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Millman R.S., Parker G.D. — Elements of Differential Geometry
Millman R.S., Parker G.D. — Elements of Differential Geometry



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Название: Elements of Differential Geometry

Авторы: Millman R.S., Parker G.D.

Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1977

Количество страниц: 266

Добавлена в каталог: 19.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$du^i$ as a linear functional      100
$u^i$-curve      84
$\epsilon$-neighborhood      88
$\Gamma_{ij\mid l}$      105
Angle between vectors      3 181
Angular excess      187
Angular variation      182
Angular velocity vector      35 (4.11)
Antipodal points      162
Arc length      20
Arc length, of a surface curve      95
Arc length, reparametrization by      21
Area      130 135
Area element      130
Asymptotic curve      134
Asymptotic curve, of a developable surface      140 (8.53)
Asymptotic direction      134 138
Atlas      204
Axis of a helix      32
Axis of revolution      86 (1.2)
Ball of radius $\epsilon$      200
Barbier's Theorem      70
Basis      2
Basis, orthonormal      3
Beltrami — Enneper Theorem      139 (8.35)
Bertrand curves      38 41 48
Bertrand curves, characterization      38 (4.31)
Bertrand mate      38
Bertrand mateof a circular helix      46 (5.6)
Binormal      26
Binormal spherical image      36
Bisect a curve      163
Bonnet's rigidity theorem      151
Bound      181
Boundary      181
Bounded set      174
Bump function      215 (3.6)
Catastrophe theory      246
Catenoid      137 (8.15) 146 147 148
Cauchy — Schwarz inequality      3
Chern — Hopf Conjecture      241
Christoffel symbol      104 105
Christoffel symbol of a connection      224
Christoffel symbol, intrinsic formula      105
Circle of latitude      86 (1.2)
Class $C^k$      11 209
Closed set      174
Codazzi — Mainardi equations      142
Coefficients of the Riemannian metric      93
Compact      174
complete      112 238
Components of a vector      2
Cone      207
Cone $$g_{ij}      101 (3.7)
Cone as a developable surface      141 (8.56)
Conjugate directions      134 138
Conjugate family of curves      139 (8.31)
Connection      224
Connection, for a submanifold      241 (8.7)
Connection, metrical      234
Connection, Riemannian      232 236
Connection, symmetric      235
Connection, torsion-free      235
Contact: of a surface      138 (8.29)
Contact: spherical      39
Continuous function: from a surface to $R^2$      89
Continuous function: in a metric space      201
Contravariant transformation      98
Convex curve      60
Convex surface      193
Coordinate chart      203
Coordinate function      212
Coordinate patch      77
Coordinate transformation      79
Coordinate transformation, effect on $du^i$      101
Coordinate transformation, effect on $g^{kl}$      97
Coordinate transformation, effect on $g_{ij}$      96
Coordinate transformation, effect on $L^i_j$      127 (7.6)
Coordinate transformation, effect on $L_{ij}$      108 (4.4)
Coordinate transformation, effect on $\Gamma_{ij}^k$      108 (4.11)
Coordinate transformation, effect on g      97
Coordinate transformation, effect on tangent vectors      85
Covariant derivative      116 (5.14) 224 225
Covariant transformation      98
Cover      89 204
Crofton's Formula      168
Cross product      6
Curvatura integra      189
Curvature: center of      39 (4.34) 41
Curvature: Gaussian      127 130 174 242
Curvature: Gaussian, formula      137 (8.6 8.7)
Curvature: Gaussian, intrinsic formula      143
Curvature: geodesic      103
Curvature: geodesic, extrinsic formula      106
Curvature: geodesic, intrinsic formula      106
Curvature: mean      127 130 242
