Greub W., Halperin S., Vanstone R. — Connections, curvature, and cohomology. Volume 1 :: Электронная библиотека попечительского совета мехмата МГУ

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Greub W., Halperin S., Vanstone R. — Connections, curvature, and cohomology. Volume 1

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Название: Connections, curvature, and cohomology. Volume 1

Авторы: Greub W., Halperin S., Vanstone R.

Аннотация:

This monograph developed out of the Abendseminar of 1958-1959 at the University of Zurich. It was originally a joint enterprise of the first author and H. H. Keller, who planned a brief treatise on connections in smooth fibre bundles. Then, in 1960, the first author took a position in the United States and geographic considerations forced the cancellation of this arrangement.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1972

Количество страниц: 467

Добавлена в каталог: 08.12.2013

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Предметный указатель

$(d\varphi)_{a}$       88
$(j^{*}\Omega)_{t}=j_{t}^{*}\Omega$ , $I_{a}^{b}$       156
$A(M) = \{\Phi, \Psi, ...\}$       120
$A^{p, q}(M \times N)$ , $\Phi \times \Psi$       122
$A_{+}(M)$ , $A_{-}(M)$       122
$A_{c}(M; E)$ , $\Phi \bullet \Psi$       152
$A_{F}(E)$ , $H_{F}(E)$       295
$A_{H}(E)$ , $A_{V}(E)$       283
$B_{P}$       177
$deg\ \varphi$       240
$deg_{a}\varphi$       260
$d\varphi$       95
$f_{M}(t)$ , $\chi_{M}$       178
$Hom(\xi;\eta)$       50
$Hom(\xi^{1}, ... , \xi^{p};\eta)$ , $A(\xi;\eta)$       51
$H_{+}(M)$ , $H_{-}(M)$       187
$H_{c}(M)$ , $\alpha \ast \beta$       189
$i_{M}(X)$       110
$j_{a}(\sigma)$       330
$j_{a}(\sigma)$ , $j(\sigma)$       369
$j_{b}:M \to M \times N$       97
$L(\xi; \eta)$ , $L_{\xi}$       56
$S(\xi; \eta)$ , $A(\xi)$       51
$Sk_{\xi}$       72
$S^{n}$       22
$T_{a}(M) = \{\xi, \eta, ...\}$       87
$T_{a}(M)^{*}$ , $\tau^{*}_{M}$       97
$V_{E}$       281
$V_{z}(E)$       280
$X\ \sim\atop\varphi\ Y$       109
$Z^{M}$ , $(Z_{1}, Z_{2})_{M}$       111
$\alpha: V_{E} \to E$ , $\beta_{\sigma}:E \to V_{E}$       291
$\alpha_{*}:A(M; E) \to A(M; F)$       151
$\alpha_{M}$ , $\dot{M} = M - \{a\}$       252
$\alpha_{\xi}$       360
$\chi_{\mathscr{B}}$ , D      320
$\Delta$       145
$\Delta:M \to M \times M$       29
$\delta_{m}$       148
$\dot{\sigma}$ , $\int_{a}^{b}\sigma$ , $\int_{a}\sigma_{t} dt$       153
$\dot{\xi} = (\dot{E}, \dot{\pi}, B, \dot{F})$       105
$\epsilon^{r}$ , $\xi \mid_{O}$ , $\xi^{1} \times \xi^{2}$       46
$\frac{d}{dt}$       116
$\int_{F}$       298 300
$\int_{F}^{\#}$       306
$\int_{M}$       160
$\int_{M}^{\#}$ , $\mathscr{P}_{M}$ , $D_{M}$       194
$\kappa$ , $\kappa_{\#}$       208
$\kappa_{c}$ , $(\kappa_{c})_{\#}$ , $H^{+}(M)$       210
$\mathbb{R}P^{n}$ , $T^{n}$       23
$\mathcal{M} \otimes_{M} \mathcal{N}$ , $Hom_{M}(\mathcal{M};\mathcal{N})$       30
$\mathcal{O}_{f}$ , $\mathcal{O}_{s}$       16
$\mathcal{X}(M) = \{X, Y, ...\}$       106
$\mathscr{I}(M)$ , carr f      30
$\mathscr{I}(M; N)$       24
$\mathscr{X}_{V}(E)$ , $H_{z}(E)$ , $H_{E}$       282
