Isometries between leaf spaces
  In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers-Steenrod's theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations. 
Marcos M. Alexandrino and Marco Radeschi
math.DG