On Horospherical Rigidity

TL;DR

This paper establishes intrinsic geometric conditions on horospheres in negatively curved manifolds that ensure the manifold's sectional curvature is constant, contributing to the understanding of rigidity phenomena in differential geometry.

Contribution

It introduces new intrinsic conditions on horospheres that imply global curvature rigidity in higher-dimensional negatively curved manifolds.

Findings

Intrinsic conditions on horospheres guarantee constant sectional curvature.

Conditions lead to rigidity results in negatively curved Riemannian manifolds.

Results apply to manifolds of dimension ≥ 3.

Abstract

We provide intrinsic conditions on the geometry of horospheres in a closed, negatively curved Riemannian manifold of dimension greater than or equal to 3, which guarantee that the sectional curvature is constant.