Brickell F., Clark R.S. — Differentiable Manifolds :: Электронная библиотека попечительского совета мехмата МГУ

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Brickell F., Clark R.S. — Differentiable Manifolds

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Название: Differentiable Manifolds

Авторы: Brickell F., Clark R.S.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Количество страниц: 389

Добавлена в каталог: 05.07.2008

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Предметный указатель

$C^{\infty}$ structure      19
      13
$M(r \cross s, R)$       15
$M(\Omega)$       200
, standard $C^{\infty}$ structure      20
, standard topology      6 8 9 10
(unit sphere)      7
, $C^{\infty}$ structure      19 26
, as a coset space      251
, orientability      120
, standard topology      7 10 11 44
, vector fields on      116
$\mathfrak{R}(G)$       244
$\mathfrak{R}(G, M)$       245
$\mathscr{F}(M)$       146
$\mathscr{F}^1(M)$       111
$\mathscr{F}^1_U$       111 153
$\mathscr{F}_U$       110
$\mathscr{L}G$       219
$\partial f/\partial x^i$       53
Anchor ring      27 77
Atlas $C^{\infty}$       15
Atlas $C^{\infty}$ , complete      18
Atlas $C^{\infty}$ , stereographic      27
Automorphism      256 260
Basis      57 161 193
Bijection      2
Boundary of a subset      5
Bracket (Lie)      111
Chart, flat      194
Chart, n-dimensional      15
Chart, normal      180
Chart, of manifold      19
Christoffel symbols      168
Circular cylinder      79
Closed set      5
Closure of a subset      5
Compact space      9
Compact space, manifold      43
Component      10
Component, of a Lie group      215
Component, of a manifold      40
Connected space      9
Connected space, manifold      44
Coordinate, curve      134
Coordinate, domain      19
Coordinate, function      15
Coordinate, neighbourhood      19
Coordinate, representative      20
Countability axioms      10 11
Covering, open      8
Covering, subcovering      8
Curve      131
Curve, canonical lift      131
Curve, coordinate      134
Dense subset      5
Derivation, on $\mathfrak{F} (m)$       55
Derivation, on $\mathfrak{F} (U)$       112
Derived linear function      58
Diffeomorphic manifolds      21
Diffeomorphism      13 21
Diffeomorphism, locally a      13 64
Differentiable function      21
Differentiable function, differential of      62 114
Differentiable function, rank      60
Differentiable manifold      19
Differentiable manifold, dimension      19
Differentiable variety      26 90
Differentiable variety, orientability      124
Differentiable variety, submanifold of       72 75
Differentiable variety, topology      35 41 43
Differential equation (first order)      131
Differential equation (first order), solution of      131
Differential equation (second order)      169
Differential equation (second order), solution of      169
Differential of a function      62 114
Differential operator      110
Distance function      4
Distance function, euclidean      7
Distribution $\Omega$       193
Distribution $\Omega$ , basis      193
Distribution $\Omega$ , fiat chart      194
Distribution $\Omega$ , integrable      194
Distribution $\Omega$ , integral manifold      198
Distribution $\Omega$ , involutive      194
Distribution $\Omega$ , leaf      202
Distribution $\Omega$ , manifold M(fl)      200
Distribution $\Omega$ , regular      204
Distribution $\Omega$ , slice      204
Elliptic plane      166
Equivalence relation, invariant of      94
Equivalent atlases      16
Exponential mapping Exp      179
Exponential mapping Exp,       180
Fibre of a function      88
Figure eight      17
Figure Eight, submanifold structures      73
Frobenius' Theorem      197
Function      1
Function, $C^{\infty}$       12
Function, continuous      4
Function, differentiate      21
Function, domain      1
Function, global      2
Function, inverse      2
Function, locally constant      10
Function, open      5
Function, range      1
Function, restriction      1
Function, value of      1
Functions, composition      2
Functions, projection      2
Fundamental domain      101
Fundamental domain, normal      102
Geodesic      171
GL(n, C), Lie group structure      212
GL(n, C), standard $C^{\infty}$ structure      38
GL(n, R), exponential mapping      228
GL(n, R), Lie algebra      220
GL(n, R), Lie group structure      210
GL(n, R), standard $C^{\infty}$ structure      36
GL(n, R), topological properties      73
Grassmann manifold       29
Grassmann manifold ,       31
Grassmann manifold , as a coset space      252
Grassmann manifold , as a quotient manifold      92
Grassmann manifold , topological properties      41 43 253
Group of transformations      254
Group of transformations, discrete      272
Group of transformations, vector field tangent to      254
Hausdorft space      8
Hausdorft space, manifold      41
Heine — Borel theorem      9
Homeomorphic spaces      5
Homeomorphism      5
Hopf mapping      87
Horizontal distribution on TM      160
Hyperbolic plane      163
Hyperboloids      28 44 45 46
Imbedding      76
Imbedding, regular      81
Immersion      68
Implicit function theorem      14
Induced topology on a manifold      22
Injection      2
Injection, natural      7
Integral curve      131
Interior of a subset      5
Inverse function theorem      13 63
Isotropy group      249

