Differential Geometric Heuristics for Riemannian Optimal Mass Transportation

We give an account on Otto’s geometrical heuristics for realizing, on a compact Riemannian manifold M , the L Wasserstein distance restricted to smooth positive probability measures, as a Riemannian distance. The Hilbertian metric discovered by Otto is obtained as the base metric of a Riemannian submersion with total space, the group of diffeomorphisms of M equipped with the Arnol’d metric, and projection, the push-forward of a reference probability measure. The expression of the horizontal constant speed geodesics (time dependent optimal mass transportation maps) is derived using the Riemannian geometry of M as a guide.