The homogeneous-spaces tag has no usage guidance.

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### 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 ...

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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
...

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### Geometric interpretation of splitting of sequence associated to a homogeneous space

Let $G$ be a Lie group acting transitively on a smooth manifold $M$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\xi : \mathfrak{g} \to \Gamma(TM)$ be the Lie algebra homomorphism sending $...

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### Fiber product of nilmanifolds

Let $M_1$ and $M_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi_i:M_i \to B_i\ \ i=1,2$. Suppose that $B_1=B_2$ and that the fibers are torus of the same dimension and ...

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### 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 ...

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### On the maximal powers of $q$ which arise in a quantum product

Let $X=G/P$ be a generalized flag variety (where $G$ denotes a connected, simply connected, semisimple complex linear algebraic group and $P$ a parabolic subgroup). In this paper by Fulton and ...

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### Two nilmanifolds of the same Lie group

By a nilmanifold I mean a quotient $M =\Gamma \backslash G$ of a connected, simply-connected nilpotent real Lie group $G$ by the left action of a maximal lattice , i.e. a discrete cocompact subgroup....

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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 \...

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### Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...

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### A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...

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### Curvature Characterization of Homogeneous Spaces

A (Riemnnian) symmetric space is locally characterized by the first covariant derivative of its curvature tensor, namely, a manifold $M$ is locally a symmetric space if and only if $\nabla R=0$ (where ...

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### Isotropically conjugate points in reductive spaces

Is it the case that a naturally reductive Riemannian homogeneous space that contains a pair of conjugate points necessarily contains a pair of isotropically conjugate points? That is, there is a pair ...

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### Geodesic Digons in Reductive Spaces

Consider a naturally reductive homogeneous space $M$ of positive curvature. Is it necessarily the case that there exists a geodesic digon whose interior angles sum to less than $2\pi$? Angles are ...

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### 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 ...

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### Mirror Symmetry for Homogeneous Spaces other than Flag Manifolds

Mirror symmetry is (reasonably) well understood for the general flag manifolds, due to the work of Kim, Givental, Rietsch, and others. Do there exist other homogeneous spaces for which mirror symmetry ...

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### 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 ...

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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....

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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
$$
...

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### Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition ...

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### Canonical class of partial flag variety

Let $F(d_1,d_2,\ldots,d_k;n )$ be the variety of all flags $\mathbb A^{d_1} \subset\mathbb A^{d_2}\subset\ldots\subset \mathbb A^{d_k}\subset \mathbb A^{n}$. This variety has the natural maps to ...

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### Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant ...

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### Equivariant Almost Complex Structures on the Full Flag Manifolds

On complex projective space ${\bf CP}^m$, there exists a unique $SU(m+1)$-equivariant almost-complex structure. What happens for the case of the full flag manifold of $SU(m+1)$, which is to say the ...

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### Poisson Manifold Structures on Even Dimensional Spheres

The $2n$-sphere, for $n=1,2,3$, possess a (non-trivial) Poisson manifold structure. Is this still true for $n > 3$? Describing the spheres as homogeneous spaces $SO(n)/S(n-1)$, are there Poisson ...

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### multiplicity free chain of Lie subgroups gives a completely integrable system on symplectic manifold

Assume we have a symplectic manifold $N$ and a Hamiltonian action of a Lie group $G$, such that the algebra of all $G$-invariant functions on $N$ is commutative under Poisson bracket.
Furthermore, ...

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### Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...

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173 views

### Smooth morphism to homogeneous spaces and fibers

Let $f:X \to Y$ be a smooth morphism between projective varieties. Suppose $Y$ is a homogeneous space. Under what additional condition on $f$, can we conclude that every fibers of $f$ are isomorphic?

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### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

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### $\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?

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### 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 $\...

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### Basics on lattice in classical groups

as a beginner,I am not sure whether this question is too basic to post here./-\。
Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...

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### 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 ...

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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}...

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### 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 ...

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### Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

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### From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of $...

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### 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 ...

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### 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?

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### Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space

Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...

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### 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 ...

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### Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation.
Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...

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### Invariant Finsler Metrics on Homogeneous Spaces

Given:
1) a Finsler metric $F_p : T_p G \rightarrow \mathbb{R}$ on $SU(N+1)$
2) a $U(N)$ subgroup of $SU(N+1)$ which is a stabilizer subgroup of some point on $CP^N$ (complex projective space) in ...

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### 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....

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### Does the 4-sphere have a nonzero Poisson structure as a Poisson homogeneous space?

It is known that the 4-sphere does not have a symplectic structure. However, it does admit Poisson structures, for example the zero Poisson structure, which is quite boring. Does it have other, more ...

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### 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 ...

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### 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^*...

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### 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 ...

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### Noncommutative Erlangen Program

Has Klein's "Erlangen Program" been generalized/extended to the noncommutative setting (say, à la Connes)? Is there a classification of "noncommutative klein geometries" at least in very low ...

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### Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 \...

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### Homogeneous Spaces and Equivariant Hodge Maps

For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one ...

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### Cross section for closed Lie subgroup in a Lie group

Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?