  We study the geometry of the space of densities $\VolM$, which is the
quotient space $\Diff(M)/\Diff_\mu(M)$ of the diffeomorphism group of a compact
manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with
a right-invariant homogeneous Sobolev $\dot{H}^1$-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 $\dot{H}^1$-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
$L^2$-metric on $\Diff(M)$. The $\dot{H}^1$ geometry we introduce in this paper
can be seen as an infinite-dimensional version of the geometric theory of
statistical manifolds.
