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Carmo M.P. — Differential geometry of curves and surfaces
Carmo M.P. — Differential geometry of curves and surfaces

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Название: Differential geometry of curves and surfaces

Автор: Carmo M.P.

Аннотация:

This book is an introduction to the differential geometry of curves and surfaces, both in its local and global aspects. The presentation differs from the traditional ones by a more extensive use of elementary linear algebra and by a certain emphasis placed on basic geometrical facts, rather than on machinery or random details.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1976

Количество страниц: 503

Добавлена в каталог: 23.09.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Acceleration vector      345
Accumulation point      457
Angle between two surfaces      87
Angle, external      266
Angle, interior      274
Antipodal map      80
ARC      462
Arc length      6
Arc length in polar coordinates      25 (Ex. 11)
Arc length, reparametrization by      22
arc, regular      266
Area      98
Area of a graph      100 (Ex. 5)
Area of surface of revolution      101 (Ex. 11)
Area, geometric definition of      115
Area, oriented      15 (Ex. 10) 166
Area-preserving diffeomorphisms      230 (Ex. 18) 231
Asymptotic curve      148
Asymptotic direction      148
Ball      118
Beltrami — Enneper, theorem of      152 (Ex. 13)
Beltrami’s theorem on geodesic mappings      296 (Ex. 13)
Bertrand curve      26
Bertrand mate      26
Binormal line      19
Binormal vector      18
Bolzano — Weierstrass theorem      112 124 466
Bonnet, O.      535
Bonnet’s theorem      352 424
Boundary of a set      459
Braunmuehl, A.      363
Buck, R.C.      43 97 132
Calabi, E.      354
Catenary      23 (Ex. 8)
Catenoid      221
Catenoid as a minimal surface      202
Catenoid, asymptotic curves of      168 (Ex. 3)
Catenoid, local isometry of, with a helicoid      213 (Ex. 14) 223
Cauchy sequence      461
Cauchy sequence in the intrinsic distance      336 (Ex. 5)
Cauchy — Crofton formula      41
Chain rule      91 (Ex. 24) 126 129
Chern and Lashof, R.      387
Chern, S.S.      318
Christoffel symbols      232
Christoffel symbols for a surface of revolution      232
Christoffel symbols in normal coordinates      295 (Ex. 4)
Cissoid of Diodes      7 (Ex. 3)
Clairaut’s relation      257
Closed plane curve      30
Closed set      458
Closure of a set      458
Compact set      112 465
Comparison theorems      369 (Ex. 3)
Compatibility equations      235
Complete surface      325
Cone      64 65 327
Cone as a ruled surface      189
Cone, geodesies of      307 (Ex. 6)
Cone, local isometry of, with plane      223
Conformal map      226
Conformal map, linear      229 (Ex. 13)
Conformal map, local      226 229
Conformal map, local, of planes      229 (Ex. 15)
Conformal map, local, of spheres into planes      230 (Ex. 16)
Conjugate directions      150
Conjugate locus      363
Conjugate minimal surfaces      213 (Ex. 14)
Conjugate points      362
Conjugate points, Kneser criterion for      370 (Ex. 7)
Connected      462
Connected, arcwise      462
Connected, arcwise, locally      464
Connected, component      469
Connected, simply      382
Connected, simply, locally      383
Connection      306 (Ex. 2) 442
Conoid      210 (Ex.5)
Contact of curves      171 (Ex. 9)
Contact of curves and surfaces      171 (Ex. 10)
Contact of surfaces      91 (Ex.27) 170
Continuous map      120
Continuous map, uniformly      468
Convergence      456
Convergence in the intrinsic distance      336 (Ex.4)
Convex curve      37
Convex hull      48 (Ex. 11)
Convex neighborhood      303
Convex neighborhood, existence of      305
Convex set      48 (Ex. 9)
Convexity and curvature      40 174 387 397
Coordinate curves      53
Coordinate neighborhood      53
Coordinate system      52
Courant, R.      115
