Consequences of ergodic frame flow for rank rigidity in negative curvature

Abstract

This paper presents a rank rigidity result for negatively curved spaces. Let $M$ be a compact manifold with negative sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-Λ^2 < K < -λ^2$ where $λ/Λ> .93$. When $a$ is extremal, i.e. $-a^2$ is the curvature minimum or maximum for the manifold, this result is analogous to rank rigidity results in various other curvature settings where higher rank implies that the space is locally symmetric. In particular, this result is the first positive result for lower rank (i.e. when $-a^2$ is minimal), and in the upper rank case gives a shorter proof of the hyperbolic rank rigidity theorem of Hamenstädt, subject to the pinching condition in even dimension. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. Our proof of the main theorem uses the ergodic theory of the frame flow developed by Brin and others - in particular the transitivity group associated to this flow.