Geometry of Fisher Information Metric and the Barycenter Map

Geometry of Fisher metric and geodesics on a space of probability measures defined on a compact manifold is discussed and is applied to geometry of a barycenter map associated with Busemann function on an Hadamard manifold X. We obtain an explicit formula of geodesic and then several theorems on geodesics, one of which asserts that any two probability measures can be joined by a unique geodesic. Using Fisher metric and thus obtained properties of geodesics, a fibre space structure of barycenter map and geodesical properties of each fibre are discussed. Moreover, an isometry problem on an Hadamard manifold X and its ideal boundary partial differential X-for a given homeomorphism phi of partial differential X find an isometry of X whose partial differential X-extension coincides with phi-is investigated in terms of the barycenter map.