Smoothness of isometric flows on orbit spaces and applications to the theory of foliations

We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M\mathrm{/}K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow on the manifold $M$. As a direct corollary we infer the smoothness of isometric actions on orbit spaces. Another relevant application of our result concerns Molino’s conjecture, which states that the partition of a Riemannian manifold into the closures of the leaves of a singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino’s conjecture for the main class of foliations considered in his book, namely orbit-like foliations.