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PrefaceContents1£®Curves in the plane and in space1£®1 What is a curve 1£®2 Arc-length1£®3 Reparametrization1£®4 Closed curves1£®5 Level curves versus parametrized curves2£®How much does a curve curve 2£®1 Curvature2£®2 Plane curves2£®3 Space curves3£®Global properties of curves3£®1 Simple closed curves3£®2 The isoperimetric inequality3£®3 The four vertex theorem4£®Surfaces in three dimensions4£®1 What is a surface 4£®2 Smooth surfaces4£®3 Smooth maps4£®4 Tangents and derivatives4£®5 Normals and orientability5£®Examples of surfaces5£®1 Level surfaces5£®2 Quadric surfaces5£®3 Ruled surfaces and surfaces of revolution5£®4 Compact surfaces5£®5 Triply orthogonal systems5£®6 Applications of the inverse function theorem6£®The flrst fundamental form6£®1 Lengths of curves on surfaces6£®2 Isometries of surfaces6£®3 Conformal mappings of surfaces6£®4 Equiareal maps and a theorem of Archimedes6£®5 Sphericalgeometry7£®Curvature of 8urfaces7£®1 The second fundamental form7£®2 The Gauss and Weingarten maps7£®3 Normal and geodesic curvatures7£®4 Parallel transport and covariant derivative8£®Gaussian£¬ mean and principal curvatures8£®1 Gaussian and mean curvatures8£®2 Principal curvatures of a surface8£®3 Surfaces of constant Gaussian curvature8£®4 Flat surfaces8£®5 Surfaces of constant mean curvature8£®6 Gaussian curvature of compact surfaces9£®Geodesics9£®1 Definition and basic properties9£®2 Geodesic equations9£®3 Geodesics on surfaces of revolution9£®4 Geodesics as shortest paths9£®5 Geodesic coordinates10£®Gauss' Theorema Egregium10£®1 The Gauss and Codazzi-Mainardi equations10£®2 Gauss' remarkable theorem10£®3 Surfaces of constant Gaussian curvature10£®4 Geodesic mappings11£®Hyperbolic geometry11£®1 Upper half-plane model11£®2 Isometries of H11£®3 Poincare disc model11£®4 Hyperbolic parallels11£®5 Beltrami-Klein model12£®Minmal surfaces12£®1 Plateau's problem12£®2 Examples of minimal surfaces12£®3 Gauss map of a minimal surface12£®4 Conformal parametrization of minimal surfaces12£®5 Minimal surfaces and holomorphic functions13£®The Gauss-Bonnet theorem13£®1 Gauss-Bonnet for simple closed curves13£®2 Gauss-Bonnet for curvilinear polygons13£®3 Integration on compact surfaces13£®4 Gauss-Bonnet for compact surfaces13£®5 Map colouring13£®6 Holonomy and Gaussian curvature13£®7 Singularities of vector fields13£®8 Critical pointsA0£®Inner product spaces and self-adjoint linear mapsA1£®Isometries of Euclidean spacesA2£®Mobius transformationsHints to selected exercisesSolutionsIndex