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Preface | |
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Contents | |
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Curves in the plane and in space | |
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What is a curve? | |
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Arc-length | |
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Reparametrization | |
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Closed curves | |
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Level curves versus parametrized curves | |
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How much does a curve curve? | |
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Curvature | |
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Plane curves | |
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Space curves | |
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Global properties of curves | |
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Simple closed curves | |
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The isoperimetric inequality | |
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The four vertex theorem | |
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Surfaces in three dimensions | |
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What is a surface? | |
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Smooth surfaces | |
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Smooth maps | |
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Tangents and derivatives | |
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Normals and orientability | |
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Examples of surfaces | |
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Level surfaces | |
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Quadric surfaces | |
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Ruled surfaces and surfaces of revolution | |
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Compact surfaces | |
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Triply orthogonal systems | |
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Applications of the inverse function theorem | |
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The first fundamental form | |
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Lengths of curves on surfaces | |
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Isometries of surfaces | |
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Conformal mappings of surfaces | |
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Equiareal maps mid a theorem of Archimedes | |
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Spherical geometry | |
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Curvature of surfaces | |
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The second fundamental form | |
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The Gauss and Weingarten maps | |
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Normal and geodesic curvatures | |
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Parallel transport and covariant derivative | |
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Gaussian, mean and principal curvatures | |
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Gaussian and mean curvatures | |
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Principal curvatures of a surface | |
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Surfaces of constant Gaussian curvature | |
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Flat surfaces | |
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Surfaces of constant mean curvature | |
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Gaussian curvature of compact surfaces | |
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Geodesics | |
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Definition and basic properties | |
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Geodesic equations | |
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Geodesics on surfaces of revolution | |
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Geodesics as shortest paths | |
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Geodesic coordinates | |
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Gauss' Theorema Egregium | |
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The Gauss and Codazzi-Mainardi equations | |
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Gauss' remarkable theorem | |
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Surfaces of constant Gaussian curvature | |
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Geodesic mappings | |
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Hyperbolic geometry | |
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Upper half-plane model | |
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Isometries of H | |
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Poincar� disc model | |
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Hyperbolic parallels | |
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Beltrami-Klein model | |
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Minimal surfaces | |
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Plateau's problem | |
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Examples of minimal surfaces | |
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Gauss map of a minimal surface | |
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Conformal parametrization of minimal surfaces | |
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Minimal surfaces and holomorphic functions | |
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The Gauss-Bonnet theorem | |
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Gauss-Bonnet for simple closed curves | |
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Gauss-Bonnet for curvilinear polygons | |
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Integration on compact surfaces | |
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Gauss-Bonnet for compact surfaces | |
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Map colouring | |
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Holonomy and Gaussian curvature | |
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Singularities of vector fields | |
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Critical points | |
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Inner product spaces and self-adjoint linear maps | |
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Isometries of Euclidean spaces | |
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M�bius transformations | |
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Hints to selected exercises | |
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Solutions | |
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Index | |