Questions tagged [homogeneous-spaces]
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231
questions
4
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Tight bound on spectral gap of compact homogeneous manifold?
This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter $d$:
$$ \lambda_1 \ge c/d^2, $$
where $c=\pi^2/4$. Can this bound be ...
7
votes
0
answers
121
views
Infinitesimal description of homogeneous supermanifolds
Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...
2
votes
1
answer
50
views
Descending central extensions to homogeneous spaces
Let $G$ be a Lie group (finite dimensional or Banach), and let $H$ be a Lie subgroup (in the Banach case we assume that $H$ is a submanifold which is also a Lie group). Let $\text{U}(1) \rightarrow \...
3
votes
1
answer
254
views
The space of complex structure compatible with metric
Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?
5
votes
2
answers
927
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Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
5
votes
1
answer
517
views
Volume of balls in homogeneous manifolds
Let $X=G/H$ be a homogeneous manifold, where $G$ and $H$ are connected Lie groups and assume there is given a $G$-invariant Riemannian metric on $X$.
Let $B(R)$ be the closed ball of radius $R>0$ ...
3
votes
0
answers
70
views
Homogeneous space for intersection of subgroups
Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $...
7
votes
2
answers
2k
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Embeddings of flag manifolds
Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
1
vote
0
answers
265
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Invariant measures on locally compact homogeneous spaces
Edit: answered in the first comment. This turned out to be really easy, as the orbits of an open $\sigma$-compact subgroup yield a partition of $X$ into open $\sigma$-compact subsets.
Let $G$ be a ...
2
votes
1
answer
92
views
Homogeneity of a projective vector bundle
Let $M=G/K$ be a $G$-homogeneous manifold and suppose that $E\to G/K$ is a homogeneous (complex) vector bundle, i.e. it is defined by a representation $\phi : K \to \text{Aut(V)}$ for some complex ...
4
votes
1
answer
265
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A homogeneous space that's not a fibre bundle
Let $G$ be a locally-compact group and $H$ a closed subgroup.
Let $X=G/H$ and let $\pi:G\to X$ be the projection.
We say that the projection $\pi$ is a fibre bundle, if for every point $x\in X$ there ...
12
votes
1
answer
252
views
Flag manifolds as incidence correspondences
Let $G$ be a reductive group, $B$ a Borel and $P_j$ the maximal parabolics, indexed by the vertices $j$ of the Dynkin diagram. Then $B = \bigcap_j P_j$, so the flag manifold $G/B$ injects into $\...
0
votes
1
answer
99
views
On a pair of continuous functions "connected" by continuous functions
Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$
...
5
votes
1
answer
221
views
Conformally flat homogeneous spaces
Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly ...
8
votes
1
answer
234
views
Compact simply-connected homogeneous symplectic manifold
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...
2
votes
0
answers
180
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The isometry groups of flag manifolds
For any sequence of integers $0<n_1<...<n_k$, there is a flag manifold of type $(n_1, ..., n_k)$, which is the collection of ordered sets of vector subspaces of $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
3
votes
0
answers
147
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Classification of Euclidian-like Klein geometries in spirit of Erlangen program
All we know about Erlangen Program approach to geometry. The main idea is to consider Lie Group $G$ acting transitively on smooth manifold $M$ (e.g. nLab). I really like such style of thinking, but ...
11
votes
2
answers
585
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Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension
Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.
I have heard that, for any real number $\...
13
votes
3
answers
865
views
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
3
votes
1
answer
152
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Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram
$$
\beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,.
$$
Let $P_{\{\beta_2\}}...
3
votes
0
answers
78
views
Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?
$\newcommand{\GL}{\operatorname{GL}}$
Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
1
vote
0
answers
74
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Construction of homogeneous space
Given a Hilbertspace $\mathcal{H}$ of dimension $n<\infty$ and all hermitian matricies $Symm(\mathcal{H})$. I'd guess that the set $M_{2,2} \subset Symm(\mathcal{H})$ of all matricies of rank 4 and ...
6
votes
0
answers
161
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Injectivity of exponential chart in a homogeneous space
Let $G$ be a finite-dimensional connected real Lie group and $\mathfrak{g}$ its Lie algebra. Let $\exp : \mathfrak{g} \to G$ denote the exponential map. Among the results proved independently by ...
7
votes
2
answers
324
views
Explicit description of SU(2,2)/U
Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...
6
votes
0
answers
227
views
Integrals of modular forms
Setup: Write $G = \text{SL}_2(\mathbf{R})$ and $\Gamma = \text{SL}_2(\mathbf{Z})$.
Let $f$ be a modular form on $\mathbf{H}$ of weight $2k$, so that
$$f(gz) = f(z) (cz + d)^{2k} \qquad \text{for} \...
8
votes
1
answer
145
views
Criterion for existence of a homogeneous space associated to a Lie pair
Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism.
This statement generalises somewhat to ...
1
vote
0
answers
136
views
Is there a class of non-symmetric spaces $G/SO(n)$ such that $Hol(G/SO(n))=SO(n)$?
Is there a certain class (or unique construction of a set) of simply connected non-symmetric spaces $G/SO(n)$ such that the Riemannian holonomy is $Hol(G/SO(n))=SO(n)$? (That is, is there a way to ...
