Questions tagged [homogeneous-spaces]

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Homogeneous representations of compact manifolds

There is a classification of effective transitive groups actions on the sphere by compact connected Lie groups, compare Besse "Einstein manifolds" 7.13 Examples. Are there similar results ...
Julian Seipel's user avatar
3 votes
2 answers
116 views

The closure of the orbit of an irrational grid contains the fiber

Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
No One's user avatar
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2 votes
2 answers
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Riemannian homogeneous equivalent to linear group orbit

Let $ M $ be a smooth manifold. Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $. Recall that a manifold $ M $ is Riemannian homogeneous if ...
Ian Gershon Teixeira's user avatar
2 votes
2 answers
285 views

Compact linear group orbit equivalent to linear compact group orbit

A corollary of the Mostow-Palais theorem is that every homogeneous space for a compact group is a linear group orbit. In other words, if $ H $ is a closed subgroup of a compact group $ K $ then there ...
Ian Gershon Teixeira's user avatar
17 votes
3 answers
2k views

Examples of the Thurston geometries with transitive Lie group action

Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries: (1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup (2) Euclidean: 3 torus $\...
Ian Gershon Teixeira's user avatar
4 votes
2 answers
287 views

Does the maximal compact subgroup always act transitively on a compact homogeneous space?

Let $ G $ be a Lie group, $ H $ a closed subgroup, and $ G/H $ compact. Under what conditions do we have that $$ G/H \cong K/(K\cap H) $$ where $ K $ is a maximal compact subgroup of $ G $? Obviously ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
80 views

Local decomposition of semisimple Lie groups at the identity into the product of centralizer, unstable and stable horospherical subgroups

First let us look at an example. Let $G=\text{SL}(d,\mathbb R)$ and $A$ be its subgroup consisting of diagonal matrices with positive entries. Let $a\in A$ be an element. We define the stable ...
No One's user avatar
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What is the precise definition of connect semisimple Lie groups "without compact factors" in the literature?

I frequently see in the literature (of homogeneous dynamics, in particular) of the notion "semisimple Lie groups without compact factors" without seeing its precise definition. I have three (...
No One's user avatar
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2 votes
0 answers
77 views

Transitive Lie group actions with uniformly bounded derivatives

Let $\phi:G\times M\rightarrow M$ be a smooth and transitive action of a real Lie group $G$ on a smooth real manifold $M$. Both $G$ and $M$ are assumed to be finite-dimensional but neither are assumed ...
Jochen Trumpf's user avatar
6 votes
1 answer
515 views

Are the quaternionic Grassmannians quaternionic Kaehler manifolds?

The complex Grassmannians $\mathrm{Gr}(n,r)$, of $r$-planes in $\mathbb{C}^n$ are Kaehler manifolds. What about the quaternionic Grassmannians of $r$-planes in $\mathbb{H}^n$ are they quaternionic ...
Martim Pereir's user avatar
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217 views

What can we say about the homogeneous spaces $E_8/E_7$ and $E_7/E_6$?

For the three exceptional compact Lie groups $E_8, E_7, E_6$ we have the inclusions $$ E_6 \subseteq E_7 \subseteq E_8. $$ What can we say about the the homogeneous spaces $$ E_8/E_7, ~~~~ E_7/E_6? $$ ...
Alain Rochefort's user avatar
6 votes
2 answers
409 views

Homogeneous symplectic manifolds

I have often heard/read a statement (see, e.g., this MathOverflow question) equivalent to the following: Let $G$ be a connected Lie group and $(M,\omega)$ a connected and simply-connected symplectic ...
José Figueroa-O'Farrill's user avatar
4 votes
1 answer
162 views

Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian $U(n)/U(k)\times U(n-k)$ with it's $U(n)$-invariant measure. The affine chart corresponding to $i_1, \ldots, i_k$ is given by $n\times k$ matrices for which the ...
Vít Tuček's user avatar
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Cartan geometry: jet space perspective on the tractor bundle

Let $G$ a Lie group and $H\subset G$ a Lie subgroup. For simplicity we assume that the adjoint action of $H$ on $\mathfrak g/\mathfrak h$ is faithful. Let $M$ a differentiable manifold of the same ...
jpdm's user avatar
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1 answer
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$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]

The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$). Because the $SO(k)$ may not be a normal ...
Марина Marina S's user avatar
5 votes
0 answers
112 views

Stronger form of countable dense homogeneity

I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove ...
Saúl RM's user avatar
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6 votes
0 answers
101 views

subalgebra of invariants for a reductive subgroup

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Spec{Spec}$Trying to understand some tannakian reconstruction, I've stumbled about the following problem in invariant theory. I guess it's something ...
Dragos Fratila's user avatar
11 votes
1 answer
412 views

