All Questions
Tagged with lie-groups homogeneous-spaces
34 questions with no upvoted or accepted answers
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
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
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
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
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
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
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
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
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
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
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
0
answers
72
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 $...
- 1,488
3
votes
0
answers
154
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 ...
- 1,307
3
votes
0
answers
82
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{...
- 6,741
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
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
$...
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
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 ...
- 1,565
2
votes
0
answers
556
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
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 ...
- 209
2
votes
0
answers
161
views
Intermediate quotient for a Hermitian Symmetric Spaces of $Sp(n)$
We know that $U(N)$ can be embedded into $SU(n+1)$ and that the quotient is isomorphic to complex projective space:
$$
SU(n+1)/U(n) \simeq {\mathbb CP}^{n}.
$$
We can split this process into two ...
- 167
2
votes
0
answers
82
views
Free S^1 action on a symmetric space of compact type
Consider a symmetric space G/H of compact type where rank(G) is greater than rank(H). The Euler characteristic of this space is known to be zero. What can be said about the existence of a fixed point ...
- 21
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 ...
- 1,763
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
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
1
vote
0
answers
137
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 ...
- 239
1
vote
0
answers
121
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:
...
- 1,959
1
vote
0
answers
149
views
Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$
The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...
- 459