SMOOTHNESS OF ISOMETRIC FLOWS ON ORBIT SPACES AND APPLICATIONS

We prove here that given a proper isometric action K × M → 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 a 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.