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