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More precisely, is there a criterion that decides the above question? I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...

$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold $$ M_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma. $$ Let $A\subset \SL(...

This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let $V_p$...