Closed, negatively curved Riemannian manifolds with a flat horosphere are hyperbolic

TL;DR

This paper proves that if a closed, negatively curved Riemannian manifold with strict quarter pinching has a flat horosphere in its universal cover, then it must be hyperbolic, revealing a strong geometric rigidity.

Contribution

It establishes a new rigidity result linking the existence of a flat horosphere to the manifold being hyperbolic in the context of strictly quarter pinched negatively curved manifolds.

Findings

Presence of a flat horosphere implies the manifold is hyperbolic.

The result applies to manifolds of dimension n ≥ 3.

Provides a rigidity criterion for negatively curved manifolds.

Abstract

We prove that the existence of one flat horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension n with n greater than or equal to 3, implies that the manifold is homothetic to a real hyperbolic manifold.