All Questions
Tagged with lie-groups homogeneous-spaces
94 questions
33
votes
1
answer
4k
views
Isometry group of a homogeneous space
Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
- 33.3k
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 $\...
- 4,081
13
votes
3
answers
882
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 ...
12
votes
1
answer
264
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 $\...
- 156k
12
votes
0
answers
477
views
Maurer-Cartan equation for Lie groups/homogeneous space vs. Maurer-Cartan of deformation theory
What is the relationship between the Maurer-Cartan equation
$$
d\theta + \dfrac{1}{2}[\theta,\theta] = 0
$$
satisfied by Maurer-Cartan forms on Lie groups, or by pullbacks of Maurer-Cartan forms along ...
- 4,070
11
votes
0
answers
336
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 ...
- 5,760
10
votes
1
answer
272
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 ...
- 29.1k
9
votes
3
answers
790
views
A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
9
votes
0
answers
203
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$ ...
8
votes
1
answer
566
views
Quotienting $SU(3)$ by $U(1)$?
As is well-known, if we quotient $SU(2)$ by the action of $U_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on ...
- 155
8
votes
1
answer
298
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 $\...
- 639
8
votes
1
answer
491
views
Does a free action always induce a diffeomorphism?
Suppose that $G$ is a Lie group with a transitive action on a smooth manifold $M$. The regular theory of Lie groups tells us that $G$ and $M$ are diffeomorphic if the isotropy group is trivial.
The ...
- 355
8
votes
1
answer
610
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 ...
- 141
8
votes
1
answer
3k
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 ...
- 449
8
votes
2
answers
649
views
How to "lift" a transitive group action on a manifold?
Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$.
QUESTION: is there a general prescription to obtain a Lie group $\widetilde{...
- 2,527
8
votes
1
answer
151
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 ...
- 33.3k
8
votes
0
answers
228
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?
$$
...
- 305
7
votes
1
answer
280
views
Non-homogeneous line bundles over a homogeneous space
Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...
- 257
7
votes
3
answers
1k
views
Is the group of isometries of a homogeneous Riemannian manifold maximal?
I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of G,...
- 71
7
votes
2
answers
833
views
Lie groups acting transitively (and isometrically) on anti de Sitter spaces
I hope this question is not deemed too localised.
Recall that anti de Sitter space is the lorentzian analogue of hyperbolic space; that is, a simply-connected lorentzian manifold of constant negative ...
- 33.3k
7
votes
1
answer
178
views
Homogeneous metric connections on 3-dimensional Lie groups
Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...
- 73
7
votes
3
answers
577
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 ...
- 1,565
7
votes
0
answers
123
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 ...
- 33.3k
6
votes
2
answers
448
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 ...
- 33.3k
6
votes
1
answer
2k
views
Connections on a Lie Group
A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...
- 1,378
6
votes
0
answers
345
views
Why can't a Lie group act transitively on a finite volume hyperbolic manifold?
In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...
- 4,081
6
votes
0
answers
163
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 ...
- 33.3k
5
votes
1
answer
261
views
Non-integrable almost complex structure for complex projective $3$-space
It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...
5
votes
1
answer
203
views
Can all hermitian symmetric spaces be realised as coadjoint orbits?
Here is what I know. Assume $M\cong G/K$ is an irreducible hermitian symmetric space. Denote the Lie-algebra of $K$ by $\mathfrak{t}$. Proposition 1.2. chapter 3 in
Wienhard - Bounded cohomology and ...
- 61
5
votes
1
answer
549
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 $R>0$ ...
user130903
5
votes
2
answers
1k
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 ...
- 51
5
votes
1
answer
472
views
Finite dimensional homogeneous spaces of $Diff(S^1)$
This question is a refined version of Representations of infinite dimensional Lie algebras as vector fields on manifolds
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^1)$. ...
- 548
5
votes
1
answer
902
views
Explicit description of the Lagrangian Grassmannian as a homogeneous space
Looking at this and this question about the Lagrangian Grassmannian, and its linked Wikipedia description as the quotient of $Sp(N)$ by the unitary group $U(n)$, I wondering what is the explicit ...
- 459
5
votes
1
answer
1k
views
G-invariant differential forms on homogeneous space of Lie Groups
Let $G$ be a connected Lie Group and $K<G$ a maximal compact subgroup.
Denote by $\Omega^q(G/K)^G$ the $G$-invariant real-valued $q$-forms on the manifold $G/K$, i.e. those forms $\omega$ s.t. $g^*...
- 115
5
votes
1
answer
134
views
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 ...
- 179
5
votes
0
answers
146
views
Is every linear Lie group of bounded geometry?
$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
- 293
5
votes
0
answers
363
views
Classification of Compact Symplectic Homogeneous Spaces
Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...
- 59
4
votes
3
answers
862
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 ...
- 177
4
votes
1
answer
252
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 ...
4
votes
2
answers
332
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 ...
- 4,081
4
votes
0
answers
226
views
Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
- 4,081
4
votes
0
answers
101
views
Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds
I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
4
votes
1
answer
324
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}...
- 1,565
3
votes
2
answers
2k
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} \...
- 1,379
3
votes
3
answers
359
views
Homogeneous spaces which are not torus bundle over flag manifolds
For a compact semisimple Lie group $G$, what is an example of a homogeneous space of $G$ which is not a torus bundle over a generalized flag manifold of $G$. Examples for $SU(N)$ would be of most ...
- 357
3
votes
1
answer
181
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\}}...
- 1,066
3
votes
2
answers
337
views
compute the automorphism of Iwasawa manifold
An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup.
We can also refer to Griffiths and Harris's Principles of Algebraic Geometry p....
- 197
3
votes
1
answer
119
views
Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?
$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...
- 321
3
votes
1
answer
203
views
Free $S^1$-action on compact homogeneous spaces
Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...
- 143
3
votes
2
answers
120
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 ...
- 1,565