  For a given manifold $M$ we consider the non-linear Grassmann manifold
$Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$
gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral,
the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$.
Using this $S^1$-bundle one obtains central extensions for certain groups of
diffeomorphisms of $M$. We can realize $Gr_{m-2}(M)$ as coadjoint orbits of the
extended group of exact volume preserving diffeomorphisms and the symplectic
Grassmannians $SGr_{2k}(M)$ as coadjoint orbits in the group of Hamiltonian
diffeomorphisms. We also generalize the vortex filament equation as a
Hamiltonian equation on $Gr_{m-2}(M)$.
