Goldberg S.I. — Curvature and homology :: Электронная библиотека попечительского совета мехмата МГУ

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Goldberg S.I. — Curvature and homology

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Название: Curvature and homology

Автор: Goldberg S.I.

Аннотация:

A systematic and self-contained treatment, this revised edition examines the topology of differentiable manifolds, curvature and homology of Riemannian manifolds, compact Lie groups, complex manifolds, and curvature and homology of Kaehler manifolds. Four appendixes deal with holomorphic bisectional curvature, Gauss-Bonnet theorem and its applications, and Bochner’s Lemma.

Язык:

Рубрика: Математика/Алгебра/Гомология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 395

Добавлена в каталог: 19.03.2005

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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, $1^{st}$       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, $\lambda$ -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 $E^{n+1}$ )      124
Locally flat (affinely connected) manifold      see Manifold
Manifold, affinely connected      24

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