A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces
Keyword(s):
In this paper we prove the following rigidity result: let ${\it\varphi}$ be a $C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are $C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, ${\it\varphi}$ is $C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.