## Description

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** **Theory of Space Curves : Curves in the planes and in space, arc length, reparametrization, curvature, Serret-Frenet formulae. osculating circles, evolutes and involutes of curves, space curves, torsion, Serret-Frenet formulae. Theory of Surfaces, smooth surfaces, tangents, normals and orientability, quadric surfaces, the first and the second fundamental forms, Euler’s theorem. Rodrigue’s formula. Gaussian Curvature, Gauss map and Geodesics : The Gaussian and mean curvatures, the pseudosphere, flat surfaces, surfaces of constant mean curvature**.**

Gaussian curvature of compact surfaces, the Gauss map, Geodesics, geodesic equations, geodesics of surfaces of revolution, geodesics as shortest paths, geodesic coordinates.Minimal Surfaces and Gauss’s Remarkable Theorem: Plateau’s problem, examples of minimal surfaces, Gauss map of a minimal surface, minimal surfaces and holomorphic functions, Gauss’s Remarkable Theorem, isometries of surfaces, The Codazzi-Mainardi Equations, compact surface of constant Gaussian curvature