Curvature: mean, formula      137 (8.3 8.11)
Curvature: normal      103
Curvature: of a plane curve      25 52
Curvature: of a regular curve      24
Curvature: of a regular curve, motivation      24
Curvature: principal      129
Curvature: radius of      34 39
Curvature: radius of, of an oval      72 (5.6 5.9)
Curvature: Ricci      239
Curvature: scalar      240
Curvature: sectional      239
Curvature: total      161 189
Curve: asymptotic      134
Curve: binormal to      26
Curve: closed      53
Curve: convex      60 63
Curve: curvature      24 (see also Curvature)
Curve: evolute      40
Curve: Fundamental Theorem      42
Curve: geometric      14
Curve: involute      40
Curve: knotted      169
Curve: length      20
Curve: length minimizing      112
Curve: maximally straight      120
Curve: parametric      84
Curve: piecewise regular      58
Curve: plane      29 (3.6 3.7) 41 45 5.3)
Curve: plane, Bertrandmate      38 (4.32)
Curve: plane, characterization      31 34 36 4.16) 37 4.20) 48
Curve: plane, curvature      25 52
Curve: plane, evolute      41 (4.43)
Curve: plane, involute      41 (4.40 4.41)
Curve: plane, normal vector field      52
Curve: plane, rotation index      55 58
Curve: plane, spherical contact      40 (4.36)
Curve: plane, tangent circular image      58
Curve: plane, tangent vector field      52
Curve: polar axis      39 (4.34)
Curve: principal normal      26
Curve: radius of curvature      34 39
Curve: radius of torsion      34
Curve: regular      15
Curve: reparametrization      17
Curve: reparametrization, by arc length      21
Curve: segment      20
Curve: segment, reparametrization      23 (2.7)
Curve: simple      54
Curve: speed      15
Curve: sphere      19 (1.7) 34 37 45
Curve: sphere, characterization      33 37 4.25 4.26) 38
Curve: sphere, normal curvature      107 108
Curve: sphere, torsion      171
Curve: spherical contact      39
Curve: tangent developable surface      88 (1.14)
Curve: tangent line      16
Curve: tangent vector field      15
Curve: unit speed      22 24
Curve: unknotted      169
Curve: velocity vector      15
Curve: vertex of      41 (4.44) 66
Curvilinear coordinate system      84
Cusp      71 (5.5)
Cylinder      115 (5.10) 140 152
Cylinder, $R_i{}^l{}_{jk}$      144 (9.4)
Cylinder, as a developable surface      141 (8.56)
Cylinder, curvature of      136 (8.1)
Cylinder, L      127 (7.2)
Darboux vector      35 (4.11)
Developable surface      140
Developable surface, characterization      140 (8.49)
Developable surface, Gaussian curvature      140 (8.52)
Diffeomorphism      223 (5.4)
Differentiable function: of a manifold to R      209
Differentiable function: of a surface to R      124 145
Differentiable function: of manifolds      209
Differentiable function: of surfaces      146
Differentiable vector field      117
Differential of a mapping      219
Dimension: of a manifold      204
Dimension: of a vector space      2
Directional derivative      124
Distance      237
Drag curve      158
Dual basis      100
Dual space      100
Dupin indicatrix      131 138
Dupin indicatrix, asymptotes      134
Eigenvalue      5 128
Eigenvector      5 128
Ellipsoid      88 (1.12) 91 92
Elliptic integral      23
Elliptic paraboloid      88 (1.11) 108
Elliptic point      132 138
Embedding      221
Euclid's Fifth Postulate      188 (4.5) 231
Euler characteristic      188 190
Euler's theorem      129
Evolute of a curve      40
Existence and uniqueness theorem: for curves      42
Existence and uniqueness theorem: for geodesies      111 231
Existence and uniqueness theorem: for parallel vector fields      118 228
Existence and uniqueness theorem: for Riemannian connections      236
Existence and uniqueness theorem: for surfaces      151
