Questions tagged [homogeneous-spaces]

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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 ...
hwlin's user avatar
  • 361
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 ...
José Figueroa-O'Farrill's user avatar
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 \...
Peter's user avatar
  • 546
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) $ ?
Ramtin.VA's user avatar
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5 votes
2 answers
927 views

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 ...
Tom1990's user avatar
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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 $R>0$ ...
user avatar
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 $...
Peter Kravchuk's user avatar
7 votes
2 answers
2k views

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 ...
user avatar
1 vote
0 answers
265 views

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 ...
Daniel Tausk's user avatar
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 ...
Mathgymnast's user avatar
4 votes
1 answer
265 views

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 ...
user avatar
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 $\...
David E Speyer's user avatar
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 $$f = u\circ g \text{ and } g= v \circ f.$$ ...
Dominic van der Zypen's user avatar
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 ...
Sergei Ovchinnikov's user avatar
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 ...
Valentino's user avatar
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2 votes
0 answers
180 views

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 $R^{(n_k)}$ $(V_1, ..., V_k)$ ...
wonderich's user avatar
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3 votes
0 answers
147 views

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 ...
Mykola Pochekai's user avatar
11 votes
2 answers
585 views

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 $\...
Kim's user avatar
  • 4,034
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 views

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\}}...
Christoph Mark's user avatar
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{...
Asaf Shachar's user avatar
  • 6,611
1 vote
0 answers
74 views

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 ...
Alpha001's user avatar
  • 143
6 votes
0 answers
161 views

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 ...
José Figueroa-O'Farrill's user avatar
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}\...
Mikhail Borovoi's user avatar
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} \...
Kim's user avatar
  • 4,034
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 ...
José Figueroa-O'Farrill's user avatar
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 ...
Sergio Charles's user avatar
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 ...
Tomasz Köner's user avatar
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 ...
Christoph Mark's user avatar
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: ...
Bedovlat's user avatar
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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 ...
Alesandro Levi's user avatar
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 ...
user avatar
2 votes
0 answers
84 views

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 ...
Ugo Iaba's user avatar
  • 209
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 ...
Vadim Ogranovich's user avatar
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 $...
Bedovlat's user avatar
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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 $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...
Vanya's user avatar
  • 581
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 ...
q123e's user avatar
  • 39
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\...
Taras Banakh's user avatar
  • 40.8k
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. ...
Taras Banakh's user avatar
  • 40.8k
8 votes
1 answer
2k views

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 ...
Lars Pettersen's user avatar
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 ...
Mikhail Borovoi's user avatar
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 ...
Johannes B.'s user avatar
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}....
Libli's user avatar
  • 7,210
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 ...
erz's user avatar
  • 5,385
1 vote
0 answers
75 views

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....
erz's user avatar
  • 5,385
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 ...
Max Reinhold Jahnke's user avatar
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 ...
Igor Belegradek's user avatar
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 $X$ ...
Mikhail Borovoi's user avatar
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 ...
L.F. Cavenaghi's user avatar
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 ...
erz's user avatar
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