# Questions tagged [homogeneous-spaces]

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### Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space

I asked this question at StackExchange, but got no answer. So I am reposting it here. I eventually want to check how the Hopf fibration $$S^{2n+1}\to {\mathbb C}P^n$$ satisfies the Riemannian ...
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### Help in understanding the singular system of linear forms and non escape of mass

I am having some trouble in understanding certain portions of the following paper by KKLM https://link.springer.com/article/10.1007/s11854-017-0033-4 So in proposition 3.1, they proved the estimate ...
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### Criteria on $f$ such that $\begin{bmatrix} 1 & f(t)\\ 0 & 1 \end{bmatrix}\mathbb Z^2$ is equidistributed on the circle (periodic orbit)

Consider $\left(\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}\mathbb Z^2 \right)_{t\in \mathbb R} \cong S^1$. Let $\mu$ denote the rotation invariant Haar measure $m$ on this orbit. I wonder if ...
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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 $\mathscr{B}$...
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### 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 (...
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### 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 ...
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### Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"

I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16. If I understood correctly, in the second ...
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### Integrating on orbits of algebraic groups

Suppose $G$ is a $\mathbb{Q}$-algebraic group (I am interested in the semisimple case) acting rationally on a vector space $V_\mathbb{Q}$. Let $x \in V_\mathbb{Q}$ be a non-zero rational vector. ...
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### Non-integrable almost complex structure for complex projective $3$-space

It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...
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### Product decomposition into semisimple and unipotent parts of an algebraic group (in Borel’s LAG)

Let $G$ be an algebraic group, i.e., an affine reduced, separated $k$-scheme of finite type with structure of a group. In Borel’s Linear Algebraic Groups Theorem III.10.6(4) says Theorem 10.6 (3): ...
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### 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 ...
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### Classification of "homogeneous" submanifolds of ℝⁿ

I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if: it is a closed connected smooth submanifold of $\mathbb R^n$, for every $p, q$ in $M$, there is a ...
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### 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, ...
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