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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 $V_p$ be the tangent space at $p$ of the leaf containing $p$, and let $V^\perp_p$ be the orthogonal complement of $V_p$. I make the hypothesis that if $G$ is abelian then the distribution $V^\perp_p$ is integrable, i.e., there exists a foliation of which the $V^\perp_p$ are the tangent spaces.

Is my hypothesis correct? If yes, can somebody give me the reference to a proof?

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  • $\begingroup$ I have added a fifth tag differential-geometry as more comprehensive, Are you contrary? $\endgroup$
    – agt
    Commented Apr 29, 2011 at 15:59
  • $\begingroup$ Is the group action isometric? $\endgroup$
    – Deane Yang
    Commented Apr 29, 2011 at 17:25
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    $\begingroup$ What about Hopf fibration? It is an isometric action of $S^1$ on $S^3$ and $V^\perp$ is not integrable. $\endgroup$
    – Anton Petrunin
    Commented Apr 29, 2011 at 17:55

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Any contact manifold with a weakly compatible metric provides a counterexample, where the action is $\mathbb{R}$ acting by the Reeb flow. "Weakly compatible" here just means that there is a metric $g$ such that if $V$ is the Reeb vector field and $\xi$ the contact distribution, $\xi \perp_g V$.

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  • $\begingroup$ Ok, thank you, I was too optimistic. $\endgroup$
    – Bruno Galvan
    Commented Apr 30, 2011 at 12:18

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