2004], Non–linear Grassmannians as coadjoint orbits

Abstract. For a given manifold M we consider the non-linear Grassmann manifold Grn(M) of n–dimensional submanifolds inM. A closed (n+2)–form on M gives rise to a closed 2–form on Grn(M). If the original form was inte-gral, the 2–form will be the curvature of a principal S1–bundle over Grn(M). Using this S1–bundle one obtains central extensions for certain groups of dif-feomorphisms of M. We can realize Grm−2(M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplec-tic Grassmannians SGr2k(M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms. 1