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| Bishop R.L., Crittenden R.J. — Geometry of manifolds |
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| Предметный указатель |
Homogeneous space, Riemannian 137
Homogeneous space, Riemannian symmetric 138
Homometry 135
Homomorphism of continuous 37
Homomorphism of Grassmann algebra 56
Homomorphism of Lie algebra 26 30
Homomorphism of Lie group 29
Hopf maps 52
Hopf — Rinow theorem 154
Horizontal lift of curve 77
Horizontal lift of vector field 75
Horizontal, differential form 76
Horizontal, vector field 75
Hyperbolic space 172
Hypersurface 207
Hypersurface of rotation 209
Hypersurface, products of spheres as 209
Hypersurface, rigidity 211
Hypersurface, type number of 211
Ideal of Lie algebra 26
Imbedding, isometric 132
Imbedding, normal bundle 48
Immersion 132 185
Immersion, curvature of 190 193
Immersion, first fundamental form 132
Immersion, Gauss map 195
Immersion, hypersurface 207
Immersion, Riemannian connexion 188
Immersion, second fundamental form 190
Immersion, totally geodesic 194
Indecomposable elements of Grassmann algebra 55
Index form 220
Index form, augmented index of 232
Index form, basic inequality 228 250
Index form, generalized 250
Index form, index of 223
Index form, null space 221
Index form, nullity of 225
Index of symmetric quadratic form 123
Index theorem 233
Induced bundle 45
Induced bundle, connexion on 80
Infinitesimal deformations 226
Integrable distribution 23
Integral, curve 14
Integral, manifold of codistribution 70
Integral, manifold of distribution 23
Interior product i{X) with vector field 69
Invariant metric on Lie group 136
Inverse function theorem 11
Involutive distribution 23
Isometry 122 160
Isometry, local 122
Isometry, theorem of Cartan 181
Isomorphism of Lie algebras 26
Isotropy group 39 169
Jacobi equation 174
Jacobi fields 173
Jacobi identity 14
Jacobi theorem (see Basic inequality for index form)
Jacobian matrix 10
Kahler manifold 142
Kahler, theorem of Cartan 205
Killing form 137
Kowalevsky, theorem of Cauchy 205
Lens space 140
Lie algebra 26
Lie algebra of 27
Lie algebra of Lie group 27
Lie algebra, abelian 26
Lie algebra-Lie group correspondence 28
Lie derivative 16
Lie derivative on differential form 69
Lie derivative on vector field 17
Lie group 25
Lie group, abelian 25 37
Lie group, complex 34
Lie subgroup 26
Lift of vector field 75
Lift, canonical, of rectangle 147
Lift, horizontal, of curve 77
Linear endomorphism, determinant of 57 61 62
Linear endomorphism, trace of 57
Linear homogeneous vector fields 176
Linear transformation fields 103
Local one parameter group of local associated vector field 15
Local one parameter group of local diffeomorphisms 15
Locally symmetric space 168
Longitudinal curves of rectangle 147
Lorentz manifold 123
Manifold 2
Manifold, almost complex 64
Manifold, complex 7 50 64
Manifold, open 4
Manifold, product 5
Manifold, submanifold 21
Manifold, topological 2
Matrix, exponential of 32
Matrix, Jabobian 10
Maurer-Cartan, equations and forms of 70 72 82 107
Mean curvature 253
Measure zero 12
Mechanical system 157
Metric, complete 154
Metric, existence of 126
Metric, Riemannian 122
Minimal, segment 237
Minimal, submanifold 248
Minimum, locus 237
Minimum, point 237
Morse index theorem 233
Myers’ theorem 256
Myers’ theorem on 256
Myers’ theorem on p-mean curvature 253
N — Jacobi fields 221
N — Jacobi fields on Riemannian sphere 180
N — Jacobi fields, focal points and 225
N — Jacobi fields, generalized 250
N — Jacobi fields, rectangles associated to 174
Negative curvature 179 184
Non-orientable 63
Normal bundle 48 151 196
Normal coordinates 110 152
Null space of index form 221
Nullity of index form 225
One-dimensional Riemannian manifold 132
Operations on vector valued forms 72
Opposite connexion 106
Opposite spaces to Grassmann manifolds 172
Opposite spaces to Grassmann manifolds, hyperbolic space 172
Orbit 39
Order of focal point 225
Orientability 63
Oriented Grassmann manifold 170
Orthogonal group 29 33
Parallel translation in associated bundle 84
Parallel translation in principal bundle 78
Parallel translation of tangents 89
Parallel vector field along curve 96
Parallelizable manifold 49 64
Parallelizable manifold, connexions (direct, opposite, torsion zero) 106
Parallelizable manifold, covariant derivative 114
Parallelizable manifold, Lie group 107
Parallelization of bundle of bases 92
Partial derivative 8
Partition of unity 4
Permutation groups 58
Permutation groups, cross-section 58
Permutation groups, shuffle permutations 58 61
Plane section 161
| Polar decomposition of quaternions 36
Principal bundle 41
Principal bundle as reduced bundle 129
Principal bundle of adapted bases of product 119
Principal bundle of adapted frames of immersion 186
Principal bundle of bases 43
