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?



Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
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, $\xi \perp\_g V$.


