Preface Contents 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 revolution 5.4 Compact surfaces 5.5 Triply orthogonal systems 5.6 Applications of the inverse function theorem 6.The flrst 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 Sphericalgeometry 7.Curvature of 8urfaces 7.1 The second fundamental form 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 as shortest paths 9.5 Geodesic coordinates 10.Gauss' Theorema Egregium 10.1 The Gauss and Codazzi-Mainardi equations 10.2 Gauss' 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 Poincare disc model 11.4 Hyperbolic parallels 11.5 Beltrami-Klein model 12.Minmal 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.Mobius transformations Hints to selected exercises Solutions Index
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