# Hyperbolic rank rigidity for manifolds of $\frac14$-pinched negative curvature

## TL;DR

This paper proves that closed manifolds with sectional curvatures in [-1,-1/4] and higher hyperbolic rank are locally symmetric of rank one, extending and complementing existing rank rigidity results using new methods.

## Contribution

It establishes a new hyperbolic rank rigidity theorem for manifolds with sectional curvatures in [-1,-1/4], partially extending previous work by Constantine.

## Key findings

- Manifolds with higher hyperbolic rank and curvatures in [-1,-1/4] are locally symmetric of rank one.
- The result extends Constantine's work with different methods.
- It provides a partial converse to Hamenstädt's hyperbolic rank rigidity theorem.

## Abstract

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenst\"{a}dt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.