Planes in Lagrangian Grassmannians

Let $LG(h,2h)$ be the Lagrangian Grassmannian of subspaces of dimension $h$ of a complex vector space of dimension $2h$. For instace, $LG(1,2)=\mathbb{P}^1$, and $LG(2,4)\subset\mathbb{P}^4$ is a ...
Elsa's user avatar
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2 answers
438 views

Subvarieties of Lagrangian Grassmannians

Let $LG(n,2n)$ be the Lagrangian Grassmannian parametrizing Lagrangian subspaces (so of dimension $n$) of $\mathbb{C}^{2n}$. Then $LG(n,2n)\subset G(n,2n)$, where $G(n,2n)$ is the Grassmannian of ...
Puzzled's user avatar
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8 votes
1 answer
274 views

How special are homogeneous spaces?

Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$? Related questions/approaches: Of course we need $\...
GFR's user avatar
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3 votes
0 answers
92 views

Is the affine Grassmannian manifold a symmetric homogeneous space?

I am interested in the manifold of affine subspaces of dimension $k$ of $\mathbb{R}^n$, which can be viewed as the homogeneous space $$ E(n)/(E(k)\times O(n-k)),$$ where $E$ refers to rigid motions ...
Chevallier's user avatar
1 vote
0 answers
175 views

Statistical manifolds with trivial statistical structure after quotienting

A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold with a torsion-free affine connection $\nabla$ such that $\nabla g$ is symmetric in all entries. Equivalently, there is a dual affine ...
user2002's user avatar
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0 answers
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Examples of group actions on statistical manifolds

A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold with a torsion-free affine connection $\nabla$ such that $\nabla g$ is symmetric in all entries. Equivalently, there is a dual affine ...
user2002's user avatar
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1 answer
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Homology of the free loop space of generalized flag varieties

Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{...
Filip's user avatar
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4 votes
1 answer
263 views

The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories

Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$. (1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...
No One's user avatar
  • 1,545
3 votes
1 answer
155 views

Picard group of $(SL(n)\times SL(m))$-orbits

Let $\mathbb{P}^N$ be the projective space of $n\times m$ matrices with complex entries modulo scalar. Consider the $(SL(n)\times SL(m))$-action on $\mathbb{P}^N$ given by $((A,B),Z)\mapsto AZB^{T}$. ...
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7 votes
3 answers
548 views

Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$

$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel ...
No One's user avatar
  • 1,545
2 votes
1 answer
389 views

Picard group of $\mathrm{GL}(n)$-orbits

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group $$ \GL(n) = \left\lbrace \left(\begin{array}{cc} A & C \\ M & B \end{array}\right) \text{ with } A\...
user avatar
2 votes
0 answers
152 views

What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?

Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...
Christoph Mark's user avatar
1 vote
0 answers
208 views

Relation between reductive homogeneous spaces and reductive groups

To start, I would like to note that my background on Lie algebras is quite basic, so this question might be trivial when seen from a Lie algebra perspective, which I lack. We have the concept of a ...
Lezkus's user avatar
  • 167
3 votes
1 answer
140 views

What is the orbit of the standard conformal structure on $S^2$ under $\operatorname{SL}(3,\mathbb{R})$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$Consider the group $\GL_+(3,\mathbb{R})$ acting on $\mathbb{R}^3$. It induces an action of $\GL_+(3,\mathbb{R})/\...
Malkoun's user avatar
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5 votes
0 answers
117 views

Kronecker's theorem on diophantine approximation for $\mathrm{SL}_2(\mathbb{Z})$

Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to ...
Jan-Willem van Ittersum's user avatar
8 votes
1 answer
530 views

Are invariant forms on homogeneous spaces necessarily closed?

Take a compact homogeneous space $G/K$, and a left $G$-invariant differential $k$-form $\omega \in \Omega^k(G/K)$. Will $\omega$ necessarily be closed? Might it even be harmonic when $G/K$ is endowed ...
Quin Appleby's user avatar
1 vote
0 answers
53 views

About Countable Dense Homogeneous spaces (CDH) and strongly locally homogeneous spaces

I am new to the study of CDH topological spaces, I wanted to study basic examples of this type of spaces, for example I could understand the demonstration that $\mathbb{R}$ is CDH, using the Cantor ...
Gabriel Medina's user avatar
6 votes
1 answer
259 views

Bounded non-symmetric domains covering a compact manifold

This question is somewhat related to this other question of mine. I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient. By a ...
diverietti's user avatar
  • 7,852
11 votes
0 answers
328 views

