Non-linear Grassmannians as coadjoint orbits

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. Mathematics Subject Classification (2000):58B20Both authors are supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT