentiable 2 3
Manifold, oriented, orientable 3
Manifold, real analytic 2
Manifold, sub- 9
Maurer — Cartan, equations of 41
Metric connection 117 158
Moebius band 223
Natural lift of a vector field 230
Non-prolongable 178
Normal coordinate system 148 162
Orbit 12
Orientation 3
Orthonormal frame 60
Paracompact 58
Parallel cross section 88
Parallel displacement 70 87 88
Parallel displacement, affine 130
Parallel tensor field 124
Partition of unity 272
Point field 131
Projection, covering 50
Properly discontinuous 43
Pseudogroup of transformations 1 2
Pseudotensorial form 75
Quotient space 43 44
Rank of a mapping 8
Real projective space 52
Recurrent curvature 305
Recurrent tensor 304
Reduced bundle 53
Reducible connection 81 83
Reducible Riemanman manifold 179
Reducible structure group 53
Reduction of connection 81 83
Reduction of structure group 53
Reduction Theorem 83
Ricci tensor (field) 248 292
Riemannian connection 158
Riemannian curvature tensor 201
Riemannian homogeneous space 155
Riemannian manifold 60 154
Riemannian metric 27 154 155
Riemannian metric, canonical invariant 155
Riemannian metric, indefinite 155
Riemannian metric, induced 154
Riemannian metric, invariant 154
Scalar curvature 294
Schur, theorem of 202
Sectional curvature 202
Segment 168
Simple covering 168
Skew-derivation 33
Space form 209
Standard horizontal vector field 119
Structure constants 41
Structure equations 77 78 118 120 129
Structure group 50
Subbundle 53
Submanifold 9
Symmetric homogeneous space 301
Symmetric locally 303
Symmetric Riemannian 302
Symmetrization 28
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