Covariant derivative      238
Covariant derivative in terms of parallel transport      305 (Ex. 1)
Covariant derivative, algebraic value of      248
Covariant derivative, along a curve      240
Covariant derivative, expression of      239
Covariant derivative, properties of      306 (Ex. 2)
Covering space      371
Covering space, number of sheets of      377
Covering space, orientable double      443 (Exs. 3 4)
Critical point      58 89
Critical point, nondegenerate      173 (Ex.23)
Critical value      58
Cross product      12
Curvature in arbitrary parameters      25 (Ex. 12)
Curvature of a plane curve      21
Curvature of a space curve      16
Curvature, Gaussian      146 155 see
Curvature, geodesic      248 253
Curvature, lines of      145
Curvature, lines of, differential equations of      161
Curvature, mean      146 156 163
Curvature, mean, vector      201
Curvature, normal      141
Curvature, principal      144
Curvature, radius of      19
Curvature, sectional      442
Curve of class $C^k$      10 (Ex. 7)
Curve, asymptotic      148
Curve, asymptotic, differential equations for      160
Curve, asymptotic, maximal      410
Curve, closed      30
Curve, closed, continuous      392
Curve, closed, piecewise regular      266
Curve, closed, simple      30
Curve, coordinate      53
Curve, divergent      336 (Ex. 7)
Curve, knotted      402
Curve, level      102 (Ex. 14)
Curve, level, parametrized      3
Curve, level, piecewise differentiable      329
Curve, level, piecewise regular      244
Curve, level, regular      6
Curve, piecewise $C^l$      35
Curve, simple      10 (Ex.7)
Cut locus      420
Cycloid      7
Cylinder      65 (Ex. 1)
Cylinder, as a ruled surface      188
Cylinder, first fundamental form of      93
Cylinder, isometries of      229 (Ex. 12)
Cylinder, local isometry of, with plane      219
Cylinder, normal sections of      144
Darboux trihedron      261 (Ex. 14)
Degree of a map      390
Developable surface      194 210
Developable surface as the envelope of a family of tangent planes      195
Developable surface, classification of      194
Developable surface, tangent plane of a      210 (Ex. 6)
Diffeomorphism      74
Diffeomorphism, area-preserving      230 (Exs. 18 19)
Diffeomorphism, local      86
Diffeomorphism, local, orientation-preserving      165
Diffeomorphism, local, orientation-reversing      166
Differentiable function      72 80 82 125
Differentiable manifold      438
Differentiable map      73 126 426
Differentiable structure      425 439
Differential of a map      86 127 430
Direction, asymptotic      148
Direction, principal      144
Directions, conjugate      150
Directions, field of      178
Directrix of a ruled surface      188
Distance on a surface      329
Distribution parameter      192
Do Carmo, M. and E. Lima      387
Domain      97
Dot product      4
Dupin indicatrix      148
Dupin indicatrix, geometric interpretation of      164
Dupin’s theorem on triply orthogonal systems      153
Edges of a triangulation      271
Efimov, N.V.      453
Eigenvalue      216
Eigenvector      216
Ellipsoid      61 80 90
Ellipsoid, conjugate locus of      263
Ellipsoid, first fundamental form of      99 (Ex. 1)
Ellipsoid, Gaussian curvature of      173 (Ex. 21)
Ellipsoid, parametrization of      66 (Ex. 12)
Ellipsoid, umbilical points of      172 (Ex. 20)
Embedding      435
Embedding of the Klein bottle into $R^4$      436
Embedding of the projective plane into $R^4$      437
Embedding of the torus into $R^4$      435
Energy of a curve      307 (Ex. 4)
Enneper’s surface      168 (Ex. 5)
Enneper’s surface as a minimal surface      205
Envelope of a family of tangent planes      195 210 212 244 308
Euclid’s fifth axiom      279 431 432
Euler formula      145
Euler — Lagrange equation      365
Euler — Poincare characteristic      272
Evolute      23 (Ex. 7)
Exponential map      284
Exponential map, differentiability of      285
