Geometry of Diffeomorphism Groups, Complete Integrability and Optimal Transport

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diffμ(M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev Ḣ-metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler-Arnold equation is a completely integrable system in any space dimension. We also prove that its smooth solutions break down in finite time. Furthermore, we show that the Ḣ-metric induces the Fisher-Rao (information) metric on the space of probability distributions, and thus its Riemannian distance is the spherical version of Hellinger distance. We compare it to the Wasserstein distance in optimal transport which is induced by an L-metric on Diff(M). The Ḣ geometry we introduce in this paper can be seen as an infinite-dimensional version of the geometric theory of statistical manifolds. AMS Subject Classification (2000): 53C21, 58D05, 58D17.