{
  "id": "math/0305089",
  "version": "v1",
  "published": "2003-05-06T08:50:51.000Z",
  "updated": "2003-05-06T08:50:51.000Z",
  "title": "Non-linear Grassmannians as coadjoint orbits",
  "authors": [
    "Stefan Haller",
    "Cornelia Vizman"
  ],
  "journal": "Math. Ann. 329(2004), 771--785.",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "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. We also generalize the vortex filament equation as a Hamiltonian equation on $Gr_{m-2}(M)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-06T08:50:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58B20"
    ],
    "keywords": [
      "coadjoint orbits",
      "non-linear grassmannians",
      "vortex filament equation",
      "exact volume preserving diffeomorphisms",
      "non-linear grassmann manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5089H"
    }
  }
}