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| Goldberg S.I. — Curvature and homology |
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| Предметный указатель |
(Ricci) scalar curvature see Curvature
Abelian differential of the first kind 178
Abstract complex 56—57
Abstract complex, dimension of 57
Abstract complex, geometric realization of 60
Affine collineation (of a Riemannian manifold) 119
Affine connections 23—24
Affine connections on a Lie group 137
Affine connections, projectively related 121—122
Affine transformation (of a Riemannian manifold) 121
Algebra of Cayley numbers 191
Almost complex manifold 156
Almost hermitian manifold 259
Almost hermitian metric 267—268
Almost Kaehler manifold 259
Anti-derivation 96
Arithmetic genus 178
Associated fibre bundle 55
Atlas 149
Automorphism 261
Automorphism, infinitesimal 261
Bar operation 96
Barycentric coordinates 61
Base space 11 see
Beltrami's differential operator of the second kind 78
Betti numbers 60
Betti numbers of a compact semi-simple Lie group 144
Betti numbers of a Kaehler manifold 177
Betti numbers of simple Lie groups 143—144
Bianchi identities 29 162
Bidegree of a homogeneous exterior differential form 148
Bilinear symmetric form, fundamental 136
Bilinear symmetric form, invariant 145
Bochner's lemma 242
Boundaries 58 61
Boundary operation see Boundary operator
Boundary operato 58 61 276
Bounding p-cycles 58
Bundle of bases see Bundle of frames
Bundle of exterior differential polynomials 168
Bundle of frames 27—28
Bundle of unitary frames 158
Canonical bundle 235
Carrier (of a differential form) 19—20
Cauchy — Riemann equations 67 189
Cauchy's theorem for multiply connected regions 65
Cauchy's theorem for simply connected regions 64
Chain of type (p, g) 283
Chain, locally finite 61
Chain, p-chain 57 276
Chain, p-dimensional 57
Chain, support of (a p-chain) 61
Characteristic class 233
Characteristic class, positive definite 235
Chern class,  166 235
Chern class, representative of 228
Circle bundle 185
Class of a differential form 14
Class of a differentiate structure on a manifold 2—3
Class of a map 3
Class of a pfaffian system 47
Class of an induced structure 4
Co-closed p-form 72
Co-exact p-form 72
Cobound 72
Coboundaries 59
Coboundary operator 59 273
Cobounding p-cycles 59
Cochain 59
Cochain of type (p, q) 283
Cochain, finite 275
Cochain, p-cochain 59
Cochain, p-cochain of the nerve 272
Cochain, p-dimensional 59
Coframe 46 159
Cohomologous cocycles 60
Cohomology class 60
Cohomology group 15
Cohomology group, p-th cohomology group (of a complex) 59—60
Cohomology ring 15
Cohomology theory 80
Compact semi-simple Lie group geometry of 136—138
Complete parallelisability 29
Complete system 74
Complete vector field 99
Completely continuous (linear) operator 299
Complex analytic manifold 147
Complex dimension 147
complex extension of 179
Complex line bundle over a Kaehler manifold 232
Complex manifold see Complex analytic manifold
Complex multi-torus 149 186
Complex operators 169
Complex parallelisable manifold 167
Complex projective space 149
Conformal geometry 84
Conformal metric 149
Conformal transformations 106
Conformal transformations, group of 106
Conformal transformations, homothetic 107
Conformal transformations, infinitesimal 106
Conformally homeomorphic Riemarinian manifolds 85
Conformally related metric 106
Connected topological space 2
Contracted tensor see Tensors
Contravariant vector (at a point) 5 see
Coordinate slices 48
Covariant differentiation 192
Covariant vector 6
Covector see Covariant vector
Covering (of a complex) 60
Covering complex 60
Covering complex of a differentiable manifold 61—62
Cross-section 11 14
Cup product 293
Curvature forms 29 161
Curvature tensor 26—27 52—53
Curvature tensor, conformal 116 131
Curvature tensor, projective 206
Curvature, constant 37
Curvature, constant holomorphic 201
Curvature, Gaussian 35 185
Curvature, mean 125
Curvature, Ricci 38
Curvature, Ricci scalar (scalar curvature) 38
Curvature, sectional 35—39
Curve, auto-parallel 25
Curve, geodesic 40
Curve, parametrized 18
d-cohomology see Cohomology theory
De Rham cohomology 63—64
De Rham cohomology group 63—64 66
de Rham's existence theorems 76 270—292
de Rham's isomorphism theorem 289—291 292
de Rham's isomorphism theorem for compact Riemann surfaces 65—66
de Rham's isomorphism theorem for simple coverings 284—289
Decomposition theorem for compact Riemann surfaces 65—68
Decomposition theorem, Hodge-de Rham 76 296
Degree of a differential form 14
Degree of a p-vector 13
Degree of an endomorphism 96
Derivation 96
Derived algebra 145
Differentiable structure 2
Differentiable structure of class k 2—3
Differentiable structure, equivalence of 4
Differentiable structure, oriented 4
Differentiable transformations see Map.
