Fisher information metric and Poisson kernels

A complete Riemannian manifold X   with negative curvature satisfying −b2⩽KX⩽−a2<0−b2⩽KX⩽−a2<0 for some constants a,ba,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 embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.