All Questions
Tagged with lie-groups homogeneous-spaces
94 questions
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}...
CommunityBot
- 1
2
votes
0
answers
46
views
Are maps between cohomology of homogeneous vector bundles morphisms of representations?
Let $X = G/P$ a rational homogeneous variety, e.g. a grassmannian. Consider a short exact sequence $$ 0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow 0$$
where $E_i$ are ...
- 41
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
$$
T(M) = A \...
- 108k
4
votes
0
answers
226
views
Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
CommunityBot
- 1
6
votes
0
answers
345
views
Why can't a Lie group act transitively on a finite volume hyperbolic manifold?
In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...
- 4,081
2
votes
0
answers
125
views
The double quotient of SU(N) by its diagonal maximal torus
$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...
7
votes
1
answer
178
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Homogeneous metric connections on 3-dimensional Lie groups
Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...
- 108k
7
votes
1
answer
280
views
Non-homogeneous line bundles over a homogeneous space
Let $G$ be a compact Lie group and $G/K$ a connected homogeneous space. A homogeneous vector bundle over $G/K$ is a vector bundle is one that is isomorphic to a vector bundle of the form
$$
G \times_{\...
- 3,625
3
votes
0
answers
80
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
- 2,527
3
votes
1
answer
119
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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 ...
- 105k
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
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 ...
- 6,247
3
votes
0
answers
50
views
Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?
Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...
- 5,215
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 ...
- 869
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 $r(G) > r(K)$ (...
- 9,580
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
1
vote
0
answers
132
views
About the classification of simply connected homogeneous 3-manifolds
I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...
- 63.9k
3
votes
0
answers
111
views
Almost two-point homogeneous spaces
I define a NICE space to be a connected Riemannian manifold $M$ such that for any two distinct points $p,q\in M$, there exists an isometry $R_{p,q}$ exchanging these two points (that is such that $R_{...
- 245
1
vote
0
answers
196
views
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,...
- 2,446
2
votes
1
answer
484
views
Sobolev norm of distance function on Riemannian manifold
$\DeclareMathOperator\SL{SL}$Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and ...
- 63.9k
3
votes
1
answer
273
views
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 ...
CommunityBot
- 1
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 ...
- 675
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
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 ...
- 7,842
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
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 ...
- 2,289
2
votes
2
answers
213
views
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 ...
- 4,081
2
votes
0
answers
35
views
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 $G$ ...
- 1,565
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 ...
- 15.5k
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 ...
- 1,908
2
votes
0
answers
95
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 ...
- 63.9k
2
votes
0
answers
557
views
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 (...
- 1,565
2
votes
0
answers
82
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 ...
- 187
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
0
votes
1
answer
254
views
$SU(k)/SO(k)$ 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 ...
- 148k
3
votes
1
answer
152
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 ...
CommunityBot
- 1
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 $\...
- 8,098
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 ...
- 26.3k
1
vote
1
answer
408
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 ...
- 309
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 ...
- 108k
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
1
vote
0
answers
517
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 ...
- 193
3
votes
0
answers
105
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 ...
- 568
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} \...
- 37.8k
9
votes
0
answers
203
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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 $D$ ...
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 ...
- 31.5k
2
votes
1
answer
148
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 ...
- 14.2k
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
2
votes
1
answer
51
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 \...
- 35.5k
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 ...