Nonlinear flag manifolds as coadjoint orbits

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 Grassmannians. 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.

Pointed vortex loops in ideal 2D fluids

“…This follows from lemma 3.7 together with the following mathematical folklore result (see for instance the appendix in [9]): Proposition 3.9. Suppose the action of G on (M, Ω) is transitive with injective equivariant moment map J : M → g * .…”

“…Under the identification Φ in (11) of Emb a (S 1 , R 2 ) with O w a , the momentum map ( 20) is exactly the map (9).…”

“…Here we show that pointed vortex loops also have natural polarizations. More precisely, the same polarization subgroup G C , that consists of diffeomorphisms that preserve the loop as a set, works for the coadjoint orbit O w a of the pointed vortex loop (C, β, (x i )) from (9). Thus the configuration space is the space of loops that enclose a fixed area, without information about the vorticity distribution and the attached points.…”

Weighted nonlinear flag manifolds as coadjoint orbits

“…Nonlinear Grassmannians have been generalized to spaces of nonlinear flags in [9]. Given a collection of closed manifolds S = (S 1 , .…”

“…Manifolds of low-dimensional nonlinear flags have appeared as shape spaces in [3,12,23]. Symplectic nonlinear flags have been used to describe coadjoint orbits of the Hamiltonian group [9].…”