3 Credits (3-0-0)

Curves in plane and space, arc-length, reparametrization, curvature of a plane cure, curvature and torsion of a space curve Simple closed curves, isoperimetric inequality, Four-vertex theorem Surfaces, smooth surfaces and examples, level surfaces, quadric surfaces, surfaces of revolution, ruled surfaces smooth maps, tangent space, derivatives, orientability of surfaces The first fundamental form, lengths of curves on surfaces, isometries, conformal mappings, equiareal maps The second fundamental form, Gauss and Weingarten maps, normal and geodesic curvatures, Gaussian and mean curvatures, principal curvatures Surfaces of constant Gaussian curvature, surfaces of constant mean curvature, flat surfaces Parallel transport, geodesics and their examples, properties, geodesic equations, geodesics as shortest paths, Gauss and Codazzi-Minardi equations, Theorema Egregium Gauss-Bonnet Theorem. Introduction to hyperbolic and spherical geometry.