  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 Butler's result in dimension two.
Using completely different techniques, we also prove an extension of Butler's
result to the finite volume case, where the value of the Lyapunov exponents
along all periodic orbits is constant, being the maximum or minimum possible.
