  We study geodesic equations for a family of right-invariant Riemannian
metrics on the group of diffeomorphisms of a compact manifold. The metrics
descend to Fisher's information metric on the space of smooth probability
densities. The right reduced geodesic equations are higher-dimensional
generalisations of the $\mu$--Hunter--Saxton equation, used to model liquid
crystals under influence of magnetic fields. Local existence and uniqueness
results are established by proving smoothness of the geodesic spray.
  The descending property of the metrics is used to obtain a novel
factorisation of diffeomorphisms. Analogous to the polar factorisation in
optimal mass transport, this factorisation solves an optimal information
transport problem. It can be seen as an infinite-dimensional version of $QR$
factorisation of matrices.
