MANIFOLDS ALL OF WHOSE FLATS ARE CLOSED

Let M be a locally irreducible Riemannian manifold such that (almost) any geodesic be contained in a compact immersed flat of dimension at l east 2. The main result in this article implies that M is a locally sy mmetric space of the compact type. Therefore, our result gives a chara cterization of locally symmetric spaces of the compact type and higher rank in terms of flats.