Riemannian geometry on non-parametric probability space

The family P-lambda of absolutely continuous probabilities w.r.t. the sigma-finite measure lambda is equipped with a structure of an infinite dimensional Riemannian manifold modeled on a real Hilbert. Firstly, the relation between the Hellinger distance and the Fischer metric is analysed on the positive cone M-lambda(+) of bounded measures absolutely continuous w.r.t. lambda, appearing as a flat Riemannian manifold. Secondly, the statistical manifold P-lambda is seen as a submanifold of M-lambda(+) and Amari-Chensov alpha-connections are derived. Some alpha-self-parallel curves are explicitely exhibited.