By the end of the course the student will learn 

-The basic notions about differential geometry of curves, local information like torsion, curvature and their computations;

-Fundamental Theorem of Local Theory of curves and surfaces;

-The basic notions about differential geometry of regular surfaces, tangent plane, differentiability of maps between surfaces, the notion of differential, orientability, metric properties, the first fundamental form;

-the notion of Gauss map and its fundamental properties and   the second fundamental form, the role of these in deriving local properties of the surface; Gaussian and mean curvatures, principal curvatures and directions, their computation;

-the role of first fundamental form understanding the intrinsic properties of surfaces; the notion of covariant derivative of a vector field on a surface; parallel transport and geodesics, the statement of Gauss-Bonnet Theorem.