Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one

TL;DR

This paper proves that closed, nonpositively curved manifolds with certain totally geodesic submanifolds have positive simplicial volume, linking geometric structures to topological invariants.

Contribution

It establishes a new connection between the existence of isolated, totally geodesic submanifolds and the positivity of simplicial volume in nonpositively curved manifolds.

Findings

Presence of a codimension one totally geodesic submanifold implies positive simplicial volume.

Universal covers with codimension one flats lead to either Euclidean factors or positive simplicial volume.

Results apply to closed, nonpositively curved manifolds of dimension at least three.

Abstract

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume.