CURVATURE AND GLOBAL RIGIDITY IN FINSLER MANIFOLDS

Abstract

Abstract. We present some strong global rigidity results for reversible Finsler manifolds. Following E ́ Cartan’s definition (1926), a locally symmet-ric Finsler metric is one whose curvature is parallel. These spaces strictly contain the spaces such that the geodesic reflections are local isometries and also constant curvature manifolds. In the case of negative curvature, we prove that the locally symmetric Finsler metrics on compact manifolds are Riemannian and this, therefore, extends A. Zadeh’s rigidity result. Our ap-proach uses dynamical properties of the flag curvature. We also give a full generalization of the Ossermann Sarnak minoration of the metric entropy of the geodesic flow. In positive curvature, we just announce some partial results and remarks concerning Finsler metrics of curvature +1 on the 2-sphere. We show that in the reversible case the geodesic flow is conjugate to the standard one. We also observe that a condition of integral geometry (of Radon type) forces such a metric to be Riemannian. This indicates a deep