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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$C^{\infty}$ structure      19

$J_f$       13

$M(r \cross s, R)$       15

$M(\Omega)$       200

$R^n$ , standard $C^{\infty}$ structure      20

$R^n$ , standard topology      6, 8, 9, 10

$S^n$ (unit sphere)      7

$S^n$ , $C^{\infty}$ structure      19, 26

$S^n$ , as a coset space      251

$S^n$ , orientability      120

$S^n$ , standard topology      7, 10, 11, 44

$S^n$ , 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 $R^n$       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, $Exp_m$       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 $G(p, R^n)$       29

Grassmann manifold $G(p, R^n)$ , $G(p, C^n)$       31

Grassmann manifold $G(p, R^n)$ , as a coset space      252

Grassmann manifold $G(p, R^n)$ , as a quotient manifold      92

Grassmann manifold $G(p, R^n)$ , 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 $J_f$       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 also Group of transformations

Lie group, component      215

Lie subgroup      215

Lie subgroup, connected      215

Lie transformation group      241, see also Transformation group

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 $R^n$       92

p-planes in $R^n$       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 $P^n(R)$       30

Projective space $P^n(R)$ , $P^n(C)$       31

Projective space $P^n(R)$ , $P^n(Q)$       33

Projective space $P^n(R)$ , orientability      123

Projective space $P^n(R)$ , 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 $T_1$ , $T_2$       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, $V'(p, R^n)$       92

Stiefel manifold, $V(p, R^n)$       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 $T_mM$       55

Tangent space $T_mM$ , alternative definitions      57, 63

Tangent space $T_mM$ , 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, $T_1$ space      8

Topological space, $T_2$ 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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