1
vote
1
answer
210
views
Torus actions on $Sp(n)$-spheres
In this old question of mine
https://math.stackexchange.com/questions/1651906/spheres-as-symplectic-homogeneous-spaces
the presentation of spheres as symplectic group homogeneous spaces was ...
5
votes
0
answers
157
views
Can we see the symmetry of the quantum Schubert polynomial of a point
Let $X=G/B$ be a homogeneous space and consider the quantization map
$$
S_W\otimes\mathbb{C}[q]\to(S(\mathfrak{h})\otimes\mathbb{C}[q])/I_W^q\,,
$$
where
$S_W$ is the coinvariant algebra of the Weyl ...
1
vote
0
answers
116
views
A section over an orbit space
Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...
2
votes
1
answer
309
views
Gelfand pairs and (self)-dual representations
For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...
4
votes
1
answer
243
views
Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$
The ideal of the $10$-dimensional Spinor variety $S^{10}\subset\mathbb{P}^{15}$ is generated by $10$ quadrics.
Does anyone know a reference where these 10 quadratic equations are written down ...
2
votes
0
answers
84
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Submanifolds of nilmanifolds coming from Lie subgroups
Let $G$ be a connected simply connected nilpotent real Lie group and $\Gamma$ a lattice in $G$, such that $M=\Gamma \backslash G$ is a compact nilmanifold. Let $p:G \to M$ be the projection. If $S$ is ...
2
votes
0
answers
67
views
evolution of Grassmannians along geodesic line
Let $p_0$, $p_1$ be two $n \times 2$ orthonormal matrices that represent two points on the real $Gr_{2,n}$, i.e. two 2-d subspaces in $\mathbb{R}^n$. Let $p(t): [0,1] \rightarrow Gr_{2,n}$ be a ...
1
vote
0
answers
109
views
Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
3
votes
0
answers
117
views
Compactification of symmetric spaces
Let $G = GL_n$. In the literature, I see that the symmetric space associated to $G$ is of the form $G(\mathbb{R})/K_\infty$ with $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...
3
votes
1
answer
167
views
Degeneration of coadjoint orbits
Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...
14
votes
1
answer
475
views
Is Bing's countable connected space topologically homogeneous?
In this paper R.H. Bing has constructed his famous example of a countable connected Hausdorff space.
The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\...
8
votes
0
answers
128
views
Local vs global homogeneity of topological spaces
I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have rigid spaces, i.e., topological spaces with trivial homeomorphism group.
Question. ...
8
votes
1
answer
2k
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The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$
The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given ...
3
votes
0
answers
204
views
A criterion for a $G$-variety to be isomorphic to $G/H$
Let $k$ be an algebraically closed field of characteristic 0.
Let $G$ be a connected linear algebraic group over $k$.
Let $H\subset G$ be an algebraic $k$-subgroup.
Let $Y$ be an algebraic variety ...
1
vote
0
answers
168
views
Decomposition of conic equation for two intersecting lines [closed]
By using homogeneous coordinates ${\bf x}=[x, y, 1]^T$, a conic section can be expressed by ${\bf x}^T {\bf C} {\bf x} = 0$ with ${\bf C}$ being a 3x3 symmetric matrix. A degenerate form of the conic ...
1
vote
1
answer
86
views
dense orbit projective dual homogeneous space
Let $\mathrm{G}$ be a semi-simple algebraic group over $\mathbb{C}$ and $V$ be a finite dimensional representation of $\mathrm{G}$. Let $x \in V$ be a non zero vector such that the variety $\mathrm{G}....
4
votes
1
answer
130
views
Criterion for homogeneity
Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and let $G=Aut(\Omega)$ be the full group of self-biholomorphisms of $\Omega$. Assume that there is $z\in \Omega$, such that the orbit of $z$ is ...
1
vote
0
answers
75
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Criterion of holomorphy
I have posted a similar question on MSE few days ago, but it received no attention.
Let $G$ be a homogeneous (or even symmetric) domain in $\mathbb{C}^{n}$ and $u:G\to \mathbb{C}\backslash\{0\}$.
Q1....
5
votes
1
answer
352
views
Holomorphic extension of an action by a compact Lie group on a complex homogeneous manifold
Let $G$ be a compact Lie group and let $M$ be a $G$-homogeneous manifold. Suppose that $M$ is endowed with a complex structure invariant by the action of $G$. Denote by $G_{\mathbb C}$ the ...
10
votes
1
answer
271
views
Compact Lie group inclusions that are trivial on all homotopy groups
Is there a classification of closed subgroups $H\le SO(n)$ such that the inclusion $H\to SO(n)$ is trivial on all homotopy groups?
This happen e.g. when the group $H$ is finite. Are there other ...
5
votes
0
answers
163
views
The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$
I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.
First let $G$ be any algebraic group over $\mathbb C$, and let $X$ ...
3
votes
2
answers
261
views
Homogenous structure on $S^2\times S^2$ and its geometry
Is a well known fact that $SO(3)$ acts transitively on $S^2$ and that the isotropy group of this action is $SO(2).$ In this case, $S^2$ has a natural structure of homogeneous space. In particular, I ...
3
votes
0
answers
94
views
Isotropy symmetric holomorphic functions
Let $G$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $z\in G$.
Assume that $f$ is a holomorphic function on $G$, which is isotropy symmetric, i.e. $f\circ \varphi=f$ for any ...