Principal bundle of frames 127
Principal bundle, affine bundle 109
Principal bundle, associated to (see Associated bundle)
Principal bundle, bundle map 49
Principal bundle, bundle of bases 43
Principal bundle, bundle of frames 127
Principal bundle, bundle space 42
Principal bundle, connexion on (see Riemannian connexion)
Principal bundle, coordinate bundle 44
Principal bundle, induced 45
Principal bundle, parallelization of 92
Principal bundle, product (trivial) 42
Principal bundle, reduction of structure group 49 87
Principal bundle, reduction to O(d) 129
Product bundle 42
Product connexion 119
Product manifold 5
Product manifold, bundle of adapted bases 119
Product manifold, tangents to 10
Product of spheres as hypersurface 209
Projective space, complex 44 52
Projective space, conjugate points 226
Projective space, Kahler structure 142
Projective space, quaternion 52
Projective space, real 5 43
Prolongation of covering map 120
Prolongation of homometry 135
Quaternion Grassmann manifold 170
Quaternion projective space 52
Quaternion projective space, conjugate points 226
Quaternions 34 37
Quaternions, adjoint representation 36
Quaternions, polar decomposition 36
Quaternions, regular left representation 34
Quotient space bundle of imbedding 48
Quotient space bundle of imbedding, normal bundle 48 151 196
Rank of homogeneous element of order two 55
Rauch’s comparison theorem 250 252
Real projective space 5 43
Rectangle,  147
Rectangle, associated vector field 147
Rectangle, broken C 213
Rectangle, canonical lifting 147
Rectangle, longitudinal curve 147
Rectangle, transverse curve 147
Reduced arc length 223
Reduction of structural group 49 87
Reductive subgroup 83 187
Regular map 14
Representation 34
Representation of quaternions 34 36
Representation, adjoint 36
Representation, faithful 34
Restricted holonomy group 87
Restriction of function 22
Restriction of vector field 22
Ricci curvature 253
Ricci transformation 250 253
Riemannian 133
Riemannian connexion 134
Riemannian connexion of immersion 188
Riemannian connexion, existence of 131
Riemannian connexion, uniqueness 131
Riemannian covering 134
Riemannian curvature 161
Riemannian curvature and geodesics 179
Riemannian curvature of immersed manifold 190 193
Riemannian curvature of sphere 162 170 180
Riemannian curvature of symmetric space 170
Riemannian curvature, computation of 165
Riemannian curvature, constant 165 184
Riemannian curvature, continuity 166
Riemannian curvature, holomorphic 171
Riemannian curvature, negative curvature 184
Riemannian curvature, zero curvature 183
Riemannian homogeneous space 137
Riemannian homogeneous symmetric space 139
Riemannian manifold 122
Riemannian manifold, complete 154
Riemannian manifold, distance function 124
Riemannian manifold, examples 132—144
Riemannian manifold, imbedding of 132
Riemannian manifold, semi 123
Riemannian metric or structure 122
Riemannian metric or structure on vector bundle 127
Riemannian metric or structure, existence 126
Riemannian metric or structure, induced by action of compact group 136
Riemannian product 133
Riemannian structure as reduction of bundle of bases 129
Riemannian symmetric space 138 168
Right action of Lie group 38
Right connexion of Lie group 107
Rigidity theorem, classical 211
Rinow, theorem of Hopf 154
Rotation, hypersurface of 209
Rotation, manifold of partial 209
Ruled surface 208
Sard’s theorem 12
Schur’s Theorem 165
Second fundamental form 190
Second fundamental form and curvature 193
Second fundamental form, interpretations of 191 192
Second variation of arc length 213
Section, holomorphic 171
Section, plane 161
Sectional curvature (see Riemannian curvature)
Semi — Riemannian manifold 123
Semi — Riemannian manifold, index of 123
Semi — Riemannian manifold, Lorentz manifold 123
Semi-direct product 39
Shuffle permutations 58 61
Solder form 90 93
Space form 184
Space form, simple 257
Special linear group 33
Sphere 4
Sphere, conjugate points 225
Sphere, curvature 162 170 180
Sphere, geodesics 133
Sphere, Jacobi fields 180
Sphere, product of spheres as hypersurface 209
Sphere, Riemannian 133
Stereographic projection 4
Stiefel manifold 137
Striped pants 21
Structural constants of Lie algebra 69
Structural equations for covariant differentiation 116
Structural equations of affine connexion 100
Structural equations of connexion 81
Structural equations of horizontal form 86
Structural equations, dual formulation 80 101
Structure group 42
Structure group to complex general linear group 51
Structure group to orthogonal group 129
Structure group, reduction of 49
Subalgebra 26
Subgroup 26
Submanifold 21
Submanifold, minimal 248
Submanifold, open 4
Submanifold, totally geodesic 194
surface 5
Surface, doubly ruled 209
Surface, hyper 207
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