Non-linear Grassmannians as coadjoint orbits

الأصل · EN

For a given manifold M we consider the non-linear Grassmann manifold Grₙ(M) of n-dimensional submanifolds in M. A closed (n+2)-form on M gives rise to a closed 2-form on Grₙ(M). If the original form was integral, 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 diffeomorphisms of M. We can realize Grₘ₋₂(M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians SGr₂ₖ(M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms. We also generalize the vortex filament equation as a Hamiltonian equation on Grₘ₋₂(M).