Non-linear Grassmannians as coadjoint orbits

2003/05/06 by Stefan Haller, Cornelia Vizman
Mathematics · #math.DG #math.SG #msc:58B20

paper · pdf

published as Math. Ann. 329(2004), 771--785.

arxiv created 2003/05/06 · arxiv updated 2009/11/30

Abstract

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 S¹-bundle over Gr_n(M). Using this S¹-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. We also generalize the vortex filament equation as a Hamiltonian equation on Gr_m-2(M).

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