Lee J.M. — Introduction to Smooth Manifolds :: Электронная библиотека попечительского совета мехмата МГУ

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Lee J.M. — Introduction to Smooth Manifolds

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Название: Introduction to Smooth Manifolds

Автор: Lee J.M.

Язык:

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 474

Добавлена в каталог: 19.04.2005

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Предметный указатель

(continuously differentiable)      425
(k times continuously differentiable)      425
(k times continuously differentiable), manifold      9
(k times continuously differentiable), structure      9
      425
$C^{\infty}$ (infinitely differentiable)      7 425
$C^{\infty}$ , structure      9
$C^{\infty}(M)$       24
$C^{\infty}(U)$       425
$C^{\omega}$ structure      9
(Riemannian volume form)      258
$F^{\ast}$ (pullback) of covectors      75
$F^{\ast}$ (pullback) of forms      213
$F^{\ast}$ (pullback) of functions      40
$F^{\ast}$ (pullback) of tensors      181
$F_{\ast}$ (push-forward) in coordinates      50
$F_{\ast}$ (push-forward) of vector fields      63
$F_{\ast}$ (push-forward) of vectors      46
$F_{\ast}$ (push-forward), induced Lie algebra homomorphism      378
$GL(n, \mathbb{C})$       see “Complex general linear group”
$GL(n, \mathbb{R})$       see “General linear group”
$GL^+(n, \mathbb{R})$       124
(identity matrix)      409
(left translation)      32
$M(m\times n,\mathbb{C})$ (space of complex matrices)      12
$M(m\times n,\mathbb{R})$ (space of matrices)      12
$M(n,\mathbb{C})$ (space of square complex matrices)      12
$M(n,\mathbb{R})$ (space of square matrices)      12
      389
(right translation)      32
$S(n, \mathbb{R})$ (symmetric matrices)      115
$SL(n, \mathbb{C})$ (complex special linear group)      150
$SL(n, \mathbb{R})$       see “Special linear group”
$Sp(n, \mathbb{R})$ (symplectic group)      227
(symmetric group)      204 206
$V^{\ast\ast}$ (second dual space)      67
$V^{\ast\ast}$ (second dual space), isomorphic to V      67
$V^{\ast}$ (dual space)      66
$X^{\flat}$ (X flat)      193
$\delta$ -close      138
$\delta_I^J$ (Kronecker delta for multi-indices)      206
$\delta_i^j$ (Kronecker delta)      66
$\mathbb{C}\mathbb{P}^n$ (complex projective space)      169
$\mathbb{C}^{\ast}$ (nonzero complex numbers)      31
$\mathbb{H}^n$ (upper half space)      19
$\mathbb{P}^n$ (real projective space)      6
$\mathbb{R}\mathbb{P}^n$ (real projective space)      6
$\mathbb{R}^4$ , fake      27
$\mathbb{R}^4$ , nonuniqueness of smooth structure      27
$\mathbb{R}^n$       see “Euclidean space”
$\mathbb{R}^{\ast}$ (nonzero real numbers)      31
$\mathbb{S}^1$       see “Circle”
$\mathbb{S}^n$       see “Sphere”
$\mathbb{T}^2$       see “Torus”
$\mathbb{T}^2$ (n-torus)      32
$\mathbb{T}^n$       see “Torus”
$\mathfrak{T}M$ (space of vector fields)      62
$\mathfrak{T}^{\ast}M$ (space of covector fields)      71
$\xi^\sharp$ ( $\xi$ sharp)      193
1-form      70
Abelian Lie algebra      372
Accessible slice, from a point      367
Action by discrete group      149
Action by left translation      148
Action by right translation      148
Action by right translation is free      160
Action by right translation is proper      160
Action by right translation is smooth      160
Action of a group      145
Action of O(n) on $\mathbb{S}^{n-1}$       148
Action, free      147
Action, group, local one-parameter      313
Action, group, one-parameter      310
Action, left      145
Action, natural, of $GL(n, \mathbb{R})$       148
Action, natural, of O(n)      148
Action, proper      147
Action, proper of a discrete group      157
Action, properly discontinuous      158
Action, right      146
Action, smooth      146
Action, transitive      147
Adapted chart      154
Adapted orthonormal frame      192 262
Addition, vector      403
Adjoint matrix      150
Ado’s theorem      398
Affine, singular simplex      292
Affine, singular subspace      407
Algebra, Lie      371
Algebra, Lie of a Lie group      373
Algebra, Lie, induced homomorphism      379
Algebra, Lie, product      372
Along a submanifold      123
Alt (alternating projection)      205
Alt convention      212
Alternating projection      205
Alternating tensor      202
Alternating tensor and orientation      231
Alternating tensor, basis for space of      207
Alternating tensor, elementary      206
Alternative definitions of the tangent space      55
angle      186 421
Angle, function      14
Antisymmetry of bracket      371
Approximation linear      41
Approximation theorem on manifolds      142
Approximation theorem, Whitney      138
Approximation, linear and the differential      74
Atlas      8
Atlas for a manifold with boundary      20
Atlas, smooth      8
Atlas, smooth, complete      9
Atlas, smooth, maximal      9
Autonomous system of ODEs      326
Backward reparametrization      81
Baker — Campbell — Hausdorff formula      392
Ball, closed      423
Ball, closed is a manifold with boundary      21 40
Ball, coordinate      4
Ball, open      423
Band, Mobius      265
Base of a covering      28
Base of a vector bundle      59
Basis, dual      66
Basis, dual for a vector space      405
Basis, dual, standard, for $\mathbb{R}^n$       66
Basis, isomorphism      11 408
Basis, ordered      406
Basis, representation of a vector      406
Basis, standard      406
Bilinear      172
Bilinear form      173
Block upper triangular      420
Boundary of a manifold with boundary      19 128
Boundary of a manifold with boundary is a manifold      21
Boundary of a manifold with boundary is a submanifold      120
Boundary of a manifold with boundary, disjoint from interior      20
Boundary of a singular simplex      292
Boundary, induced volume form      262
Boundary, manifold      20
Boundary, manifold with      19
Boundary, manifold with push-forward      53
Boundary, manifold with tangent space      53
Boundary, manifold with topological      19
Boundary, manifold with vector field on      62
Boundary, operator, singular      292
Boundary, singular      292
Boundary, topological      20
bounded linear map      424

