Geometry of Diffeomorphism Groups, Complete Integrability and Geometric Statistics

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 diffeomorphisms, 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 whose smooth solutions break down in finite time. We also show that the Ḣ-metric induces the Fisher–Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.