Bishop R.L., Crittenden R.J. — Geometry of manifolds :: Электронная библиотека попечительского совета мехмата МГУ

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Bishop R.L., Crittenden R.J. — Geometry of manifolds

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Название: Geometry of manifolds

Авторы: Bishop R.L., Crittenden R.J.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1964

Количество страниц: 284

Добавлена в каталог: 22.10.2012

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Предметный указатель

$C^{\infty}$       (see Class $C^{\infty}$ )
$C^{\infty}$ -related      2
Abelian, Lie algebra      26
Abelian, Lie group      25 37
Action by compact group, induced Riemannian metric      136
Action of $C^{\infty}$ maps on forms      68
Action of Lie groups      38
Action of Lie groups on connexion      118
Action of Lie groups to right      38
Action of Lie groups, effective      38
Action of Lie groups, free      39
Action of Lie groups, prolongation      118
Action of Lie groups, transitive      39
Action of principal bundle      49
Adjoint representation of Lie algebra      36
Adjoint representation of Lie group      36
Adjoint representation of quaternions      36
Ado’s theorem      30
Affine Bianchi identities      101
Affine bundle      109
Affine bundle, connexion on      120
Affine connexion      89
Affine connexion on Lie group      107
Affine connexion on parallelizable manifold      106
Affine connexion, classical definition      117
Affine connexion, complete      110
Affine connexion, curvature and torsion zero      95
Affine connexion, curvature transformation      96 97
Affine connexion, determined by geodesics and torsion      106
Affine connexion, symmetric connexion      (see Torsion zero)
Affine connexion, torsion form      93
Affine connexion, torsion translation      96 97
Affine connexion, torsion zero      95 103 106 117 118 166
Affine product connexion      119
Almost complex manifold      64
Almost complex manifold, structure      51 73
Alternating multilinear functions      57
Arc length of broken $C^{\infty}$ curves      123
Arc length of continuons curves      158
Arc length, first variation of      213 214
Arc length, minimizing      149 151 154
Arc length, reduced      223
Arc length, second variation of      213 214
Arc length, Synge’s formula      218
Associated bundle      45
Associated bundle, connexions on      83
Associated bundle, coordinate bundle      46
Associated bundle, tangent bundle      46
Associated bundle, vector bundles      48
Associated codistribution      70
Associated distribution      70
Associated vector field to a rectangle      147
Atlas, same as basis for $C^{\infty}$ structure      3
Augmented index      232
Automorphism of Lie algebra      35
Automorphism of Lie group      35
Bagel      210
Base space of bundle      42
Basic inequality for index form      228
Basic inequality, generalized      250
Basic vector fields      92
Basis for $C^{\infty}$ structure      3
Bianchi identity      82
Bianchi identity, affine identities      101
Bonnet’s theorem      252
Bracket of vector fields      13
Bracket of vector fields, geometrical interpretation      18
Broken $C^{\infty}$ curve      77
Broken $C^{\infty}$ rectangle      213
Bundle      (see Associated bundle)
Canonical lifting of C      00
Canonical lifting of C, rectangle      147
Cartan connexion      121
Cartan — Kahler Theorem      205
Cartan, equations and forms of Maurer      70 72 82 107
Cartan, theorem of Hadamard      184
Cartan’s criterion for closed subgroups      30
Cartan’s local isometry theorem      181
Cauchy — Kowalevsky theorem      205
Center      36
Chain rule      10
Characteristic value and vectors of generalized index form      236
Class $C^{k}$       9
Class $C^{\infty}$ of action      38
Class $C^{\infty}$ of bundles      41
Class $C^{\infty}$ of codistribution      70
Class $C^{\infty}$ of covering map      6
Class $C^{\infty}$ of curve      7
Class $C^{\infty}$ of differential form      62
Class $C^{\infty}$ of distribution      23
Class $C^{\infty}$ of function      2 3
Class $C^{\infty}$ of manifold      3
Class $C^{\infty}$ of partition of unity      4
Class $C^{\infty}$ of rectangle      147
Class $C^{\infty}$ of structure      3
Class $C^{\infty}$ of structure, basis for      3
Class $C^{\infty}$ of tensor field      47
Class $C^{\infty}$ of Urysohn function      2
Class $C^{\infty}$ of vector field      13 47
Classical coordinate definition of affine connexion      118
Classical covariant derivative      117
Classical notion of differential form      53
Classical rigidity theorem      211
Closed form      114
Closed geodesic      241
Closed geodesic, smooth      241
Codistribution      70
Coefficients of connexion      116
Comparison theorem, Rauch’s      250
Complete connexion      110
Complete connexion, negative curvature      184
Complete connexion, Riemannian manifold      154
Complete connexion, zero curvature      183
Complex differential forms of type (p, q)      73
Complex Grassmann manifold      170
Complex Lie group      34
Complex Lie group, compact      34
Complex manifold      7 50 64
Complex manifold, complex structure      51
Complex manifold, Hermitian structure      141
Complex manifold, Kahler structure      142
Complex projective space      140
Complex structure      51
Configuration space of mechanical system      157
Conformal map      136
Conjugate point      225
Conjugate point, complex projective space      225
Conjugate point, order of      225
Conjugate point, quaternion projective space      226
Conjugate point, sphere      225
Connexion of Cartan      121
Connexion on affine bundle      120
Connexion on associated bundle      83
Connexion on principal bundle      75
Connexion one form of      76
Connexion, curvature form      80(see also Affine connexion Riemannian
Connexion, existence of      83
Connexion, flat      82
Connexion, induced      80
Conservative force field      157
continuous curve      158
Continuous curve, arc length      158
Convexity      246
Convexity, local      246
Convexity, strong local      248
Coordinate bundle, associated      46
Coordinate bundle, principal      44
Coordinate bundle, transition functions of      44
Coordinate neighborhood      2
Coordinate system      2

