Consider the fiber bundle $X\longrightarrow S^1\times S^1 $ with fibers $\mathbb R^q$.

If $X$ is a $G$-homogeneous space, where $G$ is a simply-connected and connected Lie group i.e $G$ acts transitively on $X$. My questions:

  1. Can always $X$ be written as $G/H$ for $G$ solvable?
  2. Is there any semisimple Lie group who can act transitively on $X$?
  • $\begingroup$ Hint: Think of $G=SL(2,R)$ or its universal cover. Can you think of a manifold homotopy-equivalent to the circle on which $G$ acts transitively? $\endgroup$ – Moishe Kohan Feb 11 '18 at 4:16
  • $\begingroup$ $\mathbb R^2\setminus \{0\}$? What about my first question? $\endgroup$ – Ronald Feb 11 '18 at 4:20
  • $\begingroup$ I do not understand your first question. Is there any relation between $G$ in the 1st sentence and $G$ in the 1st question? $\endgroup$ – Moishe Kohan Feb 11 '18 at 4:25
  • $\begingroup$ Sorry, I meant if $X=G/H$ for any Lie group $G$. Is $X$ isomorphic to $R/J$ for a solvable Lie group. i.e is there a solvable Lie group $R$ acting transitively on $X$? $\endgroup$ – Ronald Feb 11 '18 at 4:29
  • $\begingroup$ If you give me an example of a nontrivial bundle $R^q\to E\to T^2$ whose total space is a homogeneous space, I might think about your question. $\endgroup$ – Moishe Kohan Feb 11 '18 at 4:32

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