$\omega_{M}$       201
$\omega_{\tau}$ , $[\tau, \sigma]$       326
$\partial$       181
$\partial_{c}$       193
$\pi_{M}:M \times N \to M$       29
$\rho_{V}:A(E) \to Sec \wedge V_{E}^{*}$       284
$\sigma^{*}\xi$       48
$\tau_{M}$ , $T_{M}$       94
$\theta(X)$       142
$\tilde{\mathscr{B}}$ , $\tilde{B}$       70
$\varphi \sim \psi$       33
$\varphi^{*}$       121
$\varphi^{*}$ , $\varphi^{\#}$ , $\varphi_{*}$       62 63
$\varphi_{c}^{*}$ , $\varphi_{c}^{\#}$ , $(\varphi_{c})_{*}$ , $(\varphi_{c})_{\#}$       190
$\varphi_{F}^{*}$       296
$\varphi_{F}^{\#}$ , $(\varphi_{F})_{*}$ , $(\varphi_{F})_{\#}$       297
$\varphi_{x}$       45
$\wedge\xi$ , $\wedge\varphi$ , $\vee^{p}\xi$ , $\vee^{p}\varphi$       57 58
$\xi = (E, \pi, B, F)$ , $\psi_{\alpha, x}$       44
$\xi^{*}$ , $\varphi^{*}$       52
$\xi^{1} \oplus \cdots \oplus \xi^{p}$       54
$\xi^{1} \otimes \cdots \otimes \xi^{p}$       55
$\xi_{s} = (E_{s}, \pi_{s}, B, S)$       105
$\xi_{\epsilon} = (E_{\epsilon}, \pi_{\epsilon}, B, F_{\epsilon})$       359
$\xi_{\mathbb{R}} = (E, \pi, B, F_{\mathbb{R}})$       73
A(M; E)      150
Abstract simplicial complex      217
Adjoint, Poincare      400
Affine simplex      236 388
Affine spray      135 418
Algebra of smooth functions      30
Algebra, anticommutative      4
Algebra, cohomology      11 176
Algebra, connected      4 177
Algebra, exterior      4 57
Algebra, graded      3 4
Algebra, graded differential      11
Algebra, Lie      4 107 152 173
Algebra, symmetric      4 58
Anticommutative algebra      4
Anticommutative tensor product of graded algebras      4
Antiderivation      4 141
Associated sphere bundle      105 293
Atlas      15 414
Atlas, equivalent      22 24
Atlas, finite      20
Atlas, smooth      22
Barycentre      390
Base space      38
Basis for topology      14
Betti groups      232
Betti numbers      178 205 231
Bigraded module      8
Borsuk — Ulam theorem      275
Boundary      232
Boundary, manifold-with      139 231 350 415
Bundle of skew transformations      72
Bundle, along fibre      281
Bundle, composite      311
Bundle, cotangent      97 173
Bundle, deleted      105
Bundle, disc      381
Bundle, exterior algebra      57
Bundle, fibre      38
Bundle, isomorphism      45
Bundle, jet      132
Bundle, map      45 47 84 291
Bundle, normal      138 380
Bundle, pseudo-Riemannian      66 85
Bundle, quotient      84
Bundle, Riemannian      66
Bundle, space      38
Bundle, sphere      105
Bundle, subbundle of      44 68
Bundle, symmetric algebra      58
Bundle, tangent      94 385
Bundle, vector      44
Bundle, vertical subbundle of      281
Canonical tensor product of graded algebras      4
Canonical transformation      173
carr $\Phi$ , $A_{c}(M)$       147
carr $\sigma$       59
Carrier of cross-section      59
Carrier of differential form      147
Carrier of smooth function      30
Carrier, compact      147 189 295 380
Cauchy's integral theorem      235
Cayley map      25
Cayley numbers      132 175
Cech cohomology      238
Chain      232
Chain, degenerate      236
Chain, invisible      236
Charts      15

