# Questions tagged [homogeneous-spaces]

The homogeneous-spaces tag has no usage guidance.

192
questions

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### Scalar curvature of homogeneous bounded domains

Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?

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105
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### Eigenvalues of the Casimir operator and the space $L^2(E)$

I have read something about harmonic analysis on a homogeneous vector bundle in a very interesting paper Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds by J. DeWitt in my M.Sc. ...

1
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0
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128
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### Homogeneous metrics on compact Lie groups

Given a differentiable manifold $M$, a Riemannian metric $g$ on $M$ is called homogeneous if its isometry group acts transitively on $M$. In that case, given any group $G$ acting transtitively on $(M,...

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0
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### Intersection of open orbits in homogeneous space

Let $G$ be a simple complex algebraic group. Let $P(\alpha_i),P(\alpha_k)$ be maximal standard parabolic subgroups of $G$ associated to simple roots $\alpha_i,\alpha_k$ in the root system associated ...

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2
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334
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### The (last step of the) proof that the set of badly approximable matrices has measure zero

An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have
$$ \|A q + p \| \ge c \| ...

3
votes

1
answer

179
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### Classification of "homogeneous" submanifolds of ℝⁿ

I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if:
it is a closed connected smooth submanifold of $\mathbb R^n$,
for every $p, q$ in $M$, there is a ...

11
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1
answer

394
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### Classification of homogeneous Einstein manifolds

In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, ...

4
votes

1
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191
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### An analogy of product formula for homogeneous space?

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)...

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113
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### Decomposition of fiber product of $G$-sets in $G$-orbits

I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem:
Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then
the right ...

2
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118
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### Homogeneous space and rational section

Let's embed $\operatorname{SO}_n$ inside $\operatorname{GL}_n$ through the standard representation. Does the map $\operatorname{GL}_n\rightarrow \operatorname{GL}_n/{\operatorname{SO}_n}$ admit a ...

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### Properties of the orbit $Abx_0$ when $b$ is upper or lower triangular but not diagonal

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 $A$ denote the subgroup of $G$ ...

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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 ...

3
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2
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97
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### 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 ...

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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 ...

2
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2
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227
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### 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 ...

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### 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 $\...

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2
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156
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### 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 ...

2
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### 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 ...

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100
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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 (...

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0
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### 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 ...

5
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1
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412
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### 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 ...

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### 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
$$
E_8/E_7, ~~~~ E_7/E_6?
$$
...

5
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2
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255
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### 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 ...

4
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1
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116
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### 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 ...

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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 ...

0
votes

1
answer

220
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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 ...

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0
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110
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### 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 ...

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### 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 ...

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345
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### 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 ...

5
votes

2
answers

336
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### 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 ...

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votes

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### 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 $\...

3
votes

0
answers

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### 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 ...

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vote

0
answers

156
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### 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 ...

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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 ...

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1
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144
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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{...

3
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1
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179
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### 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}...

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votes

1
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138
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### 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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3
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463
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### 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 ...

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1
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333
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### 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\...

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### 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 $f\colon\mathbb{P}^1\to X$ ...

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### 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 ...

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votes

1
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### 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})/\...

5
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### 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 ...

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votes

1
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327
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### 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 ...

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### 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 ...

6
votes

1
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209
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### 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 ...

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295
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### 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 ...

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287
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### 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 ...

3
votes

1
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254
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### 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 ...

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0
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### 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 ...