The periodic-orbit Lyapunov exponent rigidity conjecture for Anosov geodesic flows

Let $M$) be a complete Riemannian manifold of finite volume whose geodesic flow is Anosov. The unstable Lyapunov exponents are the asymptotic exponential growth rates in unstable directions along the geodesic flow, and a periodic orbit is a closed orbit of that flow.

Periodic-orbit Lyapunov rigidity conjecture. If the unstable Lyapunov exponents are constant across all periodic orbits, then $M$ has constant negative sectional curvature.

The conjecture concerns whether periodic-orbit data for the geodesic flow determines the curvature of the manifold. It was proved in dimension two by Sergio Romaña and N. Nina, and C. Butler established the compact case under negative curvature; the general finite-volume statement remains open.