The existence of a fiber sequence involving $\mathrm{Spin}(9)$ and $\mathrm{SU}(2)$

$\newcommand{\SU}{\mathrm{SU}} \newcommand{\Spin}{\mathrm{Spin}}$There is a fiber sequence $G_2\to \Spin(9) \to \Spin(9)/G_2$, and $G_2$ contains $\SU(3)$ as a subgroup. Is there a space (possibly a ...
skd's user avatar
  • 5,550
1 vote
0 answers
466 views

Horizontal lift of fundamental vector field

Suppose $\theta\colon G\times M\to M$ is a transitive smooth left action of a compact Lie group $G$ on a manifold $M$ and $\pi\colon G\to M\cong G/K$ the corresponding smooth submersion for some ...
Sven Pistre's user avatar
3 votes
1 answer
280 views

Rank 3 Lagrangian vector bundles on an elliptic curve

Let $k$ be an algebraically closed field of characteristic zero (feel free to assume $k= \mathbb{C}$) and $E$ an elliptic curve over $k$ with identity $P \in E(k)$. I am interested in certain ...
Jef's user avatar
  • 949
3 votes
2 answers
236 views

Automorphism group of formally real Jordan algebras of hermitian matrices

It is well known that the automorphism group of exceptional Jordan algebra $\mathcal{h}_{3}(\mathbb{O})$ is the exceptional Lie group $F_{4}$. I am trying to understand the automorphism group of the ...
Inácio 's user avatar
3 votes
0 answers
98 views

p-adic analogue of classification of irreducible Riemannian symmetric spaces?

For Riemannian symmetric spaces, we have a nice result that the simply-connected ones are products of irreducible symmetric spaces, of which we have a list of ten infinite families and some ...
Roger Van Peski's user avatar
3 votes
1 answer
149 views

Model geometry uniqueness

Let $ M $ be a compact connected manifold with $$ M \cong \Gamma \backslash G /H $$ where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and ...
Ian Teixeira's user avatar
4 votes
3 answers
156 views

Separability of subspaces of homogeneous topological spaces

Let $\ X\ $ be a homogenous separable topological space (i.e. for every $\ x\ y\in X\ $ there exists a homeomorphism $\ f:X\to X\ $ such that $\ f(x)=y,\ $ and there is a countable dense subset of $\ ...
Wlod AA's user avatar
  • 4,686
8 votes
2 answers
437 views

Geodesic sphere in the octonion projective plane

I am considering Laplacian eigenvalues of a geodesic sphere in the octonion projective plane and I would like to know the metric on a geodesic sphere. Does the metric on a geodesic sphere in the ...
Donghwi Seo's user avatar
2 votes
2 answers
1k views

Is a manifold paracompact? Should it be?

We will say that a Hausdorff topological space $X$ is a smooth manifold if there is an open cover $(U_{\alpha})$ of $X$ and a corresponding collection of homeomorphisms $\varphi_{\alpha} : U_{\alpha} \...
AmorFati's user avatar
  • 1,349
3 votes
3 answers
758 views

Is every homogeneous space Riemannian homogeneous?

A manifold $M$ together with a transitive $G$-action is always diffeomorphic a quotient $G/H$ for $H < G$ Lie groups. On the other hand, there might be a proper subgroup of $G$ that also acts ...
Lezkus's user avatar
  • 167
2 votes
1 answer
436 views

Euler Characteristic of $SL_m(\mathbb{C})/SO_m(\mathbb{C})$

As described in the title, what is the (topological) Euler characteristic of the homogeneous space $SL_m(\mathbb{C})/SO_m(\mathbb{C})$?
Winnie_XP's user avatar
  • 287
9 votes
0 answers
189 views

Octonionic Stiefel manifolds

The Stiefel manifolds are presented in this Wikipedia article over the division algebras $\mathbb{R,C,H}$. In fact, they are presented as homogeneous spaces, respectively for the $A,B,C$,and $D$ ...
Fofi Konstantopoulou's user avatar
3 votes
1 answer
232 views

Flag manifolds as homogeneous Kahler manifolds

In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written Flag manifolds exhaust all compact homogeneous Kähler ...
Fofi Konstantopoulou's user avatar
2 votes
1 answer
142 views

Coinvariant representative of homogeneous space cohomology

Given a compact homogeneous space $M = K/L$, consider its de Rham complex $(\Omega^*,d)$. Will every cohomology class $[\omega] \in ker(d)/im(d)$ contain a representative $\nu$ which is invariant with ...
Fofi Konstantopoulou's user avatar
1 vote
1 answer
366 views

De Rham cohomology of homogeneous spaces

Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
Fofi Konstantopoulou's user avatar