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Kühnel W., Hunt B. — Differential Geometry: Curves - Surfaces - Manifolds
Kühnel W., Hunt B. — Differential Geometry: Curves - Surfaces - Manifolds

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Название: Differential Geometry: Curves - Surfaces - Manifolds

Авторы: Kühnel W., Hunt B.

Аннотация:

Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to helpalong the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Second Edition

Год издания: 2006

Количество страниц: 376

Добавлена в каталог: 30.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Acceleration vector      141
angle      2
Angle preserving      101 128
Apex      12
Arc element      60
Arc length      9
Archimedean spiral      51
Asymptotic curve      84 128
Atlas      204
Banchoff, T.      187
Beltrami, E.      85 95
Bertrand curve      53
Bianchi identity      249 341
Binary dihedral group      309
Binary icosahedral group      309
Binary octahedral group      309
Binary tetrahedral group      309
Binormal      17
Biquadratic form      252
Bivector      337
Bonnet, O.      154
Boost      276
Canal surface      78
Cardan angles      206
Cartan, E.      166
Catalan, E. C.      113 129
Catenary      11 112 195
Catenoid      112 157 196
Cauchy-Riemann equations      103
Cayley map      205
Cayley plane      336
Chain rule      214
Chart      5 202 203
Christoffel symbols      139 160 168 228
circle      9
Clifford torus      32
Codazzi-Mainardi equation      147 154 170 198 248
Cohn-Vossen, S.      190
Complex manifold      207
Complex projective space      335
Complex structure      209
Cone      90 91 93
conformal      101 105 107 128 221 278
Conformal curvature      346
Conformally flat      126 351 353
Conic type      82
Conjugate point      312
Conjugate surface      112
Connection      226
Connection form      168
Constant curvature      25 36 81 191 193 255 273 294 316
Constant Gaussian curvature      162
Contact of kth order      12
Contraction      257
Contravariant tensor      242
Convergence      208
Convex      42 183
Convex hull      45 183
Coordinate transformation      204
Cornu spiral      16
Cosmological constant      331
Countability axiom      222
Covariant derivative      136 138 168 226 245 272
Covariant tensor      242
Covector field      243
Covering      298 307 316
Coxeter, H.S.M.      308
CR equations      103
Cubical parabola      20
Curvature      14 17 20 36 71
Curvature tensor      150 171 249 253 321
Curve      7 8
Curve, closed      37
Curve, length of      8
Curve, simply closed      37 45
Cyclic group      307 309
Cycloid      50
Cylinder      90 91
Darboux equations      26 52
Darboux vector      26 52
Derivative      3 6 214
Developable surface      89 121
Dicyclic group      309
Diffeomorphic      208
Differentiable      3
Differentiable manifold      203
Differentiable structure      204
Differential      6 214 246
Differential form      167
Dihedral group      307
Dini, U.      95
Directional derivative      100 135 210 211 226
Directional vector      63
Directrix      85
Distance      2
Divergence      257 258
Double point      48
Double tangent      48
Dual basis      217
Duality      356
Dupin indicatrix      76
Eigenvalue      72
Eigenvector      72
Einstein field equations      330
Einstein space      261 334 357
Einstein tensor      263 330
Einstein, A.      318 323 357
ellipse      76
Ellipsoid      131
Elliptic point      73 196
Elongated sphere      82
Energy functional      284
Enneper, A.      85 113
Equations of Gauss and weingarten      140 146
Euler angles      206
Euler characteristic      179—181 190 222
Evolute      15
Exponential mapping      231 285
Exterior derivative      168
Fabricius-Bjerre, Fr.      48
Fenchel, W.      46
Fermi coordinates      161
First fundamental form      59
Flow      235
Focal curve      15
Four Vertex Theorem      45
Free motion      275
Frenet curvature      27
Frenet curve      13
Frenet equations      14 17 36
Frenet matrix      26 27
Frenet n-frame      13
Frobenius, G.      155
Gauss equation      147 150 154 170 198 267 272 365
Gauss formula      140
Gauss lemma      287
Gauss map      64 67
Gauss, C. F.      148 326
Gauss-Bonnet formula      173 326
