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# Questions tagged [homogeneous-spaces]

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### Monodromy action on homogeneous spaces

If $H$ is a Lie subgroup of $G$, then there is a fibration sequence $$G/H\to BH\to BG.$$ By choosing a model for $EG$ we can promote this into a fibre bundle. My question is about how to understand ...
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### Is every linear Lie group of bounded geometry?

$\newcommand\norm{\lVert#1\rVert}$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 ...
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### Decomposition of fiber product of $G$-sets in $G$-orbits

I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem: Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then the right ...
131 views

### Homogeneous space and rational section

Let's embed $\operatorname{SO}_n$ inside $\operatorname{GL}_n$ through the standard representation. Does the map $\operatorname{GL}_n\rightarrow \operatorname{GL}_n/{\operatorname{SO}_n}$ admit a ...
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### 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 $G$ ...
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### 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 ...
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### 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 ...
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### 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 ...
A corollary of the Mostow-Palais theorem is that every homogeneous space for a compact group is a linear group orbit. In other words, if $H$ is a closed subgroup of a compact group $K$ then there ...