Official Course Description
We show that a curve in R3 is, up to Euclidean motions, totally determined by its curvature and torsion. We study the second fundamental form of a surface, describing its shape in the the ambient space R3. This leads to a fundamental object the curvature of the surface. Amongst many interesting results we prove the famous "Theorema Egregium" of Gauss which tells us that the curvature is an intrinsic object i.e. determined by the way we measure distances on the surface. Furthermore we prove the astonishing Gauss-Bonnet theorem. This implies that for a compact surface the curvature integrated over it is a topological invariant.