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Kobayashi S., Nomizu K. — Foundations of Differential Geometry, Volume 1
Kobayashi S., Nomizu K. — Foundations of Differential Geometry, Volume 1

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Название: Foundations of Differential Geometry, Volume 1

Авторы: Kobayashi S., Nomizu K.

Аннотация:

This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. The second volume continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1963

Количество страниц: 329

Добавлена в каталог: 17.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
1-parameter, group of transformations      12
1-parameter, subgroup      39
Absolute parallclism      122
Adjoint representation      40
Affine connection      129
Affine connection, generalized      127
Affine frame      126
Affine holonomy group      130
Affine mapping      225
Affine parallel displacement      130
Affine parameter      138
Affine space      125
Affine space, tangent      125
Affine transformation      125 226
Affine transformation, infinitesimal      230
Alternation      28
Analytic continuation      254
Arc-length      157
Atlas      2
Atlas, complete      2
Automorphism of a connection      81
Automorphism of a Lie algebra      40
Automorphism of a Lie group      40
Bianchi's identities      78 121 135
Bundle of affine frames      126
Bundle of linear frames      56
Bundle of orthonormal frames      60
Bundle, associated      55
Bundle, holonomy      85
Bundle, homomorphism of      53
Bundle, induced      60
Bundle, principal fibre      50
Bundle, reduced      53
Bundle, sub-      53
Bundle, tangent      56
Bundle, tensor      56
Bundle, trivial      50
Bundle, vector      113
Canonical 1-form on a group      41
Canonical decomposition (= de Rham decomposition)      185 192
Canonical flat connection      92
Canonical form on L(M)      118
Canonical invariant connection      110 301
Canonical invariant Riemannian metric      155
Canonical linear connection      302
Canonical metric      155
Canonical parameter of a geodesic      162
Chart      2
Christoffel’s symbols ($\Gamma^{i}_{jk}$)      141
Compact-open topology      46
Complete linear connection      139
Complete Riemannian manifold      172
Complete Riemannian metric      172
Complete vector field      13
Components of a 1-form      6
Components of a linear connection      141
Components of a tensor (field)      21 26
Components of a vector (field)      5
Conformal transformation      309
Conformal transformation, infinitesimal      309
Connection      63
Connection, affine      129
Connection, canonical      110 301
Connection, canonical flat      92
Connection, canonical linear      302
Connection, flat      92
Connection, flat affine      209
Connection, form      64
Connection, generalized affine      127
Connection, induced      82
Connection, invariant      81 103
Connection, invariant, by parallelism      262
Connection, Levi-Civita      158
Connection, linear      119
Connection, metric      117 158
Connection, Riemannian      158
Connection, universal      290
Constant curvature      202
Constant curvature, space of      202 204
Contraction      22
Contravariant tensor (space)      20
Convex neighborhood      149 166
Coordinate neighborhood      3
Covariant derivative      114 115 122
Covariant differential      124
Covariant differentiation      115 116 123
Covariant tensor (space)      20
Covector      6
Covering space      61
Cross section      57
Cross section adapted to a normal coordinate system      257
Cubic neighborhood      3
Curvature      132
Curvature, constant      202
Curvature, form      77
Curvature, recurrent      305
Curvature, scalar      294
Curvature, sectional      202
Curvature, tensor (field)      132 145
Curvature, tensor (field), Riemannian      201
Curvature, transformation      133
Cylinder      223
Cylinder, Euclidean      210
Cylinder, twisted      223
de Rham decomposition      185 192
Derivation of $\mathfrak{D}(M)$      33
Derivation of $\mathfrak{T}{M)$      30
Derivation of the tensor algebra      25
Development      131
Diffeomorphism      9
Differential covariant      124
Differential of a function      8
Differential of a mapping      8
Discontinuous group      44
Discontinuous group, properly      43
Distance function      157
distribution      10
Distribution, involutive      10
Divergence      281
Effective action of a group      42
Einstein manifold      294
Elliptic      209
Euclidean cylinder      210
Euclidean metric      154
Euclidean motion      215
Euclidean subspace      218
Euclidean tangent space      193
Euclidean torus      210
Euclidean, locally      197 209 210
Exponential mapping      39 140 147
Exterior, covariant derivative      77
Exterior, covariant differentiation      77
Exterior, derivative      7 36
Exterior, differentiation      7 36
Fibre      55
Fibre, bundle, principal      50
