Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds

TL;DR

This paper investigates the structure of totally geodesic submanifolds in products of non-positively curved manifolds, showing that their abundance implies the manifold is locally symmetric.

Contribution

It establishes a link between the existence of many such submanifolds and the manifold being locally symmetric, under mild irreducibility conditions.

Findings

Infinite or dense totally geodesic submanifolds imply the manifold is locally symmetric.

The result holds under mild irreducibility assumptions.

A stronger version is proven in the universal cover setting.

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 $\widetilde M \times \widetilde M$.