Non-linear Grassmannians as coadjoint orbits

By Stefan Haller, Cornelia Vizman. 2003

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Abstract.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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dff77e77d6078916db31a68afc6bb92a1ced08ae