Fibre, metric      116
Fibre, transitive      106
Flat, affine connection      209
Flat, connection      92
Flat, connection, canonical      92
Flat, linear connection      210
Flat, Riemannian manifold      209 210
Form, 1-form      6
Form, curvature      77
Form, r-form      7
Form, tensorial      75
Form, tensorial, pseudo-      75
Form, torsion      120
Frame, affine      126
Frame, linear      55
Frame, orthonormal      60
Free action of a group      42
Frobenius, theorem of      10
Fundamental vector field      51
G-structure      288
Geodesic      138 146
Geodesic, minimizing      166
Geodesic, totally      180
Green’s Theorem      281
Holomorphic      2
Holonomy bundle      85
Holonomy group      71 72
Holonomy group, affine      130
Holonomy group, homogeneous      130
Holonomy group, infinitesimal      96 151
Holonomy group, linear      130
Holonomy group, local      94 151
Holonomy group, restricted      71 72
Holonomy theorem      89
Homogeneous Riemannian manifold      155 176
Homogeneous space (quotient space)      43
Homogeneous space (quotient space), symmetric      301
Homomorphism of fibre bundles      53
Homothetic transformation      242 309
Homothetic transformation, infinitesimal      309
Homotopic      284
Homotopic, $C^{k}-$      284
Horizontal component      63
Horizontal curve      68 88
Horizontal lift      64 68 88
Horizontal subspace      63 87
Horizontal vector      63
Hyperbolic      209
Hypersurface      9
Imbedding      9 53
Imbedding, isometric      161
Immersion      9
Immersion, isometric      161
Indefinite Riemannian metric      155
Induced bundle      60
Induced connection      82
Induced Riemannian metric      154
Inner product      24
Integral curve      12
Integral manifold      10
Interior product      35
Invariant, by parallelism      262
Invariant, connection      81 103
Invariant, Riemannian metric      154
Involutive distribution      10
Irreducible, group of Euclidean motions      218
Irreducible, Riemannian manifold      179
Isometric      161
Isometric imbedding      161
Isometric immersion      161
Isometry      46 161 236
Isometry, infinitesimal      237
Isotropy, group, linear      154
Isotropy, subgroup      49
Killing vector field      237
Killing — Cartan form      155
Klein bottle      223
Lasso      73 184 284
Leibniz’s formula      11
Levi-Civita connection      158
Lie algebra      38
Lie derivative      29
Lie differentiation      29
Lie group      38
Lie subgroup      39
Lie transformation group      41
Lift      64 68 88
Lift, horizontal      64 68 88
Lift, natural      230
Linear connection      119
Linear frame      55
Linear holonomy group      130
Linear isotropy group      154
Local basis of a distribution      10
Local coordinate system      3
Locally affine      210
Locally Euclidean      197 209 210
Locally symmetric      303
Lorentz manifold (metric)      292 297
Manifold      2 3
Manifold, complex analytic      3
Manifold, differentiable      2 3
Manifold, oriented, orientable      3
Manifold, real analytic      2
Manifold, sub-      9
Maurer — Cartan, equations of      41
Metric connection      117 158
Moebius band      223
Natural lift of a vector field      230
Non-prolongable      178
Normal coordinate system      148 162
Orbit      12
Orientation      3
Orthonormal frame      60
Paracompact      58
Parallel cross section      88
Parallel displacement      70 87 88
Parallel displacement, affine      130
Parallel tensor field      124
Partition of unity      272
Point field      131
Projection, covering      50
Properly discontinuous      43
Pseudogroup of transformations      1 2
Pseudotensorial form      75
Quotient space      43 44
Rank of a mapping      8
Real projective space      52
Recurrent curvature      305
Recurrent tensor      304
Reduced bundle      53
Reducible connection      81 83
Reducible Riemanman manifold      179
Reducible structure group      53
Reduction of connection      81 83
Reduction of structure group      53
Reduction Theorem      83
Ricci tensor (field)      248 292
Riemannian connection      158
Riemannian curvature tensor      201
Riemannian homogeneous space      155
Riemannian manifold      60 154
Riemannian metric      27 154 155
Riemannian metric, canonical invariant      155
Riemannian metric, indefinite      155
Riemannian metric, induced      154
Riemannian metric, invariant      154
Scalar curvature      294
Schur, theorem of      202
Sectional curvature      202
Segment      168
Simple covering      168
Skew-derivation      33
Space form      209
Standard horizontal vector field      119
Structure constants      41
Structure equations      77 78 118 120 129
Structure group      50
Subbundle      53
Submanifold      9
Symmetric homogeneous space      301
Symmetric locally      303
Symmetric Riemannian      302
Symmetrization      28
1 2
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