Boothby W.M. — An introduction to differentiable manifolds and riemannian geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Boothby W.M. — An introduction to differentiable manifolds and riemannian geometry

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Название: An introduction to differentiable manifolds and riemannian geometry

Автор: Boothby W.M.

Аннотация:

The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 1986

Количество страниц: 430

Добавлена в каталог: 14.02.2006

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Предметный указатель

Hopf — Rinow theorem      347
Hyperbolic space      168 404
I(P)      134
Ideal, of an exterior algebra      225
Imbedding, of manifolds      69—74
Imbedding, of manifolds, in $\mathbb{R}^n$ , compact case      196
Imbedding, of manifolds, in $\mathbb{R}^n$ , general case      197
Immersion of manifolds      69—74
Index of vector field      140
Infinitesimal generator      see “One-parameter group action”
Initial conditions for differential equations      131
Inner product      2 184
Int M      253
Integrable functions      231 236
Integrable n-form      236
Integral curve      126
Integral manifold of distribution      159
Integral manifold of distribution, maximal      163
Integral of function on Riemannian manifold      240
Integral of n-form on manifold      239
Integral of n-form on manifold, on $\mathbb{R}^n$       240
Integration on manifolds, Lie groups      244—250
Integration on manifolds, manifolds with boundary      251—255
Integration on manifolds, Riemannian manifolds      240
Integration on manifolds, surfaces      241
Invariance of 1-parameter subgroup (under diffeomorphism)      142
Invariance of domain (Brouwer)      10 28
Invariant forms on $M\vee(M)$       285
Invariant forms on Lie groups      246 285
Invariant metric on Lie group      247 353
Invariant vector field      120
Invariant vector field, G-invariant      124
Inverse function theorem      41—46
Irreducible representation      250
Isometric surfaces      377—379
Isometries, group of      353
Isometries, group of, of       361—362 409—413
Isometries, group of, of       406
Isometries, group of, of Euclidean space (rigid motions)      92167 408
Isometry of Euclidean vector spaces      3
Isometry of Riemannian manifolds      191 327
Isometry of Riemannian manifolds, local isometry      402
Isotropic Riemannian manifold      386
Isotropy subgroup      94
Iterated integral theorem      232
Jacobi identity of Lie algebras      152
Jacobian of a mapping      25—26
Jordan content      230
k(s), curvature of plane curve      305
k(s), curvature of space curve      302
K, Gaussian curvature of surface      375
K, quaternions      407
k-frame of $\mathbb{R}^n$       63
Klein bottle      13
l, m, n, coefficients of second fundamental form      369
Lattice, integral      85 99
Leibniz rule      33
Length of curve      see “Arc length”
Lie algebra      151
Lie algebra, group of automorphisms of      245
Lie algebra, homomorphism of      156
Lie algebra, of Lie group      156
Lie algebra, of subgroup      156
Lie algebra, of vector fields on manifold      151—157
Lie derivative      154 226
Lie group      81—89
Lie group, compact, bi-invariant metric      247
Lie group, compact, bi-invariant volume      247
Lie group, compact, integration on      244—250
Lie group, de Rham groups of      285
Lie group, homomorphism of      85
Lie group, left invariant metric      247
Lie group, representation of      246
Lie group, subgroup of      87—88 142
Lift of mapping      289
Line integral      264
Line integral, independence of path      271
Linear fractional transformations      410
Linear mapping, dual of      177
Linear transformations, field of      206
Local one-parameter group action      127
Loops, product of      268—269
m(A), Lebesgue measure of set      230
Manifold with boundary      11 253
Manifold with boundary, double of      255
Manifolds, abstract      14
Manifolds, differentiable      52—59
Manifolds, imbedding in $\mathbb{R}^n$       196—197
Manifolds, orientable      13 215—219
Manifolds, topological      6—11
Manifolds, two-dimensional, classification of      14
Mappings, differentiable ( $C^{\infty}$ ) on manifolds      65
Mappings, of Class       26 65
Mappings, on $\mathbb{R}^n$       25
Mappings, smooth      26 67
Mean value theorem      23
Mean value theorem, for mappings      26
Measure zero, set of, in $\mathbb{R}^n$       230
Measure zero, set of, in $\mathbb{R}^n$ , on a manifold      235
Metric, Riemannian      186
Minimal surface      376
Mobius band      13
Monkey saddle      376
Motions, rigid, (isometries) on $\mathbb{R}^n$       92 167 408
Motions, rigid, (isometries) on hyperbolic plane      410
N, unit normal to surface      366
Natural basis of $T_a(\mathbb{R}^n)$       2 30
Natural isomorphism, tangent spaces to $\mathbb{R}^n$       29
Neighborhood      see also “Coordinate neighborhoods”
Neighborhood, admissible neighborhood      101
Norm, of vector      3
Normal coordinates      339
Normal section of surface      371
Normal space to submanifold      308
Normal vector to curve      303
Normal vector to surface      366
O(n), orthogonal group      85
One parameter subgroups of Lie groups      147—150 355
One-parameter group action, basic theorem      136
One-parameter group action, examples of      139—146
One-parameter group action, global      123 128
One-parameter group action, infinitesimal generator of      123
One-parameter group action, local      127
Orbit of group action      92
Orbit of group action, of one-parameter group      125 128
Orbit space of group action      93
Order of differentiation, interchange of      24 325
Ordinary differential equations      131—138 172—174
Orientation of manifold      216
Orientation of manifold, of vector space      215
Oriented basis      215
Paracompact space      11 193
Parallel curvature tensor      401
Parallel displacement of vector field      311 323
Parallelizable manifold      119
Parametrization of manifolds      68
Parametrization of submanifold      312
Parametrization of surface      113
Partial derivatives      21

