Related papers: The index of equidimensional flag manifolds
In this paper we study the $\Z/2$ action on real Grassmann manifolds $G_{n}(\R^{2n})$ and $\widetilde{G}_{n}(\R^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\Z/2$…
This paper computes the Fadell-Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian $p$-groups. The results are…
Using the cohomology of the $G_2$-flag manifolds $G_2/U(2)_{\pm}$, and their structure as a fiber bundle over the homogeneous space $G_2/SO(4)$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of such fiber bundles, for the…
We study the index of symmetry of a compact generalized flag manifold M=G/H endowed with an invariant Kaehler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces and we…
We study the Fadell-Husseini index of the configuration space F(R^d,n) with respect to different subgroups of the symmetric group S_n. For p prime and d>0, we completely determine Index_{Z/p}(F(R^d,p);F_p) and partially describe…
We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$,…
The main result of the paper is a Borel type description of the $Sp(1)^n$-equivariant cohomology ring of the manifold $Fl_n(\mathbb{H})$ of all complete flags in $\mathbb{H}^n$. To prove this, we obtain a Goresky-Kottwitz-MacPherson type…
We introduce the concept of $r$-equilateral $m$-gons. We prove the existence of $r$-equilateral $p$-gons in $\mathbb R^d$ if $r<d$ and the existence of equilateral $p$-gons in the image of continuous injective maps $f:S^d\to \mathbb…
We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest…
The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an…
A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix…
We study the real Grassmann manifold $G_{2n}^n$ (of $n$-subspaces in $\mathbb R^{2n}$), and the action of $Z_2$ on it by taking the orthogonal complement. The homological index of this action is estimated from above and from below. In case…
We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection…
We consider manifolds of oriented flags SO(n)/SO(2)xSO(n-3) (n>=4) as 4- and 6-symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonical f-structures on these homogeneous…
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…
Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ enjoys two very peculiar geometric properties: It carries a transitive action of the (noncompact) Lie group $G$, and it is embedded in…
In this note, we compute the upper characteristic rank of the projective Stiefel manifolds over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$ and the flip Stiefel manifolds. We also provide bounds for the $\mathrm{cup}$ lengths of these spaces.…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
We express the diagonals of projective, Grassmann and, more generally, flag bundles of type (A) using the zero schemes of some vector bundle sections, and do the same for their single point subschemes. We discuss diagonal and point…
A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Frechet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear…