Non-linear Grassmannians as coadjoint orbits

Abstract

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