Fisher information metric and poisson kernels, Differential Geometry and its

A complete Riemannian manifold X with negative curvature sat-isfying −b2 ≤ KX ≤ −a2 < 0 for some constants a, b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is em-bedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to con-stant provided X is a rank one symmetric space of noncompact type. Furthermore, we give a geometric meaning of the embedding