Abstract: Ultralimits provide a way to talk about the "large-scale geometry" of metric spaces. In the case of the universal cover of a closed Riemannian manifold of non-positive curvature, I will explain how existence of flats in the ultralimit imply existence of parallel Jacobi fields along certain geodesics. As an application, I will show the following rigidity result: if such an ultralimit is isometric to the ultralimit of a higher rank symmetric space of non-compact type, then the original closed Riemannian manifold is itself isometric to a locally symmetric space. This is work in progress with Stefano Francaviglia (UAB - Spain).