"Geodesics on Groups of Diffeomorphisms"
We begin by constructing a Banach manifold structure on the set of maps from a compact manifold M to another manifold N. Given Riemannian metrics on the two manifolds, we can find geodesics on this space of maps. If M = N, we can look at the group of diffeomorphisms, Diff, of M, which is an open subset of this space of maps.
Next we look at SDiff, the diffeomorphisms that preserve the volume element of M, a subgroup and submanifold of Diff. Using the submanifold structure and its 2nd fundamental form, we find geodesics on SDiff. These correspond to motions of an incompressible inviscid fluid in M.
Finally we assume that M has a symplectic form and using it we define SymDiff, the group of symplectic diffeomorphisms. We construct geodesics on this group as well.
In the case that M is two-dimensional, SDiff and SymDiff coincide and we can show geodesic completeness. Geodesic completeness for SymDiff in the higher dimensional case is work in progress.
The work on SymDiff is new, but the other work goes back over several decades and involves a number of authors.