Nonlinear flag manifolds as coadjoint orbits
- PDF / 2,863,124 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 88 Downloads / 190 Views
Nonlinear flag manifolds as coadjoint orbits Stefan Haller1 · Cornelia Vizman2 Received: 11 February 2020 / Accepted: 29 June 2020 © The Author(s) 2020
Abstract A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmann‑ ians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags. Keywords Nonlinear flag manifolds · Nonlinear Grassmannians · Groups of diffeomorphisms · Spaces of embeddings · Fréchet manifold · Moment map · Coadjoint orbits Mathematics Subject Classification 58D10 · 37K65 · 53C30 · 53D20 · 58D05
1 Introduction Let M be a smooth manifold, and suppose S1 , … , Sr are closed smooth manifolds. A non‑ linear flag of type S = (S1 , … , Sr ) in M is a sequence of nested embedded submanifolds N1 ⊆ ⋯ ⊆ Nr ⊆ M such that Ni is diffeomorphic to Si for all i = 1, … , r . The space of all nonlinear flags of type S in M can be equipped with the structure of a Fréchet manifold in a natural way and will be denoted by FlagS (M) . The aim of this paper is to study the geometry of this space using the convenient calculus of Kriegl and Michor [18]. Nonlinear flag manifolds provide a natural generalization of nonlinear Grassmannians which correspond to the case r = 1 . Nonlinear Grassmannians (a.k.a. differentiable Chow manifolds) play an important role in computer vision[1, 24] and continuum mechanics[25]. They have also been used to describe coadjoint orbits of diffeomorphism groups. Nonlin‑ ear Grassmannians of symplectic submanifolds have been identified with coadjoint orbits * Stefan Haller [email protected] Cornelia Vizman cornelia.vizman@e‑uvt.ro 1
Department of Mathematics, University of Vienna, Oskar‑Morgenstern‑Platz 1, 1090 Vienna, Austria
2
Department of Mathematics, West University of Timişoara, Bd. V.Pârvan 4, 300223 Timisoara, Romania
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
of the Hamiltonian group in[12]. Codimension two Grassmannians have been used to describe coadjoint orbits of the group of volume-preserving diffeomorphisms[12, 16]. Let us also point out that every closed k-fold vector cross-product on a Riemannian manifold induces an almost Kähler structure on the nonlinear Grassmannians of (k − 1)-dimensional submanifolds[20]. In some applications, decorated nonlinear Grassmannians have been considered, that is, spaces of submanifolds equipped with additional data supported on the submanifold. Functional shapes (fshapes), for instance, may be described as signal functions supported on shapes[3–5]. Weighted nonlinear Grassmannians of isotropic submanifolds have been used to describe coadjoint orbits of the Hamiltonian group[9, 19, 28]. Recently, weighted nonlinear Grassmannians of isotropic submanifolds have been identified with coadjoint orbits of the contact group[13]. Decorated codimension one Grassmann
Data Loading...
