Elementary Differential Geometry

Indeholder "Preface", " Preface to the Second Edition", "1. Curves in the plane and in space", " 1.1 What is a curve?", " 1.2 Arc-length", " 1.3 Reparametrization", " 1.4 Closed curves", " 1.5 Level curves versus parametrized curves", "2. How much does a curve curve?", " 2.1 Curvature", " 2.2 Plane curves", " 2.3 Space curves", "3. Global properties of curves", " 3.1 Simple closed curves", " 3.2 The isoperimetric inequality", " 3.3 The four vertex theorem", "4. Surfaces in three dimensions", " 4.1 What is a surface?", " 4.2 Smooth surfaces", " 4.3 Smooth maps", " 4.4 Tangents and derivatives", " 4.5 Normals and orientability", "5. Examples of surfaces", " 5.1 Level surfaces", " 5.2 Quadric surfaces", " 5.3 Ruled surfaces and surfaces of show more revolution", " 5.4 Compact surfaces", " 5.5 Triply orthogonal systems", " 5.6 Applications of the inverse function theorem", "6. The first fundamental form", " 6.1 Lengths of curves on surfaces", " 6.2 Isometries of surfaces", " 6.3 Conformal mappings of surfaces", " 6.4 Equiareal maps and a theorem of Archimedes", " 6.5 Spherical geometry", "7. Curvature of surfaces", " 7.1 The second fundamental form. 158. 8 Gaussian mean and principal curvatures", " 7.2 The Gauss and Weingarten maps", " 7.3 Normal and geodesic curvatures", " 7.4 Parallel transport and covariant derivative", "8. Gaussian, mean and principal curvatures", " 8.1 Gaussian and mean curvatures", " 8.2 Principal curvatures of a surface", " 8.3 Surfaces of constant Gaussian curvature", " 8.4 Flat surfaces", " 8.5 Surfaces of constant mean curvature", " 8.6 Gaussian curvature of compact surfaces", "9. Geodesics", " 9.1 Definition and basic properties", " 9.2 Geodesic equations", " 9.3 Geodesics on surfaces of revolution", " 9.4 Geodesics on shortest paths", " 9.5 Geodesic coordinates", "10. Gauss' Theorema Egregium", " 10.1 The Gauss and Codazzi-Mainardi equations", " 10.2 Gauss's remarkable theorem", " 10.3 Surfaces of constant Gaussian curvature", " 10.4 Geodesic mappings", "11. Hyperbolic geometry", " 11.1 Upper half-plane model", " 11.2 Isometries of H", " 11.3 Poincaré disc model", " 11.4 Hyperbolic parallels", " 11.5 Beltrami-Klein model", "12. Minimal surfaces", " 12.1 Plateau's problem", " 12.2 Examples of minimal surfaces", " 12.3 Gauss map of a minimal surface", " 12.4 Conformal parametrization of minimal surfaces", " 12.5 Minimal surfaces and holomorphic functions", "13. The Gauss-Bonnet theorem", " 13.1 Gauss-Bonnet for simple closed curves", " 13.2 Gauss-Bonnet for curvilinear polygons", " 13.3 Integration on compact surfaces", " 13.4 Gauss-Bonnet for compact surfaces", " 13.5 Map colouring", " 13.6 Holonomy and Gaussian curvature", " 13.7 Singularities of vector fields", " 13.8 Critical points", "A0. Inner product spaces and self-adjoint linear maps", "A1. Isometries of Euclidean spaces", "A2. Möbius transformations", "Hints to selected exercises", "Solutions", "Index".

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