Università Roma Tre

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.