UNIFORM FINSLER HADAMARD MANIFOLDS

The subject of this work are reversible uniform Finsler Hadamard manif olds, the Finsler analogues of simply connected Riemannian manifolds o f nonpositive curvature. We introduce asymptotic geodesics, the geodes ic ray boundary and study visibility, introduced by P. Eberlein, and d elta-hyperbolicity in the sense of M. Gromov. In Finsler geometry shar p comparison statments, such as the Aleksandrov-Toponogov comparison t heorem, do not exist. Hence, the synthetic methods developed for Aleks androv spaces of bounded curvature can not be used to study Finsler ma nifolds. To apply techniques developed in Riemannian geometry we face the problem to integrate Jacobi field estimates. Unfortunately, this i ntegration process only leads to ''coarse'' estimates of the Finsler d istance. However, under the hypothesis of nonpositive curvature these ''coarse'' distance estimates are sufficient to establish a satisfacto ry theory of uniform Finsler Hadamard manifolds, extending thereby man y results already known in the Riemannian situation.