Katok's entropy rigidity conjecture

Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.

Manuscript

Entropy equality and local symmetry in negative curvature

September 23, 2026 · source · cite as \cite{OAI:Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026}

For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok's entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.

Result 339 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.