Faces of a triangulation      271
Fary — Milnor Theorem      402
Fenchel’s theorem      399
Fermi coordinates      306 (Ex. 3)
Field of directions      178
Field of directions, differential equation of      179
Field of directions, integral curves of      178
Field of unit normal vectors      104
First fundamental form      92
Flat torus      435
Focal surfaces      210 (Ex. 9)
Folium of Descartes      8 (Ex. 5)
Frenet formulas      19
Frenet trihedron      19
Function, analytic      207
Function, component      120
Function, continuous      119
Function, differentiable      72 125
Function, harmonic      201
Function, height      72
Function, Morse      173 (Ex. 23)
Fundamental theorem for the local theory of curves      19 309
Fundamental theorem for the local theory of surfaces      236 311
Gauss formula      234
Gauss formula in orthogonal coordinates      237 (Ex. 1)
Gauss lemma      288
Gauss map      136
Gauss theorem egregium      234
Gauss — Bonnet theorem (global)      274
Gauss — Bonnet theorem (global), application of      276
Gauss — Bonnet theorem (local)      268
Gaussian curvature      146 155
Gaussian curvature for graphs of differentiable functions      163
Gaussian curvature in terms of parallel transport      270 271
Gaussian curvature, geometric interpretation of      167
Genus of a surface      273
Geodesic circles      287
Geodesic coordinates      306 (Ex. 3)
Geodesic curvature      248 253
Geodesic flow      440
Geodesic mapping      296 (Ex. 12)
Geodesic parallels      306 (Ex. 3)
Geodesic polar coordinates      286
Geodesic polar coordinates, first fundamental form in      287
Geodesic polar coordinates, Gaussian curvature in      288
Geodesic polar coordinates, geodesies in      295 (Ex. 7)
Geodesic torsion      153 (Ex. 19) 261
Geodesics      307
Geodesics as solutions to a variational problem      345
Geodesics of a cone      306 (Ex. 6)
Geodesics of a cylinder      246 247
Geodesics of a paraboloid of revolution      258—260
Geodesics of a sphere      246
Geodesics of the Poincare half-plane      431 432 444
Geodesics, differential equations of      254
Geodesics, existence of      255
Geodesics, minimal      303 332
Geodesics, minimizing properties of      292
Geodesics, of surfaces of revolution      255-258 356
Geodesics, radial      287
Geppert, H.      407
Gluck, H.      41
Gradient on surfaces      101 (Ex. 14)
Graph of a differentiable function      58
Graph of a differentiable function, area of      100 (Ex. 5)
Graph of a differentiable function, Gaussian curvature of      163
Graph of a differentiable function, mean curvature of      163
Graph of a differentiable function, second fundamental form of      163 164
Graph of a differentiable function, tangent plane of      88 (Ex. 3)
Green, L.      363
Gromov, M.L. and V.A. Rokhlin      454
Group of isometries      229 (Ex. 9)
Hadamard’s theorem on complete surfaces with $K\leqO$      387 390
Hadamard’s theorem on ovaloids      387
Hartman, P. and L. Nirenberg      408
Heine — Borel theorem      112 124
Helicoid      94
Helicoid as a minimal surface      204
Helicoid as the only minimal ruled surface      204
Helicoid, asymptotic curves of      168 (Ex. 2)
Helicoid, distribution parameter of      209 (Ex. 1)
Helicoid, generalized      101 (Ex. 13) 186
Helicoid, line of striction of      209 (Ex. 1)
Helicoid, lines of curvature of      168 (Ex. 2)
Helicoid, local isometry of, with a catenoid      213 (Ex. 14) 223
Helicoid, tangent plane of      89 (Ex. 9)
Helix      3 22
Helix, generalized      26 (Ex. 17)
hessian      164 173
Hilbert, D.      318 446
Hilbert’s Theorem      446
Holmgren, E.      446
Holonomy group      297 (Ex. 14)
Homeomorphism      123
Homotopic arcs      378
Homotopy of arcs      378
Homotopy of arcs, free      390 (Ex. 10)
Homotopy of arcs, lifting of      379
Hopf — Rinow’s theorem      333
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