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

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Издание: Second Edition

Год издания: 1986

Количество страниц: 440

Добавлена в каталог: 23.04.2010

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Предметный указатель

$B^{k}(M)$ , exact forms      274
$B^{n}_{\mathcal{E}}(x)$ , $B_{\mathcal{E}}(c)$ , open ball of $R^{n}$       2
$C^{n}_{\mathcal{E}}(x)$ , $C_{\mathcal{E}}(x)$ , open cube of $\mathbf{R}^{n}$       2
$C^{r}$ , $C^{\infty}$ differentiability class      21—22 66
$C^{r}(U)$ , $C^{\infty}(U)$ , $C^{\infty}(M)$ differentiable functions on U or M      22 107
$C^{\infty}$ function      22 66
$C^{\infty}$ mapping      66
$C^{\infty}$ -compatibility      52
$C^{\infty}(a)$ , $C^{\infty}(p)$ , germs of $C^{\infty}$ -functions at a, p, etc.      32 107
$C^{\omega}$ , real analytic functions      24
$ds^{2}$ , metric tensor      188
$dx^{i}$ , coordinate coframes      182
$Exp X_{p}$ , exponential mapping      337
$e^{tX}$ , one-parameter group of matrices      148
$e^{X}$ , exponential of a matrix      147
$E_{i}$ , $E_{ia}$ , $E_{ip}$ coordinate frames      30 110 118
$f \ast g$ , product of paths      268—269
$F_{*}$ , F*, linear mappings induced by F      109 121 180 184 202
$F_{*}(d/dt)$ , tangent vector to curve      112
$g_{ij}$ , coefficients of metric tensor      188 320
$H^{2}$ , half-plane as hyperbolic space      167 361 409—412
$H^{k}(M)$ , H*(M), de Rham groups      274
$H^{n}$ half space of $\mathbf{R}^{n}$       11 252
$H^{n}$ hyperbolic space      404
$I_{a}$ , inner automorphism      245
$i_{X}$       226
$I_{\delta}$ , $I_{\varepsilon}$       131
$K(\pi)$ , sectional curvature      385
$k_{1}$ , $k_{2}$ , principal curvatures to surface      370
$k_{A}$ , characteristic function of A      231
$L_{a}$ , $R_{a}$ , $L_{g}$ , left and right translations on Lie group      84 121
$L_{X}$ on $\Lambda(M)$       226
$L_{X}Y$ , Lie derivative of vector field      154
$P^{n}(\mathbf{R})$ , real projective space      15 61
$S(X_{p})$ , shape operator      367—368
$sgn\ \sigma$ , sign of a permutation      203
$Sl(2, \mathbf{R})$ , acting on $H^{2}$       361
$Sl(n, \mathbf{R})$ , special linear group      84
$S^{1}$ , $S^{2}$ , $S^{n}$ circle, 2—sphere, n-sphere      7 57 80
$T^{2}$ , $T^{n}$ , torus      7 57 80 82
$T_{p}(M)$ , $T_{p}(\mathbf{R}^{n})$ , tangent space at point      8 29 32 107—108
$V^{n}$ , vector space of n-tuples      2
$X^{*}_{a}f$ , directional derivative      32
$Z^{k}(M)$ , closed forms      274
$\chi(M)$ , Euler characteristic      14 415
$\delta^{j}_{i}$ , Kronecker delta      179
$\dot{F}$ , tangent vector to curve      126
$\Gamma$ , discrete group      97
$\Gamma^{k}_{ij}$ , $\Gamma_{ijk}$ , Christoffel symbols      313 321—322
$\hat{f}$ , f expressed in local coordinates      65
$\mathbf{E}^{n}$ , Euclidean space      4
$\mathbf{R}$ , real numbers      1
$\mathbf{R}$ , real numbers as additive group      123
$\mathbf{R}^{j}_{iki}$ , $\mathbf{R}_{ijki}$ , coefficients of curvature tensor      327 384
$\mathbf{R}^{n}$ , n-tuples of real numbers      1—3
$\mathfrak{S}$ , symmetric group      166 203 215
$\mathfrak{X}(U)$ , $\mathfrak{X}(M)$ , $C^{\infty}$ vector fields on U or M      40 122 151
$\mathscr{A}$ , alternating mapping      204
$\mathscr{D}(a)$ , $\mathscr{D}(U)$ , algebra of derivations      33 40 108
$\mathscr{I}$ , homotopy operator on $\wedge(I \times A)$       277
$\mathscr{M}(\mathbf{R})$ , $n \times n$ real matrices      56 59
$\mathscr{M}_{mn}(\mathbf{R})$ , $m \times n$ real matrices      27 56 59
$\mathscr{S}$ , symmetrizing mapping      204
