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

2013/01/12 by Alexandrino, Marcos M., Radeschi, Marco
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1301.2735

Abstract

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 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.

Related

• Isometries between leaf spaces
• On smoothness of isometries between orbit spaces
• Singular Riemannian Foliations and the prescribing scalar curvature problem
• The closure of linear foliations
• Smooth shifts along flows