On Finsler manifolds of negative flag curvature

Abstract

One of the key differences between Finsler metrics and Riemannian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p, q) is not necessarily equal to d(q, p). In this paper, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan–Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let φ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenstädt.