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 \(M/K\) 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.