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Preface | |
Contents | |
Curves in the plane and in space | |
What is a curve? | p. 1 |
Arc-length | p. 9 |
Reparametrization | p. 13 |
Closed curves | p. 19 |
Level curves versus parametrized curves | p. 23 |
How much does a curve curve? | |
Curvature | p. 29 |
Plane curves | p. 34 |
Space curves | p. 46 |
Global properties of curves | |
Simple closed curves | p. 55 |
The isoperimetric inequality | p. 58 |
The four vertex theorem | p. 62 |
Surfaces in three dimensions | |
What is a surface? | p. 67 |
Smooth surfaces | p. 76 |
Smooth maps | p. 82 |
Tangents and derivatives | p. 85 |
Normals and orientability | p. 89 |
Examples of surfaces | |
Level surfaces | p. 95 |
Quadric surfaces | p. 97 |
Ruled surfaces and surfaces of revolution | p. 104 |
Compact surfaces | p. 109 |
Triply orthogonal systems | p. 111 |
Applications of the inverse function theorem | p. 116 |
The first fundamental form | |
Lengths of curves on surfaces | p. 121 |
Isometries of surfaces | p. 126 |
Conformal mappings of surfaces | p. 133 |
Equiareal maps mid a theorem of Archimedes | p. 139 |
Spherical geometry | p. 148 |
Curvature of surfaces | |
The second fundamental form | p. 159 |
The Gauss and Weingarten maps | p. 162 |
Normal and geodesic curvatures | p. 165 |
Parallel transport and covariant derivative | p. 170 |
Gaussian, mean and principal curvatures | |
Gaussian and mean curvatures | p. 179 |
Principal curvatures of a surface | p. 187 |
Surfaces of constant Gaussian curvature | p. 196 |
Flat surfaces | p. 201 |
Surfaces of constant mean curvature | p. 206 |
Gaussian curvature of compact surfaces | p. 212 |
Geodesics | |
Definition and basic properties | p. 215 |
Geodesic equations | p. 220 |
Geodesics on surfaces of revolution | p. 227 |
Geodesics as shortest paths | p. 235 |
Geodesic coordinates | p. 242 |
Gauss' Theorema Egregium | |
The Gauss and Codazzi-Mainardi equations | p. 247 |
Gauss' remarkable theorem | p. 252 |
Surfaces of constant Gaussian curvature | p. 257 |
Geodesic mappings | p. 263 |
Hyperbolic geometry | |
Upper half-plane model | p. 270 |
Isometries of H | p. 277 |
Poincaré disc model | p. 283 |
Hyperbolic parallels | p. 290 |
Beltrami-Klein model | p. 295 |
Minimal surfaces | |
Plateau's problem | p. 305 |
Examples of minimal surfaces | p. 312 |
Gauss map of a minimal surface | p. 320 |
Conformal parametrization of minimal surfaces | p. 322 |
Minimal surfaces and holomorphic functions | p. 325 |
The Gauss-Bonnet theorem | |
Gauss-Bonnet for simple closed curves | p. 335 |
Gauss-Bonnet for curvilinear polygons | p. 342 |
Integration on compact surfaces | p. 346 |
Gauss-Bonnet for compact surfaces | p. 349 |
Map colouring | p. 357 |
Holonomy and Gaussian curvature | p. 362 |
Singularities of vector fields | p. 365 |
Critical points | p. 372 |
Inner product spaces and self-adjoint linear maps | |
Isometries of Euclidean spaces | |
Möbius transformations | |
Hints to selected exercises | |
Solutions | |
Index | |
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