Differentiable transformations, (global) l-parameter group of 99
Differentiable transformations, generated by a vector field 99
| Differential form of bidegree (q, r) 148
Differential form of degree p and class l 14
Differential form of degree p and class l, integral of 20
Differentia] system of dimension r see r-distribution
Direct limit of groups 275
Directional derivative 6
Displacement vector 34
Domain (with regular boundary) 21
Domain (with regular boundary), boundary point of 21
Domain (with regular boundary), interior point of 21
Domain (with regular boundary), regular boundary of 21
Dual (linear) operators 74
Effective form 179—182
Einstein manifold 38
Einstein summation convention 3
Endomorphism of degree r 96
Euclidean p-simplex 61
Euler — Poincare characteristic (of an abstract complex) 60
Exterior algebra 12—14 44—45
Exterior differential polynomial 14
Fibre 11 54
Fibre, standard 55
Field of frames 215
Field of frames, parallel 215
Field of r-planes 47
Field of subspaces 156
Frame (at a point) 27
Frobenius' Theorem 45—49 127
Fubini merric 188
Function see also Mapping
Function, differentiable 4
Function, differential of 14
Function, directional derivative of 98
Function, directional derivative of olomorphic, 67, 147, 189—190
Function, partial derivative of 4
Fundamental form 165
Fundamental tensor field 30
Gauss — Bonnet formula 228
General linear group 3
Genus of a compact Riemann surface 178
Genus, arithmetic see Arithmetic genus
Geodesic see Curve
Geodesic coordinate system see Local coordinates
Global scalar product 70—71
Graded algebra (over a field) 95—96
Grassman algebra see Exterior algebra
Grassman manifold 48
Green's operator 80 296
Harmonic field see Harmonic form
Harmonic form 67 73
Harmonic form on a compact semi-simple Lie group 139—141
Harmonic function 67
Hausdorff topological space 2
Hermitian (linear) operator 81
Hermitian geometry 158
Hermitian manifold 158
Hermitian manifold, hermitian structure on 158
Hermitian metric 154 158
Hermitian vector space 154
Hodge (star) operator see Star operator
Hodge existence theorem 76—77
Hodge's theorem 76
Hodge's theorem for a compact Riemann surface 68
Holomorphic connection 167
Holomorphic differential 68
Holomorphic form (of degree p) 169
Holomorphic function see Function holomorphic
Holomorphic isomorphism 183
Holomorphic section 200
Holomorphic sectional curvature 200
Holomorphic vector field see Infinitesimal holomorphic transformation
Holonomy see Holonomy group
Holonomy group 50—51
Holonomy group, restricted 51
Homogeneous space 241
Homogeneous space, symmetric 241
Homologous cycles 58
Homology class 58
Homology sphere 83
Hyperbolic space 84
Incidence number 56
Induced map see also Map
Induced map, dual of 18
Infinitesimal holomorphic transformation 247
Infinitesimal isometry or motion 106
Infinitesimal transformation 11—101
Infinitesimal transformation, differentiable of class k-1 99
Infinitesimal transformation, left invariant 103
Infinitesimal transformation, right invariant 104
Inner automorphism 104
Inner product 6
Inner product of a cochain and a chain 283
Integral p-chains 58—59
Interior product 97
Interior product operator 97 171
Invariant element see L-invariant element
Invariant vector field 100
Inverse limit of groups 277
Isometric n-dimensional Riemannian manifolds 84
Isometry 84 244
Isomorphic complexes 60
Isothermal parameters 149
J-basis 152—153
Jacobi identities 163 218
Jacobi identity 100 133
Kaehler manifold 163
Kaehler manifold,  -holomorphically pinched 238
Kaehler manifold, elliptic 197
Kaehler manifold, hyperbolic 197
Kaehler manifold, parabolic 197
Kaehler metric 163
Kaehler — Einstein manifold 210
Kaehler — Einstein metric 255
Killing vector field 108
Kodaira, vanishing theorems of 232—237
Kronecker symbol 16
L-invariant element 136
Laplace — Beltrami operator 78 176
Laplace's equation 69
Left invariant differential form 133
Left translation 103
Levi-Civita connection 32
Lie algebra (of a Lie group) 104
Lie algebra (of a Lie group), adjoint representation of 134
Lie algebra (of a Lie group), reductive 252
Lie algebra (of a Lie group), structure constants of 133 136
Lie bracket 96 106
Lie derivative 101—103 108 261
Lie group 103
Lie group, adjoint representation of 104 134
Lie group, Grassman algebra of 132—134
Lie group, l-parameter subgroup of 104
Lie group, semi-simple 136
Lie group, simple 143
Lie subalgebra 104
Lie subalgebra, reductive 252
Lie subgroup 104
Lie transformation groups 103—105
Lie transformation groups, effective 105
Lie transformation groups, transitive 241
Lie, third fundamental theorem of 128—130
Linear (differential) form 7
Local complex coordinates 147
Local coordinate system, conformally related 66
Local coordinates 3
Local coordinates, complex geodesic 173—174
Local coordinates, geodesic 40—41
Local hypersurface (of  ) 124
Locally flat (affinely connected) manifold see Manifold
Manifold, affinely connected 24
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