$\mathscr{T}^{r}_{2}(V)$ , $\mathscr{T}^{r}(V)$ , $\mathscr{T}^{r}(M)$       200 202 208
$\nabla$ , connection on a manifold      317
$\nabla^{U}$ , restriction of a connection      319
$\nabla_{x}Y$ , covariant derivative      314—316
$\Omega$ , volume element      215—219
$\omega^{i}$ , $\omega^{J}_{i}$ , connection forms      329 390—392
$\omega^{i}$ , coframes, dual basis      178
$\Omega_{ij}$ , curvature forms      390
$\overset{\circ}{D}$ , interior of D      231
$\overset{\circ}{K}$ , interior of K      193
$\partial H^{n}$ , $\partial M$ , boundary of $H^{n}$ , M      253—254
$\partial(f^{1}, ..., f^{m})/\partial(x^{1}, ..., x^{n})$ , Jacobian matrix      26
$\partial/\partial x^{i}$ , natural frames of $\mathbf{R}^{n}$       36
$\Phi^{\sigma}$       204
$\pi_{1}(M, b)$ , fundamental group      270
$\sigma$ —compact space      193
$\sigma_{p}$ , involutive isometry      351
$\tau(s)$ , torsion of space curve      303
$\tau_{c}$ , translation along geodesic      354
$\theta(t, p)$ , $\theta_{i}(p)$ , $\theta_{p}(t)$ , action of $\mathbf{R}$ on $\mathbf{M}$       123
$\theta^{j}_{i}$ connection forms      328
$\tilde{f}$ , $\tilde{g}$ , lift of a mapping      289
$\tilde{H}^{k}(M)$ , invariant de Rham groups      285
$\tilde{k}(s)$ , curvature of plane curve      305
$\tilde{M}$ , covering manifold of M      101 289—295
$\triangle$ , distribution      159
(D/dt)(dp/dt) = 0, equation of geodesic      330—331
A', $^{t}A$ , transpose of matrix      85 150 184 357
A*, transpose conjugate of complex matrix      357
Acceleration of moving particle      304
Action of group on manifold      90—96 123 165
Action of group on manifold, (properly) discontinuous      96
Action of group on manifold, effective      91
Action of group on manifold, free      94
Action of group on manifold, transitive      92—93 165—172
Ad g, adjoint homomorphism      245
Adjoint representation of Lie group      246
Admissible neighborhood of covering      101
Algebra of differential forms on M, $\wedge(M)$       214
Almost continuous function      230
Alternating tensor      202
Antipodal map of $S^{n-1}$       283
Approximation theorems      197 289
Approximation theorems, Weierstrass      197
Arc length      187—188
Arc length as parameter      301
Asymptotic direction on surface      374
Atlas      59
Automorphisms of Lie algebras      245
Automorphisms of Lie groups      245
Autonomous system of differential equations      131
Basis of covariant tensors $\mathscr{T}^{r}(V)$       200—201
Basis of vector space      1—3
Basis, canonical (natural)      2
Basis, dual      177
Basis, oriented      215
Bilinear form      183—187
Bilinear form, induced mapping of      185
Bilinear form, skew symmetric      184
Bilinear form, symmetric, positive definite      184
Binormal to space curve      303
Brouwer fixed point theorem      282
Bundle      see "Tangent bundle"
c(A), Jordan content of a set      230
C*, multiplicative group of complex numbers      82
Canonical basis of $\mathbf{R}^{n}$       2
Center of a Lie algebra      289 388
Chain rule      23
Chain rule for mappings      27
Change of variables in integration      233
Characteristic function of a set      231
Chart      59
Closed differential form      274
Coframes      179
Complete integrability      160
Complete vector field      142
Components of a bilinear form      184
Components of a covector      179
Connected sum of manifolds      258
Connection      317—323
Connection forms      328 390
Connection, restriction of      319
Connection, Riemannian      318
Constant curvature, manifold of      386 406—413
Constant vector field      323
Content zero in $\mathbf{R}^{n}$       230
Content zero on a manifold      235

