| 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 | |
| Table of Contents provided by Ingram. All Rights Reserved. |
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