Rigidity of flat holonomies

TL;DR

This paper demonstrates that under certain geometric conditions involving horospheres and holonomies, a negatively curved manifold must be hyperbolic, revealing a rigidity phenomenon in geometric structures.

Contribution

It establishes a new rigidity result linking horosphere holonomy properties to the hyperbolic nature of the manifold.

Findings

Existence of a horosphere with specific holonomy conditions implies hyperbolicity.

Stable holonomy coinciding with parallel transport characterizes hyperbolic manifolds.

The result applies to strictly quarter pinched negatively curved manifolds of dimension at least 3.

Abstract

We prove that the existence of one horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension $n\geq 3$ on which the stable holonomy along minimizing geodesics coincide with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.