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Hsiung C.-C. — A first course in differential geometry
Hsiung C.-C. — A first course in differential geometry

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Название: A first course in differential geometry

Автор: Hsiung C.-C.

Аннотация:

The origins of differential geometry go back to the early days of the differential calculus, when one of the fundamental problems was the determination of the tangent to a curve. With the development of the calculus, additional geometric applications were obtained. The principal contributors in this early period were Leonhard Euler (1707- 1783), GaspardMonge(1746-1818), Joseph Louis Lagrange (1736-1813), and AugustinCauchy (1789-1857). A decisive step forward was taken by Karl FriedrichGauss (1777-1855) with his development of the intrinsic geometryon a surface. This idea of Gauss was generalized to n( > 3)-dimensional spaceby Bernhard Riemann (1826- 1866), thus giving rise to the geometry that bears his name. This book is designed to introduce differential geometry to beginning graduate students as well as advanced undergraduate students (this introduction in the latter case is important for remedying the weakness of geometry in the usual undergraduate curriculum). In the last couple of decades differential geometry, along with other branches of mathematics, has been highly developed. In this book we will study only the traditional topics, namely, curves and surfaces in a three-dimensional Euclidean space E3. Unlike most classical books on the subject, however, more attention is paid here to the relationships between local and global properties, as opposed to local properties only. Although we restrict our attention to curves and surfaces in E3, most global theorems for curves and surfaces in this book can be extended to either higher dimensional spaces or more general curves and surfaces or both. Moreover, geometric interpretations are given along with analytic expressions. This will enable students to make use of geometric intuition, which is a precious tool for studying geometry and related problems; such a tool is seldom encountered in other branches of mathematics.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1981