Contractible space      270
Contracting mapping theorem      43
Coordinate coframes      179
Coordinate frames      110
Coordinate function      52
Coordinate neighborhoods      52—56
Coordinate neighborhoods open balls and cubes      55
Coordinates, local      10
Coordinates, oriented      215
Coset space      94
Coset space, group action on      165
Covariant derivative      see "Differentiation of vector fields"
Covariant tensor field      201—205
Covariant tensor field, induced mapping of      202
Covector, field      178
Covector, tangent      178—179
Covering manifolds      101 289—296
Covering manifolds, isomorphism of      293
Covering map      101
Covering transformation      101
Covering, locally finite      11 193
Covering, refinement of      11 193
Covering, regular, by spherical (cubical) neighborhoods      194
Cube, on a manifold      237
Curvature of plane curve      305
Curvature of space curve      302
Curvature of surface      18 374
Curvature, Riemannian      325—327
Curvature, Riemannian forms      389
Curvature, Riemannian sectional curvature      385 393
Curvature, Riemannian symmetries of      383
Curve differentiable $(C^{r})$       22
Cutting and pasting of manifolds      11—14 258
d(p, q) metric on Riemannian manifold      189
d, $d_{M}$ , exterior derivative      220
de Rham group      274
de Rham's theorem      275
Deck transformation      101
Dependent functions      50
Derivation(s) into $\mathbf{R}$       33
Derivation(s) on $C^{\infty}(U)$       39
Derivative of vector field      314
Derivative, exterior      220—227
Derivative, exterior properties of      220
DF, DF(x), Jacobian matrix      27
df, differential of function      180
Diffeomorphism      67
Diffeomorphism on open sets of $\mathbf{R}^{n}$       41
Differentiable functions in weak sense      21
Differentiable functions on Euclidean space      21—25
Differentiable functions on manifolds      65—66
Differentiable manifold      see "Manifolds"
Differentiable mapping, weak sense      26 67
Differentiable mappings composition of      28 67
Differentiable mappings on Euclidean space      25—28
Differentiable mappings on manifolds      65—67
Differentiable structure      53
Differentiable submanifold      see "Submanifold"
Differential equations, existence theorem, proof      174
Differential equations, systems of      131—137
Differential forms      213
Differential forms, closed      226 274
Differential forms, exact      226 274
Differential forms, exterior      213
Differential of function      180
Differential of mapping      109
Differentiation of vector fields along curves in $\mathbf{R}^{n}$       298
Differentiation of vector fields covariant differentiation      309 314—316
Differentiation of vector fields Lie derivative      154
Differentiation of vector fields on submanifolds of $\mathbf{R}^{n}$       307—316
Directional derivative as tangent vector      32
Discrete group, action of      96—100
Distribution, involutive      159—224
Distribution, local basis of      159
Distribution, on a manifold      159
Divergence theorem      262
Domain of integration in $\mathbf{R}^{n}$       230
Domain of integration on a manifold      235
Double of manifold with boundary      255
dp/dt, tangent vector to curve p(t)      126
Dual basis      177
Dual vector space      177
DX/dt, DY/dt, covariant derivative      309
Dynamics of moving particle      304
dZ/dt, derivative of vector field      300
E, F, G coefficients of first fundamental form      242 373
Effective action of a group      91
Einstein manifold      387
Equations of structure      391
Equivalence relation, open      60
Euclidean space      4—6
Euclidean vector space      2
Euler characteristic      14 415
Euler's formula for surfaces      371
Exact differential form      274
Exponential mapping on Lie groups      150
Exponential mapping on Riemannian manifolds      337
Exponential of matrix      147
Exterior algebra      211
Exterior algebra of V, $\wedge(V)$       211
Exterior algebra, induced mapping of      213
Exterior differential form      213
Exterior differentiation      219—223
F(k, n), k-frames in $V^{n}$       63
Fixed point of group action      92
Fixed point of mapping      282
Flags, space of      172
Flat coordinate neighborhood      160
Flow      127
Foliation      164
Forms connection      328—329 390
Forms curvature      390
Forms exterior differential      213—214
Forms, bilinear      183—187
Frames, coordinate      118
Frames, field of      38
Frames, orthonormal      329
Frames, parallel      299 323
Frames, space of      93
Free action of group      94
Frenet — Serret formulas      303
Frobenius' Theorem      161 233—236
Fundamental forms of a surface      370
Fundamental group      269—270
G(k, n), Grassmann manifold of k-planes in $\mathbf{R}^{n}$       63 168 362
G/H, homogeneous space, coset space      94 165
Gauss — Bonnet theorem      415
Gaussian curvature      18 375
Genus (of surface)      14
Geodesic sphere      344
Geodesics      191 312 330—335
Geodesics as one-parameter subgroups      355
Geodesics, minimal      346
Germs of $C^{\infty}$ functions in $\mathbf{R}^{n}$       36
Germs of $C^{\infty}$ functions on manifold      115
Gl(n, $\mathbf{R}$ ), general linear group      56
Grassman algebra      see "Exterior algebra"
Grassman manifolds      63 168 362—363
Green's theorem      262
groups      see "Action of group on manifold" "Discrete "Lie
G—invariance      124
H, mean curvature of surface      374—375
Homogeneous space      165—172
Homomorphism      see "Lie group"
Homotopy      266—267
Homotopy of mapping      266
Homotopy of paths and loops      268
Homotopy operator      277
Homotopy, relative      267
Hopf — Rinow theorem      347

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