15
votes
Accepted
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
Where to begin!
The ergodicity of non-compact subgroups (singular tori) was used by Margulis to prove that higher rank lattices $\Gamma$ are arithmetic.
Once you have that $\Gamma $ is arithmetic, ...
14
votes
Accepted
Planes in Lagrangian Grassmannians
This is, indeed, true.
To prove this, assume we have an embedding $\mathbb{P}^2 \to \operatorname{LGr}(V)$ (where $V$ is a symplectic vector space). Let $U \subset V \otimes \mathcal{O}$ be the ...
- 39.3k
14
votes
Accepted
Do all homogeneous spaces have homogeneous compactifications?
Since you want a connected example:
A surface of infinite genus has no homogeneous compactification.
Indeed first observe a dense locally compact subset has to be open.
So the surface has to be open, ...
- 63.9k
13
votes
Accepted
Is a manifold paracompact? Should it be?
every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components.
A manifold is paracompact if and only ...
- 37.8k
13
votes
Examples of the Thurston geometries with transitive Lie group action
Here is a cool fact about $\mathrm{SL}(2, \mathbb{R}) / \mathrm{SL}(2, \mathbb{Z})$; it is homeomorphic to the complement of the trefoil knot in the three-sphere. Apparently this was first proved by ...
- 28.1k
13
votes
Accepted
Classification of homogeneous Einstein manifolds
Yes, the question is still open.
I suggest to read this quite recent paper by Kerr and Böhm. It reviews some of the most important advances in the problem and includes several open problems.
- 2,446
12
votes
Accepted
Compact Lie group inclusions that are trivial on all homotopy groups
Well, the identity component $H^0$ of $H$ would have to be abelian, since, otherwise, it would have a compact simple component, and the induced map on $\pi_3$ would then be nontrivial.
If $H^0$ has ...
- 108k
12
votes
Do all homogeneous spaces have homogeneous compactifications?
The countable discrete space $\omega$ is a counterexample.
Suppose $Y$ is a homogeneous compactification of $\omega$, with $X \subset Y$ being homeomorphic to $\omega$. As $Y$ is infinite, it ...
- 29.7k
11
votes
Examples of the Thurston geometries with transitive Lie group action
These days, many popular quotients of ("real") hyperbolic three-space are obtained as $\Gamma\backslash SL_2(\mathbb C)/SU(2)$, where $\Gamma$ is the group of units in a quaternion division ...
- 23k
11
votes
A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
Remark: I assume that you want $A$ to be a non-trivial bundle. Otherwise, of course, any parallelizable compact manifold would be an example. In particular, any compact Lie group would be an ...
- 108k
10
votes
Embeddings of flag manifolds
In general there is a more efficient way: $a_1,\ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${\mathbb P}(U)$, ...
- 1,602
10
votes
How special are homogeneous spaces?
I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space $...
- 8,098
10
votes
Accepted
Non-homogeneous line bundles over a homogeneous space
Yes. This happens whenever $G$ admits nontrivial vector bundles $E$ which can be equipped with an equivariant structure for the $K$-action. Then $E$ descends in the same way to $G \times_{\rho} E \to ...
- 3,625
10
votes
Accepted
Non-integrable almost complex structure for complex projective $3$-space
There are lots of non-integrable almost-complex structures on $\mathbb{CP}^3$, but the one you are looking for is, I suspect, the following.
First I will explain the twistor fibration, which is a ...
- 6,247
10
votes
Accepted
A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
You are asking about manifolds whose projective span is equal to $k$, according to the main definition of Grant and Schutte - Projective span of Wall manifolds.
Any manifold with zero Euler ...
- 35.9k
9
votes
Accepted
Are invariant forms on homogeneous spaces necessarily closed?
Note that the answer depends on the pair $(G,K)$.
For example, if $K=\{e\}$, then one is asking whether the ring of left-invariant forms on $G$ consists only of closed forms. This only happens when $...
- 108k
9
votes
Accepted
Homology of the free loop space of generalized flag varieties
Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (...
- 148k
9
votes
Accepted
Examples of the Thurston geometries with transitive Lie group action
This is an answer to questions 7 and 8 (I have to say, having 8 questions in one post is way too much for my taste):
Suppose that $M$ is a finite-volume quotient of $H^3$ or a compact quotient of $H^...
- 12.2k
8
votes
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem.
Actually if you will investigate Ratner's original ...
8
votes
Flag manifolds as incidence correspondences
The "incidence" relation (at least for types $E_6$ and $E_7$) for all pairs of vertices $i$, $j$ is described in S. Garibaldi, M. Carr "Geometries, the principle of duality, and algebraic groups", ...
- 1,602
8
votes
Is a manifold paracompact? Should it be?
Another good gift of paracompactness:
A Hausdorff $C^k$ manifold ($k\ge0$) is metrizable iff it is
paracompact.
This is also true for infinite dimensional manifold modelled on a Banach space (...
- 60.5k
8
votes
Separability of subspaces of homogeneous topological spaces
This fails even for topological groups. For example, the separable topological group $\mathbb{Z}^{\mathfrak{c}}$ contains a subgroup of uncountable cellularity (hence certainly not separable!). This ...
- 7,125
8
votes
Are the quaternionic Grassmannians quaternionic Kaehler manifolds?
Perhaps the OP really wants to know why quaternionic Grassmannians other than the quaternionic projective spaces are not considered to be 'quaternion-Kähler'.
The reason goes back to Berger's ...
- 108k
8
votes
Accepted
Can all hermitian symmetric spaces be realised as coadjoint orbits?
This is true. One can use a few facts from Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces to show that, indeed, $K = \mathrm{Stab}_G(Z)$.
Since $M=G/K$ is an irreducible Hermitian ...
- 108k
8
votes
A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
The product of a 2-sphere with a torus is an example. There are complex analytic examples due to Beauville (Complex manifolds with split tangent bundle). Every 3-manifold has trivial tangent bundle, ...
- 26.3k
7
votes
Accepted
Ideal of the Spinor variety $S^{10}\subset\mathbb{P}^{15}$
Mukai, CURVES AND SYMMETRIC SPACES, I, (0.1)
- 39.3k
7
votes
Accepted
Torus actions on $Sp(n)$-spheres
The presentation $\mathbb{S}^{2n-1} = \mathit{U}_n/\mathit{U}_{n-1}$ is tantamount to considering $\mathbb{S}^{2n-1}$ as the unit sphere in $\mathbb{C}^n$ (as $\mathit{U}_n$ is the group of $\mathbb{C}...
- 32.5k
7
votes
Accepted
Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
The following paper says that your map $\mu$ is indeed surjective; the author attributes the result over $\mathbb C$ to Tits.
...
- 11.2k
7
votes
Accepted
k-flats in homogeneous spaces
If the k-flats are compact, then the space must be symmetric
(Molina-Olmos, J. Differential geometry
45 (1997) 575-592; see also Proc. Amer. Math. Soc. 129 (2001), 3701-3709).
Homogeneous spaces (non-...
- 86
7
votes
Accepted
The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$
Yes, the branching is multiplicity free. See e.g. Theorem 8.1.3 and Theorem 8.1.4 in Symmetry, Representations, and Invariants by Nolan Wallach and Roe Goodman. The quotient space is a sphere $S^{n-1}$...
- 8,597
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