Some remarks on the simplicial volume of nonpositively curved manifolds

TL;DR

This paper proves that certain nonpositively curved manifolds have positive simplicial volume, using geometric conditions, and provides a new proof of Gromov's conjecture in three dimensions.

Contribution

It establishes positivity of simplicial volume under specific curvature or rank conditions and offers a differential geometric proof of Gromov's conjecture in dimension three.

Findings

Manifolds with certain curvature conditions have positive simplicial volume.

A differential geometric proof of Gromov's conjecture in dimension three is provided.

The results connect geometric properties with topological invariants.

Abstract

We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature condition has positive simplicial volume. We use this to provide a differential geometric proof of a conjecture of Gromov in dimension three.