Question

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?

Answer 1

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, $\xi \perp_g V$.