The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits

Non-linear Grassmannians as coadjoint orbits

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…

非线性格拉斯曼流形：普通、装饰和增强型

Nonlinear Grassmannians: plain, decorated, augmented

本文提出了一种通用框架，用于构建装饰和增强非线性格拉斯曼流形的光滑结构，从而赋予对应的共伴随轨道以光滑辛弗雷歇特流形的结构。

限制Grassmannian上部分等距的Banach Lie groupoid

Banach Lie groupoid of partial isometries over restricted Grassmannian

本文构造了限制Grassmannian上部分等距集的微分结构，使其成为Banach Lie groupoid。

Nonlinear flag manifolds as coadjoint orbits

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet 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.

Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

Using a stability criterion due to Kröncke, we show, providing ${n\neq 2k}$, the Kähler--Einstein metric on the Grassmannian $Gr_{k}(\mathbb{C}^{n})$ of complex $k$-planes in an $n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf--Sesum on the instability of the Fubini--Study metric on $\mathbb{CP}^{n}$ for $n>1$. The key to the proof is using the description of Grassmannians as certain coadjoint orbits of $SU(n)$. We are also able to prove that Kröncke's method will not work on any of the oth…

受限的Schatten类Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ 作为仿射共共轭轨道

The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit

本文研究了受限的$p$-Schatten类Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$，证明对于$1\leq p\leq 2$，该Grassmannian是无限维受限酉群$\operatorname{U}_{\mathrm{res}, p}(\mathcal{H})$的仿射(共)共轭轨道，并具有自然的弱辛结构。