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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 | |