Jacobi identity      112
Jacobian matrix       13
Klein bottle      107
Kronecker delta      54
Leaf      202
Left translation      211
Leibniz's formula      66
Length, of a curve      184
Length, of a vector      161
Lens space      108
lg      219
Lie algebra      112
Lie algebra, of a Lie group      219
Lie algebra, structure constants      220
Lie bracket [X, Y]      111
Lie group      210
Lie group of transformations      254 see
Lie group, component      215
Lie subgroup      215
Lie subgroup, connected      215
Lie transformation group      241 see
Lindeloef's theorem      10
Lineal operator on $\mathfrak{F}(m)$       54
Linear connection      152
Linear connection, Bianchi's identity      157
Linear connection, coefficients      156
Linear connection, curvature      156
Linear connection, geodesic spray      173
Linear connection, symmetric      156
Linear connection, torsion      155
Locally affine structure      158
Locally finite collection      46
Manifold      19
Manifold, chart of      19
Manifold, compact      43
Manifold, component      40
Manifold, connected      44
Manifold, Hausdorff      41
Manifold, orientable      120
Manifold, paracompact      51
Manifold, parallelizable      117
Manifold, product      19 35
Manifold, topology      22
Metric space      4
Metric, Euclidean      7
Moebius strip      107
Moebius strip, non-orientability      122
Neighbourhood      4
Neighbourhood, cubic      8
Noose      20
Normal chart      180
One-parameter subgroup      227
Open ball      4
Open set      3
Orbit      249
Orientation      120
Orthogonal group O(n, R)      217 239
Orthonormal basis      161
p-frames in       92
p-planes in       28
Parallel vector field      257
Parallelization      117
Parallelization, associated metric structure      263
Parallelization, associated spray      173
Parallelization, automorphism of      260
Parallelization, Exponential mapping for      258
Partition of unity      46
Partition of unity, subordinate to a covering      51
Piecewise differentiable curve      183
Piecewise differentiable curve, length      184
Poincare half-plane      163
Product, cartesian      2
Product, Lie group      211
Product, manifold      19 35
Product, topological      6
Projection, of a vector field      126
Projection, of an invariant      94
Projective space       30
Projective space ,       31
Projective space ,       33
Projective space , orientability      123
Projective space , topological properties      45
Quotient manifold      91
Quotient manifold, topology      93
Quotient space      7
R (real numbers) standard topology      4
RG      244
Riemannian connection      168
Riemannian manifold      183
Riemannian spray      176
Riemannian structure      162
Riemannian structure, energy function      176
Riemannian structure, invariant      165
Riemannian structure, positive-definite      162
Riemannian structure, projectable      164
Right translation      211
S(n, R)      73
Scalar product      161
Scalar product, indefinite      161
Scalar product, non-degenerate      161
Scalar product, positive-definite      161
Section of a function      87
Separation axioms ,       8
SEQUENCE      5
Sequence, limit of      5
Sequence, subsequence      6
SL(n, R)      216 239
Slice of a chart      204
Sp(n, R)      218
Spray      171
Spray, exponential mapping      179
Spray, geodesic      171
Spray, normal chart      180
Spray, of linear connection      173
Spray, Riemannian      176
Stereographic atlas      27
Stiefel manifold,       92
Stiefel manifold,       253
Structure, $C^{\infty}$       19
Structure, topological      3
Submanifold      72
Submanifold, open      35 75 78
Submanifold, regular      75
Submanifold, vectors tangent to      73 129
Submersion      84 87
subspace      7
Support of a function      46
Surjection      2
Surjection, natural      7
t (identity chart on R)      131
Tangent bundle TM      62
Tangent bundle TM, $C^{\infty}$ structure      114
Tangent bundle TM, standard chart      113
Tangent space       55
Tangent space , alternative definitions      57 63
Tangent space , canonical basis      57
Tangent vector      55
Topological group      212
Topological group, closed subgroup      213
Topological group, open subgroup      213
Topological group, topological subgroup      213
Topological space      3
Topological space, space      8
Topological space, space (Hausdorff)      8
Topological space, compact      9
Topological space, component      10
Topological space, connected      9
Topological space, locally compact      9
Topological space, locally connected      10
topology      3

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