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Non-linear Grassmannians as coadjoint orbits

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090, Vienna, Austria

    Stefan Haller

  2. West University of Timisoara, Department of Mathematics, Bd. V.Parvan 4, 1900, Timisoara, Romania

    Cornelia Vizman

Authors
  1. Stefan Haller
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  2. Cornelia Vizman
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Correspondence to Cornelia Vizman.

Additional information

Mathematics Subject Classification (2000):58B20

Both authors are supported by the ‘Fonds zur Förderung der wissenschaftlichen Forschung’ (Austrian Science Fund), project number P14195-MAT

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Haller, S., Vizman, C. Non-linear Grassmannians as coadjoint orbits. Math. Ann. 329, 771–785 (2004). https://doi.org/10.1007/s00208-004-0536-z

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  • DOI: https://doi.org/10.1007/s00208-004-0536-z

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