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Elementary Differential Geometry



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Springer Verlag
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Customer Reviews

Good Introduction  August 17, 2011

After trying several others, I found this textbook the best for individual study. The worked out solutions to all exercises gives you a good way to check your understanding. The writing is clear and the textbook has adequate illustrations to help you see what's going on. I also liked presentation in Banchoff's book, Differential Geometry of Curves and Surfaces, but it has no answers or solutions to the exercises.

Elementary Differential Geometry: 5 out of 5 stars based on 1 user reviews.


The Springer Undergraduate Mathematics Series (SUMS) is designed for undergraduates in the mathematical sciences. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully-worked solutions.

Table of Contents

Curves in the plane and in space
What is a curve?p. 1
Arc-lengthp. 9
Reparametrizationp. 13
Closed curvesp. 19
Level curves versus parametrized curvesp. 23
How much does a curve curve?
Curvaturep. 29
Plane curvesp. 34
Space curvesp. 46
Global properties of curves
Simple closed curvesp. 55
The isoperimetric inequalityp. 58
The four vertex theoremp. 62
Surfaces in three dimensions
What is a surface?p. 67
Smooth surfacesp. 76
Smooth mapsp. 82
Tangents and derivativesp. 85
Normals and orientabilityp. 89
Examples of surfaces
Level surfacesp. 95
Quadric surfacesp. 97
Ruled surfaces and surfaces of revolutionp. 104
Compact surfacesp. 109
Triply orthogonal systemsp. 111
Applications of the inverse function theoremp. 116
The first fundamental form
Lengths of curves on surfacesp. 121
Isometries of surfacesp. 126
Conformal mappings of surfacesp. 133
Equiareal maps mid a theorem of Archimedesp. 139
Spherical geometryp. 148
Curvature of surfaces
The second fundamental formp. 159
The Gauss and Weingarten mapsp. 162
Normal and geodesic curvaturesp. 165
Parallel transport and covariant derivativep. 170
Gaussian, mean and principal curvatures
Gaussian and mean curvaturesp. 179
Principal curvatures of a surfacep. 187
Surfaces of constant Gaussian curvaturep. 196
Flat surfacesp. 201
Surfaces of constant mean curvaturep. 206
Gaussian curvature of compact surfacesp. 212
Definition and basic propertiesp. 215
Geodesic equationsp. 220
Geodesics on surfaces of revolutionp. 227
Geodesics as shortest pathsp. 235
Geodesic coordinatesp. 242
Gauss' Theorema Egregium
The Gauss and Codazzi-Mainardi equationsp. 247
Gauss' remarkable theoremp. 252
Surfaces of constant Gaussian curvaturep. 257
Geodesic mappingsp. 263
Hyperbolic geometry
Upper half-plane modelp. 270
Isometries of Hp. 277
Poincaré disc modelp. 283
Hyperbolic parallelsp. 290
Beltrami-Klein modelp. 295
Minimal surfaces
Plateau's problemp. 305
Examples of minimal surfacesp. 312
Gauss map of a minimal surfacep. 320
Conformal parametrization of minimal surfacesp. 322
Minimal surfaces and holomorphic functionsp. 325
The Gauss-Bonnet theorem
Gauss-Bonnet for simple closed curvesp. 335
Gauss-Bonnet for curvilinear polygonsp. 342
Integration on compact surfacesp. 346
Gauss-Bonnet for compact surfacesp. 349
Map colouringp. 357
Holonomy and Gaussian curvaturep. 362
Singularities of vector fieldsp. 365
Critical pointsp. 372
Inner product spaces and self-adjoint linear maps
Isometries of Euclidean spaces
Möbius transformations
Hints to selected exercises
Table of Contents provided by Ingram. All Rights Reserved.

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