Partial derivatives, differentiability and      21—22
Partial derivatives, independence of order      24
Partition of unity      193—198
Partition of unity, applications of      195—198
Path      268
Planar point      371
Poincare lemma      278
Positive curvature, spaces of      406
Principal curvatures, of surface      370—373
Principal directions      372
Projective space, real      15 61
Proper function      199
Proper mapping      81
Properly discontinuous action of a group      96 104
Quaternions      407
Quotient space      60
Quotient topology      60
R(X, Y), $R(X, Y)\dot\mathbb{Z}$ , curvature operator      326
R(X, Y, Z, W), curvature tensor      327
Rank of differentiable mapping      47 69 111
Rank of linear transformation      47
Rank of matrix      46
Rank theorem      47—49
Real analytic function      24
Regular covering, by spherical (cubical) neighborhoods      194
Regular domain      254
Representation, of Lie group      249—250
Representation, of Lie group, orthogonal      249
Representation, of Lie group, semisimple      250
Restriction of a differential form      260
Restriction of a differential form, of covector      182
Riemann integral, properties of      231
Riemannian geometry, fundamental theorem of      318
Riemannian manifold      186 192
Riemannian manifold, as metric space      189
Riemannian manifold, differentiation on      317
Riemannian manifold, volume element of      219
Riemannian metric, existence of      195
Rigid motions, group of      89 92
Section of tangent bundle      341
Section on coset space      166
Sectional curvature      385—386
Sectional curvature, geometric interpretation      393
Semisimple Lie group      388
Shape operator      367—368
Simple connectedness      268
Slice of coordinate neighborhood      74 160
Slice of cubical neighborhood      74
Smooth mapping      26
Smooth structure      54
SO(n), special orthogonal group      353
Sphere as manifold      57 80
Starlike set      23
Stokes’s theorem      257
Subgroup of Lie group      87 89
Subgroup of Lie group, closed      88
Subgroup of Lie group, one-parameter      143
Subgroup, discrete      98
Subgroup, isotropy      94 166
Submanifold      73 78
Submanifold property      75
Submanifold, imbedded      73
Submanifold, immersed      70
Submanifold, open      56
Submanifold, regular      77
Submersion      70 74 122
Support of function      194
Surface in Euclidean space      14
Surface in Euclidean space, geometry of      366 374
Symmetric Riemannian manifold      351 357
Symmetric Riemannian manifold, examples of      357—363
Symmetries of curvature tensor      382 384
Symmetrizing mapping      204
System of differential equations      131—139
System of differential equations, autonomous      131
System of differential equations, existence of solutions      131 174
System of differential equations, general case      137
System of differential equations, with parameters      137
T(M), , $T(\mathbb{R}^n)$       16 116—117 335—336
T, N, B, tangent, normal, binormal      303
Tangent bundle      16 116—117 335—336
Tangent bundle, sphere bundle      18
Tangent covectors      177—183
Tangent space, as derivations into S      35
Tangent space, at point of $\mathbb{R}^n$       29 32
Tangent space, at point of M      107—108
Tangent space, to surface      114
Tangent vector to $\mathbb{R}^n$       29—36
Tangent vector to curve      112
Tangent vector to manifold      107—116
Tensor algebra      208
Tensor field      201—202
Tensor field, covariant derivative of      396—401
Tensor field, invariant      246
Tensor field, parallel      400
Tensors      199—205
Tensors, alternating (skew symmetric)      202
Tensors, alternating mapping      204
Tensors, components of      200
Tensors, exterior product of alternating      209 214
Tensors, linear space of      200
Tensors, multiplication of      208—209
Tensors, symmetric      202
Tensors, symmetrizing mapping      204
Theorema Egregium (Gauss)      18 377
Tietze — Urysohn extension theorem      289
Topological manifold      6—10
Toral group      82
Torsion, of space curve      303
Torus      7 57 80 92
Transitive action of group      92—93 165
Triangulable manifold      243
U, $\varphi$ , coordinate neighborhood      10 52
Umbilical point of surface      371
Universal covering space      296
Vector      1—2
Vector fields      116—122
Vector fields, along submanifold      307
Vector fields, complete      142
Vector fields, constant      311
Vector fields, f-related      120 121
Vector fields, invariant      120
Vector fields, left invariant on Lie group      142 156
Vector fields, Lie algebra of      151
Vector fields, on submanifold      308
Vector fields, on subsets of $\mathbb{R}^n$       37—39
Vector fields, parallel      311
Vector fields, restriction to submanifold      118
Vector fields, singular points of      140
Vector, norm of      3
Vector, tangent to manifold      107
Velocity vector of moving particle      112 304
Vol D, volume of D, domain of integration      230
Volume element      219
Volume, Riemannian      240
Wedge product      209—214
Weierstrass approximation theorem      197
Whitney imbedding theorem      197
X, Y, Z, vector fields      116—122
[X, Y], (Lie) bracket of vector fields      151

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