Rigidity of Lyapunov exponents for geodesic flows

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then M is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes C. Butler's result [5] in dimension two. Using completely different techniques, we also prove an extension of [5] to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.