Gauss-Kronecker curvature      125
Gaussian curvature      73 119 148 195 248
Geodesic      72 123 141 220 229 230 285 312
Geodesic coordinates      277
Geodesic curvature      23 72 127 172
Geodesic parallel coordinates      160
Geodesic polar coordinates      288 295
Geodesic torsion      127
Geodesic triangle      177
Geometric linearization      8 55
Golden ratio      308
Gradient      246 319
Gram determinant      183
Gram-Schmidt orthogonalization      13
Graph      58 74
Groups of motion      274
Harmonic function      102
Hausdorff separation axiom      208
Helicoid      87 112 157
Helicoidal ruled surface      87 95
Helix      9 20
Henneberg, L.      113
Hesse tensor      246
hessian      74 246 350
Hessian matrix      74
Hexagonal torus      299
Hilbert, D.      192 318 323
Hilbert-Einstein functional      318
Hodge operator      356 359
Holomorphic      103 107
Holonomy group      232
Homogeneous space      332
Hopf, H.      41 181
Hyperbola      35 76
Hyperbolic plane      121 122 198
Hyperbolic point      73
Hyperbolic space      271 273 297
Hyperboloid      74 84 86 115 129 271
Hyperboloid type      82
Hyperplane      126
Hypersphere      126
Hypersurface element      124
Icosahedral group      307
icosahedron      308
Immersion      3 5 8 55
Implicit function      3
INDEX      126
Index form      282
Inflection point      14 48
Inner product      2 218
Instantaneous speed      8
Integrability conditions      146 147 150 155 170 353
Inverse mapping      4
Irreducible      333
Isometric      162
Isometry      221 295
Isometry group      332
Isothermal      101
Isotropic      34 116 132
Isotropy group      332
Jacobi determinant      65
Jacobi equation      289
Jacobi field      289 291 312
Jacobi identity      224 249
Jacobian      3
Klein bottle      205 222
Koszul formula      349
Kuiper, N. H.      187
Lagrange multiplier      72
Laplace-Beltrami operator      258
Laplacian      258
Length preserving      162
Lens space      310 314
Level point      73 106 107
Levi-Civita connection      226
Lie algebra      232
Lie bracket      138 223 236
Lie derivative      224
Lie group      232
Lie, S.      223
Liebmann, H.      46 191 193
Light-cone      34 115
Light-like      34
Light-like line      35
Line      9 85 123
Line of curvature      77
Lines of curvature parameters      77 105
Locally compact      208
Locally isometric      256
Logarithmic spiral      51
Lorentz group      274
Lorentz rotation      119
Lorentz space      34 274
Lorentz transformation      276
Lorentzian metric      219 222
Manifold      203
Maurer-Cartan equations      170
Mean curvature      73 99 125 130
Mean curvature vector      102
Measure tensor      240
Mercator projection      128
Meridian curve      78
Meromorphic      107 209
Metric tensor      240
Meusnier, M.      72
Minimal surface      99 132
Minkowski space      33 114 219 274
Mobius strip      65
Mobius, A.      66
Monge coordinates      74
Monge surface      132
Monge, G.      132
Monkey saddle      74
Multilinear      242
Multiplicity      312
Neil parabola      20
Non-Euclidean geometry      122
Norm      2
Normal coordinates      285
Normal curvature      72 172
Normal plane      20
Normal section      72
Normal space      6 56
Normal variation      98
Normal vector      14 57
Null vector      34 270
Null-cone      115
Oblate sphere      82
Octahedral group      307
Octahedral space      310
octahedron      308
Orientability      64
Orthogonal group      205 274
Osculating plane      20 72
Osculating sphere      21
Ovaloid      184
parabola      20
Parabola of contact      49
Parabolic point      73
Paraboloid      74 129
parallel      141 229 230
Parallel displacement      142 230
Parallel surface      130
PARAMETER      56
Parameter of distribution      87
Parameter transformation      65
Parametrization      5 56 202
Parametrized curve      8
Partition of unity      221
Petrov type      364
Pfaffian form      167
Poincare upper half-plane      196 220 312
Polar angle      42
Polar angle function      38
Polar coordinates      37 277
Polarization      252
Position vector      63
Potential equation      351
primitive      105
Principal curvature      72 125 259 265
Principal normal      17 195
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