Curvature Free Rigidity for Higher Rank Three-Manifolds

TL;DR

This paper establishes that complete three-manifolds with higher spherical or hyperbolic rank are precisely spherical or hyperbolic space forms, respectively, providing new rigidity results without curvature assumptions.

Contribution

The paper proves rigidity theorems characterizing higher rank three-manifolds as space forms, extending rigidity results to cases without curvature restrictions.

Findings

Higher spherical rank implies the manifold is a spherical space form.

Higher hyperbolic rank implies the manifold is a hyperbolic space form.

Rigidity results hold for complete finite volume three-manifolds.

Abstract

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.