A conjecture concerning the exponential map on Dμ(M)

It is known that solutions of the Euler equations of hydrodynamics correspond to geodesics on the group of volume–preserving diffeomorphisms of a compact manifold. We conjecture that, regardless of the dimension of the manifold, the associated Riemannian exponential map on the group is nonlinear Fredholm of index zero. Such a result has been established for the Riemannian exponential maps of natural Sobolev metrics on loop spaces and loop groups.