Bracket in a Lie algebra      371
Bracket, commutator      372
Bracket, Lie      329
Bracket, Lie, antisymmetry      330
Bracket, Lie, bilinearity      330
Bracket, Lie, coordinate expression      329
Bracket, Lie, equals Lie derivative      333
Bracket, Lie, is smooth      329
Bracket, Lie, Jacobi identity      330
Bracket, Lie, naturality      331
Bracket, Lie, tangent to submanifold      331
Bracket, Poisson      348
Bracket, Poisson on $\mathbb{R}^n$       348
Bredon, Glen E.      vi
Bump function      35 39
Bump function, existence      38
Bundle of tensors      179
Bundle, cotangent      69
Bundle, cotangent is a vector bundle      69
Bundle, isomorphic      60
Bundle, isomorphism      60 127
Bundle, map      60 192
Bundle, map, bijective      127
Bundle, Mobius      265
Bundle, normal      139
Bundle, normal, is a submanifold      139
Bundle, tangent      57
Bundle, tangent is a vector bundle      59
Bundle, tangent, projection      57
Bundle, tangent, smooth structure      57
Bundle, tangent, standard coordinates      58
Bundle, tangent, trivial      62
Bundle, tensor      180
Bundle, trivial      59
Bundle, trivial, tangent      62
Bundle, vector      58
Bundle, vector, projection is a submersion      95
Bundle, vector, restriction of      59
Bundle, vector, section of      59
C(M) (space of continuous functions)      40
Calibrated submanifold      304
Calibration      304
Canonical coordinates      349
Canonical form for commuting vector fields      337
Canonical form for nonvanishing vector field      331
Canonical form for symplectic tensor      221
Canonical symplectic structure on $T^{\ast}M$       223
Cayley numbers      401
Centered at a point      4
Chain rule for partial derivatives      431
Chain rule for total derivatives      429
Chain, complex      286
Chain, complex, singular      293
Chain, group, singular      292
Chain, group, smooth      296
Chain, singular      292
Chain, smooth      296
Change of variables      438
Characterization of homogeneous spaces      162
Chart      4
Chart with corners      252
Chart with corners, smooth      252
Chart with corners, smoothly compatible      252
Chart, centered at a point      4
Chart, generalized      19
Chart, negatively oriented      232
Chart, oriented      232
Chart, positively oriented      232
Chart, slice      97
Chart, smooth      7 10
Chart, smooth, on a manifold with corners      252
Chart, smooth, with corners      252
circle      6 14
Circle as a Lie group      31
Circle, de Rham cohomology      281
Circle, group      31
Circle, group, Lie algebra of      374
Circle, group, subgroup of $C^{\ast}$       124
Circle, homeomorphic to square      7
Circle, not diffeomorphic to square      127
Class       425
Class       425
Class $C^{\infty}$       425
Closed 1-form      85
Closed 1-form, coordinate independence      85
Closed ball      423
Closed ball is a manifold with boundary      21 40
Closed covector field      85 214
Closed covector field, coordinate independence      85
Closed covector field, vs. exact      85–87
Closed curve segment      82
Closed form      219
Closed form, local exactness      279
Closed rectangle      433
Closed subgroup      124
Closed subgroup is embedded      394
Closed subgroup theorem      392
Closed submanifold      126
Closest point to a submanifold      144
Cochain, complex      286
Cochain, homotopy      274
Cochain, map      286
Codimension      97
Cofactor      419
Coframe, coordinate      71
Coframe, dual      71
Coframe, global      71
Coframe, local      71
Cohomologous      273
Cohomology de Rham      272
Cohomology de Rham of nonorientable manifold      281
Cohomology de Rham of orientable manifold      279
Cohomology de Rham with compact support      284
Cohomology of a complex      285
Cohomology, class      273
Cohomology, group, de Rham      272
Cohomology, map, induced      273
Cohomology, singular      295
Coisotropic, immersion      222
Coisotropic, submanifold      222
Coisotropic, subspace      220
Column, operations, elementary      417
Column, rank      413
Combination, linear      404
Commutator bracket      372
Commute      335
Commuting vector fields      335 336
Commuting vector fields, canonical form      337
Compact support      34
Compact support, de Rham cohomology with      284
Compactly supported forms      282
Compactly supported function      34
Compactly supported section      59
Compatible, smoothly      7
Compatible, smoothly, charts with corners      252
Complement, orthogonal      423
Complement, symplectic      220
Complementary subspace      407
Complete smooth atlas      9
Complete vector field      316
Complete vector field on a compact manifold      317
Completely integrable      359
Completely integrable vs. involutive      359
COMPLEX      285
Complex, analytic structure      9
Complex, chain      286
Complex, cochain      286
Complex, general linear group      31
Complex, general linear group, Lie algebra of      402
Complex, manifold      9

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