Metric spaces and their role in geometry

Karsten Grove

We will define and discuss a coarse notion of distance between
compact metric spaces introduced by Gromov. Among the few properties preserved by the process of taking limits relative to this distance is a notion of a lower curvature bound and the notion of length space. Both notions will be explained. We will see that (finite
dimensional) length spaces with a lower curvature bound -Alexandrov spaces - have a surprisingly rich structure. I will indicate how Alexandrov geometry in several ways has strong applications to classical Riemannian geometry.