Charts, identification map of      22
Classifying map      86
Closed differential form      176
Coboundary      9 176
Cochain      218 348
Cocycle      9 176
Cohomology of $\mathbb{R}^{n}$ with compact supports      190
Cohomology of real projective spaces      187
Cohomology of sphere bundles      316ff. 344
Cohomology of spheres      185
Cohomology of vector bundles      352ff.
Cohomology with compact supports      189
Cohomology, algebra      11
Cohomology, algebra of compact manifolds      218
Cohomology, algebra of manifolds      176
Cohomology, algebra of nerve      218
Cohomology, axioms      178 190
Cohomology, space      9
Coincidence point      405 409 416
Coincidence, numbers      400ff.
Compact Kuenneth homomorphism      210
Compact manifolds      41 203 205 218 228
Compact supports      141 189 295 380
Complex functions      274
Complex projective space      42 415
Complex structure      73
Complex vector bundle      73 86
Complexification of vector space      2 27
Composite bundle      311
Composition map      57
Conjugate parallelism      175
Connected algebra      4 177
Connected manifold      177
Connected sum of manifolds      140
Connecting homomorphism      10 181 193
Constant map      24
Constant rank      84
Continuous homotopy      41
Continuous local degree      387
Continuous vector field      389
Contractible manifold      86 183
Contracting homotopy      183
Contraction      183
Contravariant tensor field      119
Coordinate functions      131
Coordinate representation for fibre bundles      38 40
Coordinate representation for vector bundles      44 45 70
Coordinate representation, Riemannian      68
Coordinate transformation      44
Cotangent bundle      97 173
Cotangent space      96
Cotangent vector      96
Covariant tensor field      118
Cover(ing), open      14
Covering transformation      71
Critical point      136 416
Critical point, nondegenerate      138
Critical value      136 245
Cross-section(s) of exterior power      81
Cross-section(s) of fibre bundle      38
Cross-section(s) of sphere bundle      337
Cross-section(s) of tangent bundle      106
Cross-section(s) of tensor product      80
Cross-section(s), index of      330 367
Cross-section(s), Lie algebra of      107
Cross-section(s), mappings of      62
Cross-section(s), module of      60 78
Cross-section(s), normed      66
Cross-section(s), pull-back of      325
Cross-section(s), smooth family of      153
CYCLE      232
Cycle, fundamental      237
Dashed degree      385 386
De Rham cohomology algebra      176
De Rham existence theorem      233
De Rham isomorphism      218 228
de Rham theorem      218ff.
Definite integral of smooth family      153
Deformation retract      184
Degenerate chain      236
Degree, (mod 2)      274
Degree, global      240 264 408 414
Degree, global dashed      386
Degree, local      259 260 264 382 383 387
Degree, local dashed      385
Degree, mapping      240ff.
Deleted bundle      105
density      171 233
Derivations as tangent vector fields      106
Derivations in algebra      2
Derivations, Lie product of      107
Derivative of map      12 88ff. 95
Derivative of smooth family      153
Derivative, exterior      145
Derivative, Lie      142
Determinant function in real vector space      1 124
Determinant function in vector bundle      64 70
Diagonal map      29
Diffeomorphism      12 24 35ff.
Diffeomorphism, local      99
Difference class      325
Differentiable map      12
Differential algebra      9
Differential equations      13 112ff.
Differential forms      115 119ff. 283ff.
Differential forms, closed      176
Differential forms, cochain of      218 348
Differential forms, components of      131
Differential forms, exact      176
Differential forms, exterior derivative of      145
Differential forms, harmonic      231
Differential forms, horizontal      283
Differential forms, invariant      144 158
Differential forms, invariant, with noncompact carrier      164
Differential forms, smooth family of      153
Differential forms, vector valued      149 163
Differential operator      9
Differential operator of order p      133 134
Differential space      9
dim M      15 17
Dimension theory      17ff. 239
Direct limit      238
Directed system of vector spaces      238
Disc bundles      381
Disjoint union axiom      179 232
distribution      134
Divergence      171 234
Double cover      71 123 399
Double of manifold      140
Dual of module      7
Dual of strong bundle map      52
Dual vector bundles      52 67 80
Ehresmann connection      314
Eigenvalue, eigenvector      85
Embedded manifold      102
Endomorphism of vector bundle      85
Euclidean half-space      139
Euclidean space      2
Euclidean space, maps between      260
Euclidean space, one-point compactification of      23 25
Euler class      320 328 334 391
Euler class of Whitney sum      345
Euler class, index sum and      372
Euler class, relative      349
Euler class, Thom class and      364
Euler — Poincare characteristic      178 186 205ff. 228 391 408 414 416
Euler — Poincare formula      11
Evaluation map      56
Exact differential form      176
Exact sequence      8 84

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