Katok's entropy rigidity conjecture in dimension at least three

Theorem 1.1: for a closed connected smooth Riemannian manifold of dimension $n\ge3$ with strictly negative sectional curvature, normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric (real, complex, quaternionic or Cayley hyperbolic, any scale). When the metric is not locally symmetric, the measure of maximal entropy and Liouville measure are mutually singular. It does not treat nonpositive curvature, Anosov geodesic flows without negative curvature, or the related volume-entropy (Besson-Courtois-Gallot) setting.

Theorem 1.1: for a closed connected smooth Riemannian manifold of dimension $n\ge3$ with strictly negative sectional curvature, normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric (real, complex, quaternionic or Cayley hyperbolic, any scale). When the metric is not locally symmetric, the measure of maximal entropy and Liouville measure are mutually singular. It does not treat nonpositive curvature, Anosov geodesic flows without negative curvature, or the related volume-entropy (Besson-Courtois-Gallot) setting.