Fary — Milnor Theorem      73 169
Fenchel's Theorem      72 165 170
Field of metrics      233
Field of vectors      216
First fundamental form      94
Flat Euclidean space: connection of      224 227
Flat Euclidean space: covariant derivatives      226
Flat Euclidean space: curvature      242 (8.11)
Flat Euclidean space: geodesies      232 (7.1)
Flat Euclidean space: parallel vector fields      228
Flat Euclidean space: Riemannian connection      241 (8.4)
Flat point (Planar point)      132
Four-vertex theorem      66
Frenet — Serret apparatus      26
Frenet — Serret equations      30
Frenet — Serret Theorem      30
Fundamental lemma of Riemannian geometry      231
Fundamental Theorem of Curves      42
Fundamental Theorem of Surfaces      151
Gauss — Bonnet formula      173 185
Gauss — Bonnet theorem      188
Gauss — Bonnet Theorem, generalization      191
Gauss's equations      141
Gauss's formulas      104 241
Gauss's Theorema Egregium      143 149
Gaussmap      131 180
Genus      190
Geodesic      109
Geodesic coordinate patch      115 (5.5) 176
Geodesic, as an asymptotic curve      138 (8.28)
Geodesic, characterization      109 110
Geodesic, differential equation      109 115 5.8)
Geodesic, existence and uniqueness theorem      111
Geodesic, image under isometries      152 (10.5)
Geodesic, on a manifold      230
Geodetic torsion      137 (8.13)
GL(n, R)      205
Graph of a function      78
Green's theorem      51
Hadamard's Theorem      193
Helicoid      87 (1.4) 138 146 147 148 153
Helicoid, as a ruled surface      139 (8.36)
Helicoid, lineofstriction      140 (8.41)
Helix      46 (5.5) 48 115
Helix, axis      32 35
Helix, characterization      35 (4.3) 37
Helix, circular      17 23 2.3) 26 32
Helix, circular, Bertrand mate      46 (5.6)
Helix, circular, characterization      46 (5.4 5.6)
Helix, general      32
Helix, general form      44—45
Helix, involute      41 (4.40)
Helix, Lancret's characterization      32
Helix, pitch      32 35
hemisphere      162
Hilbert's theorem      238
Homogeneous coordinates      200
Hopf — Rinow theorem      238
Hyperbolic cosine      12
Hyperbolic disk      see Poincare disk
Hyperbolic half plane      179 (2.4) 188 4.3 4.5)
Hyperbolic half plane, connection      225
Hyperbolic half plane, covariant derivatives      227 (6.3)
Hyperbolic half plane, distance      241 (8.8)
Hyperbolic half plane, geodesies      231
Hyperbolic half plane, parallel translation      229
Hyperbolic half plane, parallel vector fields      228
Hyperbolic half plane, Riemannian connection      241 (8.5)
Hyperbolic half plane, Riemannian metric      234
Hyperbolic paraboloid      87 (1.5) 91 111
Hyperbolic paraboloid, as a ruled surface      139 (8.37)
Hyperbolic paraboloid, asymptotic curves      138 (8.27)
Hyperbolic paraboloid, geodesies      115 (5.9) 116
Hyperbolic point      132 138 8.23)
Hyperbolic secant      231
Hyperbolic sine      12
Hyperbolic space form      242 (8.13)
Hyperbolic tangent      231
Hyperboloid of one sheet      91
Hyperboloid of one sheet, as a ruled surface      139 (8.38)
Hyperboloid of one sheet, geodesies      115 (5.11) 116
Hyperboloid of two sheets      92
Hypersurface defined by f      203 206 221 241
Hypersurface defined by f, tangent space      222
Immersion      223
Implicit function theorem      202
Index of a vector field      195
Inner product      2
Intrinsic normal      103
Inverse function theorem      202
Involute of a curve      40
Isometric      147
Isometric, locally      147
Isometry      147 234
Isometry, local      147
Isometry, orientation preserving      152 (10.8)
Isomorphism      4
Isoperimetric inequality      64
Jacobi's identity      218
Jacobi's theorem      162
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