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What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:

There are two non zero vector fields $X, Y \in g$, the lie algebra of $G$, such that the corresponding one dimensional foliations $F_{X}$ and $F_{Y}$ are not topological equivalent?

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    $\begingroup$ Such a Lie group $G$ is always isomorphic to $G\times\{e\}$, its product with the trivial group. Perhaps there are some refinements to be made to the question. $\endgroup$
    – Peter Crooks
    Commented Jan 19, 2014 at 16:21
  • $\begingroup$ Yes. The simplest example: take a torus $T:={\mathbb R}^2/{\mathbb Z}^2$ and a couple of tangent vectors at $0$. One can provide the foliation with compact leafs, the other may have all leafs dense. $\endgroup$
    – Sasha Anan'in
    Commented Jan 19, 2014 at 16:24
  • $\begingroup$ @Sasha torous does not satisfies the condition in my question. it is product of $S^{1}$ by $S^{1}$ $\endgroup$
    – Ali Taghavi
    Commented Jan 19, 2014 at 16:27
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    $\begingroup$ Take Sasha's example, and embed it into the simple Lie group $SU(3)$ (where $\mathbb{T}^2$ embeds as a maximal torus). $\endgroup$
    – Alain Valette
    Commented Jan 19, 2014 at 16:45
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    $\begingroup$ @Alain @ Ali Yes. Many compact groups which are not products contain a compact one-parameter subgroup as well as non-compact one. The cosets with respect to those provide desired foliations. $\endgroup$
    – Sasha Anan'in
    Commented Jan 19, 2014 at 16:55

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