The pressure of Ricci curvature

Let (M-n; g) be a closed Riemannian manifold and let k(0) be any positive upper bound for the sectional curvature. We prove that P(r(g)/2rootk(0)) less than or equal to n-1/2 rootk(0), where P(f) stands for the topological pressure of a function f on the unit sphere bundle SM and r(g)(v) is the Ricci curvature in the direction of v is an element of SM. This result gives rise to several estimates for the various entropies of the geodesic flow which in turn have several consequences. One of them is entropy rigidity for those metrics in a hyperbolic manifold whose normalized total scalar curvature is bigger than that of the hyperbolic metric.