Coordinate system, normal      110
Covariant differentiation      111
Covariant differentiation of Riemannian connexion      166
Covariant differentiation, examples      112
Covariant differentiation, parallelizable manifold      114
Covariant differentiation, structural equation      116
Critical point of function      197
Critical point of function, Hessian      197
Cross-sections of bundles      42 48 49
Curvature form of connexion      80
Curvature form of immersion      190
Curvature form, vanishing of      82
Curvature transformations      96
Curvature transformations, geometric interpretation of      97
Curvature transformations, symmetries of      162
Curvature, Gauss      162 211
Curvature, mean      253
Curvature, p-mean      253
Curvature, Ricci      253
Curvature, Riemannian or sectional      (see Riemannian curvature)
Curve, arc length      123
Curve, broken C      77
Curve, continuous      158
Curve, development of      98
Curve, horizontal lift      77
Cut locus      237
Decomposable elements of Grassmann algebra      55
Deformation, infinitesimal      226
Derivation of Grassmann algebra, as extension of linear endomorphism      56
Derivative, Lie      16 69
Derivative, partial      8
Determinant of linear endomorphism      57 61 62
Development of curve      98
Diffeomorphism      11
Difference forms      102
Difference forms and geodesics      105
Difference forms and torsion      103
Differentiable      (see Class $C^{\infty}$ )
Differential form      62
Differential form as cross-section      63
Differential form, action of $C^{\infty}$ maps on      68
Differential form, action of Lie derivative on      69
Differential form, classical notion      53
Differential form, closed      114
Differential form, complex, of type (p, q)      73
Differential form, difference      102
Differential form, equivariant      75 85 100
Differential form, exact      143
Differential form, holomorphic      73
Differential form, vector valued      71
Differential of map      9 12
Differential of map, chain rule      10
Differentiation, covariant      (see Covariant differentiation)
Direct connexion on parallelizable manifold      106 114
Distance on Riemannian manifold      124
distribution      22
Distribution, $C^{\infty}$       23
Distribution, associated codistribution      70
Distribution, Frobunius’ theorem      23
Distribution, integrable      23
Distribution, integral manifold of      23
Distribution, involutive      23
Doubly ruled surface      209
Dual forms of structural equations      80 101
Effective action      38
Embedding      (see Imbedding)
Endomorphism, extension as derivation to Grassmann algebra      56
Endomorphism, extension as homomorphism to Grassmann algebra      56
Endomorphism, linear determinant      57 61 62
Endomorphism, trace      57
Equivariant forms      76 85 86 90 100 101
Euclidean space      2 108 132
Exact form      143
Exponential map, affine connexion into M      108
Exponential map, into B(M)      108
Exponential map, Lie group      30
Exponential map, matrices      32
Extension of function on submanifold      22
Extension of vector field on submanifold      22
Exterior derivative      64
Exterior derivative, intrinsic formula for      65
Faithful representation      34
Fibre bundle      (see Associated bundle Principal
Fibre, action of principal bundle on      49
Fibre, of associated bundle      46
First variation of arc length      213 214
First variation of arc length, unintegrated      214
Flag manifold      137 187
Flat, affine connexion      (see Curvature and Torsion zero)
Flat, complete Riemannian manifold      183
Flat, connexion      82
Flat, tori      133
Focal point      224
Focal point, order of      225
Focal values      232
Force field of mechanical system      157
FORM      (see Differential form Index
Frames, bundle of      (see Bundle of frames)
Free action      39
Frobenius’ theorem      23 71
Function      (see also class $C^{\infty}$ )
Function, alternating      57
Function, differential of      9 12
Function, extension of      22
Function, restriction of      22
Function, Urysohn      2
Fundamental forms of immersion, first      185
Fundamental forms of immersion, second      190
Fundamental vector fields      91
Gauss curvature      162 211
Gauss map      195
Gaussian curvature      (see Gauss curvature)
Gauss’ Lemma      147
General linear group      4 25 27
General linear group, complex      7 34
Geodesics      96 104 113
Geodesics and curvature      179
Geodesics on sphere      133
Geodesics, closed      241
Geodesics, connexions with same      105
Geodesics, existence of      97
Geodesics, local minimizing property on Riemannian manifold      149 217
Geodesics, plete Riemannian manifold      154
Grassmann algebra      54
Grassmann algebra, decomposable elements      55
Grassmann algebra, derivation of      56
Grassmann algebra, existence of      57—62
Grassmann algebra, homogeneous elements      55
Grassmann algebra, homomorphisms      56
Grassmann manifold      139 170 186
Grassmann manifold, complex      170
Grassmann manifold, hyperbolic space      172
Grassmann manifold, opposite spaces      172
Grassmann manifold, oriented      170
Grassmann manifold, quaternion      170
Grassmann manifold, real      170
Hadamard — Cartan, theorem of      184
Hermitian manifold      141
Hessian of function      197
Hessian of function of normal coordinate function      198
Holomorphic, curvature      171
Holomorphic, forms      73
Holomorphic, section      171
Holonomy group of connexion      87
Holonomy group of connexion, restricted      87
Homogeneous elements of Grassmann, algebra      55
Homogeneous elements of Grassmann, rank      55
Homogeneous space      39 43
Homogeneous space, connexions on      82
Homogeneous space, reductive subgroup      83

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