Questions tagged [homogeneous-spaces]
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231
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33
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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 $M$ ...
17
votes
3
answers
2k
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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 $\...
17
votes
2
answers
1k
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Is the wonderful compactification of a spherical homogeneous variety always projective?
Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer
...
14
votes
3
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Is there a "unique" homogeneous contact structure on odd-dimensional spheres?
Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in S^{2n-1}...
14
votes
1
answer
475
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Is Bing's countable connected space topologically homogeneous?
In this paper R.H. Bing has constructed his famous example of a countable connected Hausdorff space.
The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\...
13
votes
3
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Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...
13
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2
answers
865
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References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians
Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
...
12
votes
1
answer
252
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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 $\...
12
votes
0
answers
462
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 ...
11
votes
1
answer
534
views
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, ...
11
votes
1
answer
412
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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 ...
11
votes
2
answers
585
views
Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension
Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.
I have heard that, for any real number $\...
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 ...
11
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0
answers
666
views
Big tangent bundle
Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
10
votes
1
answer
271
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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 ...
9
votes
1
answer
326
views
k-flats in homogeneous spaces
In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.
Question. Are ...
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 $D$ ...
8
votes
2
answers
741
views
Is $G/T$ a projective variety?
Let $G$ be a semisimple Lie group and $T$ be its maximal torus. Can we say that $G/T$ is a projective variety?. Is there any proof or counterexample for it?
8
votes
1
answer
553
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 ...
8
votes
2
answers
342
views
Spherical roots, restricted roots, and the dual group of a symmetric variety
Let $k$ be an algebraically closed field of characteristic $0$ and $G$ a semisimple simply-connected group over $k$. Consider a symmetric variety of the form $X=G/H$, for $H=G^\theta$ the fixed point ...
8
votes
2
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486
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When is an orbit spherical?
I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, ...
8
votes
1
answer
274
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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 $\...
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 ...
8
votes
1
answer
473
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 ...
8
votes
1
answer
529
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 ...
8
votes
1
answer
2k
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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 ...
8
votes
1
answer
234
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Compact simply-connected homogeneous symplectic manifold
I was reading a paper in which the authors use the fact that any compact simply-connected homogeneous symplectic manifold has non-zero Euler characteristic. They prove it by quoting a theorem by ...
8
votes
1
answer
1k
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Wonderful compactification
Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...
8
votes
2
answers
627
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{...
8
votes
1
answer
145
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 ...
8
votes
0
answers
224
views
Lattice point counts on the determinantal variety
I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...
8
votes
0
answers
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
$$
E_8/E_7, ~~~~ E_7/E_6?
$$
...
8
votes
0
answers
128
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Local vs global homogeneity of topological spaces
I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have rigid spaces, i.e., topological spaces with trivial homeomorphism group.
Question. ...
8
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0
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514
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example of an n-transitive but not infinitely transitive group action on a space
Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ ...
7
votes
1
answer
265
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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
$$
G \times_{\...
7
votes
2
answers
2k
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Embeddings of flag manifolds
Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
7
votes
1
answer
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Tangent bundle of a homogeneous space and the euler exact sequence
Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$:
$$0 \to \mathfrak{h} \to \mathfrak{g} \to \...
7
votes
3
answers
987
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,...
7
votes
2
answers
324
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Explicit description of SU(2,2)/U
Consider the real diagonal $4\times 4$ - matrix
$$I_{2,2}={\rm diag}(1,1,-1,-1)$$
and the corresponding special unitary group
$$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\...
7
votes
2
answers
811
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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 ...
7
votes
1
answer
164
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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 $\mathscr{B}$...
7
votes
3
answers
548
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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 ...
7
votes
0
answers
121
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 ...
7
votes
0
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251
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What is the status of this fifty-year-old conjecture of Kostant?
On page 3.27 of his 1963 thesis on the cohomology of homogeneous spaces as approached through the Eilenberg–Moore spectral sequence, Paul Baum states the following conjecture, which he attributes to B....
6
votes
2
answers
818
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Do all homogeneous spaces have homogeneous compactifications?
Let $X$ be a separable metric space which is homogeneous, i.e. for every two points $x,y\in X$ there is a homeomorphism $h$ of $X$ onto itself such that $h(x)=y$.
A compactification of $X$ is a ...
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 ...
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 ...
6
votes
3
answers
213
views
Self-Intersecting Geodesics in Homogeneous Spaces
A result of Helgason's is that any self-intersecting geodesic in a Riemannian globally symmetric space is simple and closed. To what extent does this generalise to Riemannian naturally reductive ...
6
votes
2
answers
408
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 ...
6
votes
1
answer
612
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Combinatorics of the Cohomology Ring of the Lagrangian Grassmannians
The Lagrangian Grassmannian is an important example in symplectic geometry, see here or here for details. It shares many similarities with the ordinary Grassmannians (as one would expect from the name)...