Количество страниц: 361

Добавлена в каталог: 17.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian group      24
Accumulation point      5
Admissible arc      301
Affine group      53
Affine transformation      52
Affine transformation, isometric      56
Angle, between curves on surface      199
Angle, between vectors      25
Angle, extenor      252
Angle, interior      252
Angle, interior, of geodesic triangle      264
Antipodal, mapping      246
Antipodal, point      303
Arc length      83 84 88
Arc length, of curve on surface      176
Arc length, reparametrization of curve by      84
Area, element of      271
Associated Bertrand curves      93
Asymptotic curve      192—193
Asymptotic curve, of developable surface      208
Asymptotic direction      184
Axiom of Completeness      9
Axis of helix      79
Axis of revolution      158
Basis, natural      28
Basis, of a space      28
Basis, orthonormal      28
Beltrami — Enneper Theorem      197
Bertrand curve      92—96 97 98
Binormal      91
Binormal, vector      92
Bolzano, intermediate value theorem of      14
Bonnet’s theorems      212 272 305 309
Boundary of set      3
Boundary point of set      3
Bounded set      9
Buffon’s needle problem      128 130
Cartesian product      3
Catenary      168 220
Catenoid      168 220
Catenoid, as minimal surface of revolution      220
Cauchy — Crofton formula      128
Cauchy — Riemann equations      202
Cauchy, convergence condition      10
Cauchy, sequence      10
Cauchy’s formula      115
Center of curvature      95 103 104 185
Chain rule      20
Christoffel symbols      210
Christoffel’s problem      286
Christoffel’s problem, uniqueness theorem for      289
Circular helix      95 104
Closed form      71
Closed plane curve      85 86
Closed plane curve, diameter of      113 123
Closed plane curve, exterior of      110
Closed plane curve, interior of      110
Closed set      4
Closure of set      5
Cluster point      5
Cohn — Vossen's theorem      275
Compact space      11
Complement of set      5 38
Complete surface      249 294 298 299 305
Completely integrable system      212
Component, of set      8
Component, of vector      23
Cone      161
Cone, as ruled surface      215
Cone, local isometry of, to plane      198
Confocal quadrics      204 206
Conformal mapping      198
Conformal mapping, Liouville’s theorem for      206
Conformal mapping, local      199
Conformal mapping, local, of planes      202
Conformal mapping, local, of Poincare half-plane to Euclidean plane      251
Conformal mapping, local, of spheres to planes      200 201
Conformal mapping, local, of surfaces      200
Conjugate directions      194
Conjugate directions, net      194 197
Conjugate point      302—305
Connected space      7
Conoid      202
Contact of order $k$ with curve      101
Continuous mapping      6
Convergence of sequence      10
Convex hull      137 289
Convex set      137
Convex surface      275
Convex surface, in direction      284
Convex surface, of constant Gaussian curvature      280
Convex surface, of constant mean curvature      285
Coordinate functions, of curve on surface      172
Coordinate functions, of vector field on $E^3$      32
Coordinate system      151
Coordinate system, isothermal      200 222
Coordinate system, spherical      204
Coordinate, neighborhood      151
Covariant differential      225
Covering      11
Covering, finite      11
Covering, open      11
Covering, sub      11
Critical point      19 180
Crofton's theorem      143
Cubic parabola      104
Curvature, center of      see Center of curvature
Curvature, Gaussian      189 196 211 270
Curvature, Gaussian, intrinsic property of      211 272
Curvature, Gaussian, of torus      194
Curvature, geodesic      299
Curvature, lines of      192 204 206
Curvature, lines of, differential equation of      189
Curvature, mean      189 196 270
Curvature, normal      177 183 184
Curvature, of curve      90 91 96
Curvature, principal      184 189
Curvature, radius of      91 116
Curve, asymptotic      see Asymptotic curve
Curve, Bertrand      see Bertrand curve
Curve, closed      85 86
Curve, continuous      85
Curve, convex      1 13 134 135 137
Curve, curvature of      90 91 96
Curve, Mannheim      98
Curve, of class $C^k$      85
Curve, of constant width      113 114 138
Curve, on surface      171—172
Curve, oriented      89
Curve, periodic      86
Curve, piecewise (sectionally) regular (smooth)      82 86 133
Curve, plane      92 93 136
Curve, rectifiable      83
Curve, regular      82
Curve, reparametrzation of      80—81 83
Curve, reparametrzation of, by arc length      84
Curve, simple      85
Curve, smooth      82 86
Curve, spherical      104
Curve, torsion of      91 92 96
Curve, vertex of      123 138 139
Cusp      79
Cycloid      87
Cylinder      4 157 244
Cylinder, as ruled surface      215
Cylinder, first fundamental form of      179
Cylinder, geodesics of      239
Cylinder, local isometry of, to plane      198
Cylinder, parametrization of      158
Cylindrical helix      97 107
Darboux frame      176
Decomposition of space      38
Deformation of curve      147
Derivative mapping      41
Determinant of three vectors      25
Developable surface      208
Diameter, of closed plane curve      113 123
Diameter, of set      9
Diffeomorphism      43
Diffeomorphism, area-preserving      202
Diffeomorphism, orientation-preserving or exteriorreversing      245
Differential, form      see Form differential
Differential, mapping      41
Differential, of function      64
Dimension of space      28
Direct sum of spaces      38
Direction,asymptotic      184
Direction,principal      184 189
Directional derivative      33 173
Disconnected space      7
Distance, in $E^3$      2 24
Distance, on surface      293—294
Divergence of sequence      10
Dual basis      40 65
Dual space      39
Dupin, indicatrix      184
Dupin, theorem      204
Element, of arc      176
Element, of area      218 271
Ellipsoid      157 186 244
Empty set      4
Enneper's minimal surface      223
Envelope, of family of curves      112
Envelope, of family of tangent planes      227
Equator      154
Equivalent knots      86
Erdmann’s theorem      76
Euclidean coordinate functions, of form      65
Euclidean coordinate functions, of mapping      35
Euclidean coordinate functions, of vector field      30 85
Euclidean group of rigid motions      56
Euclidean space      1
Euler angles      50
Euler characteristic      255 256
Euler equation      76
Euler’s formula      184
Evolute      98
ex      4
Exact form      71
Existence and uniqueness theorem, for system of partial differential equations of first order      309
Existence and uniqueness theorem,for system of ordinary differential equations of first order      307
Existence theorem for curves      105 307
Extenor multiplication      67
Exterior angle      252
Exterior angle, derivative      69
Exterior differential form      64 267
Exterior point      3
Exterior product      69
Exterior, of closed plane curve      110
Extremal curve of integral      76
Extremum      18
Extremum, absolute      20
Extremum, relative (local)      18 20
Extrinsic property      227
Fary — Milnor theorem, H      5
Fenchel's Theorem      140
Field, frame      31
Field, Frenet frame      90
Field, vector      see Vector field
Finite covering      11
First fundamental form      175 176
First fundamental form, coefficients of      177
First variation, of integral      75
First variation, of length      300
Form, closed      71
Form, exact      71
Form, exterior differential      64 267
Form, fundamental      175—178 269
Form, on $E^3$      64
Form, on surface      173 174
Form, structural equations for      75 269
Four-vertex theorem      123
Frame      31
Frame, Darboux      176
Frame, moving      88
Frame, positively or negatively oriented      60 85
Frame, right- or left-handed oriented      60 85
Frenet formulas      90 109
Frenet frame Held      90
Function, continuous      6
Function, harmonic      222
Function, height      145
Function, Laplacian of      222
Function, of class $C^k$      170
Function, periodic      19
Function, regular value of      155 244
Fundamenial forms      175—178 269
Fundamenial forms, relation among      208
Fundamental theorem, for curves      105
Fundamental theorem, for surfaces      212 272 309
Gauss — Bonnet formula, global      257
Gauss — Bonnet formula, local      256
Gauss — Bonnet theorem      256
Gauss — Bonnet theorem, applications of      261—264
Gauss, equations      210 211
Gauss, mapping      176
Gauss, trihedron      176
Gaussian curvature      see Curvature
Gaussian Generalized uniqueness theorem for curves      105
General linear group      47
Genus of surface      256
Geodesic circles      236
Geodesic curvature      229—231 239
Geodesic disk      293
Geodesic parallels      232
Geodesic polar coordinates      234
Geodesic polar coordinates, area of geodesic circle in      237
Geodesic polar coordinates, first fundamental form in      233
Geodesic polar coordinates, Gaussian curvature in      236
Geodesic polar coordinates, perimeter of geodesic circle in      236
Geodesic torsion      237—238
Geodesic torsion, of asymptotic curves      240
Geodesic triangle      264
Geodesic triangle, excess of      264
Geodesics      229
Geodesics, as shortest arcs, in large      298 299
Geodesics, as shortest arcs, in small      233 304
Geodesics, as shortest arcs, of sphere      239
Geodesics, characterizations of      231
Geodesics, closed      261 262
Geodesics, differential equations of      231
Geodesics, existence and uniqueness of      231
Geodesics, field of      232
Geodesics, of circular cylinder      239
Geodesics, of plane      239
Geodesics, of Poincare half-plane      251
Gradient, on $E^3$      72
Gradient, on surface      181
Graph      13
Graph, of $C^3$ function      155 160
Graph, of $C^3$ function, area of      218
Graph, of $C^3$ function, Gaussian curvature of      196
Graph, of $C^3$ function, mean curvature of      196
Graph, of $C^3$ function, principal curvature of      196
Graph, of $C^3$ function, tangent plane of      180
Greatest lower bound      9
Green’s formula      258 268
Group      47
Group, abelian      24
Group, affine      5 3
1 2 3
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