We announce several results on the structure of the group of diffeomorphisms
of a compact
n
-manifold
M
, possibly with boundary. The group
has the structure of a differentiable manifold modelled on a Fréchet space and with this structure, the group operations are smooth. See Leslie [5] and Omori [8], for the proof in case
M
has no boundary. Following Omori, we call
an ILH Lie group.
We shall show that several infinite dimensional subgroups of
are actually (ILH) submanifolds and hence also have the structure of ILH Lie groups. Also, we construct certain (weak) Riemannian structures on
(and on certain subgroups) and find the geodesic flows associated to them.