Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds

Abstract

We study non-positively curved closed manifolds $M$ and $n$-dimensional totally geodesic submanifolds of $M \times M$ which satisfy a transversality condition. We prove that, under some mild irreducibility requirements on $M$, if $M \times M$ admits infinitely many such submanifolds or just a single dense such submanifold, then $M$ is a locally symmetric space. In proving this, we prove a stronger version which only requires such submanifolds to exist in the universal cover $\widetilde M \times \widetilde M$.