Smoothness of submetries

We prove that a Riemannian submersion between smooth, compact, non-negatively curved Riemannian manifolds has to be smooth, resolving a conjecture by Berestovskii–Guijarro. We show that without any curvature assumption, the smoothness of the base is implied by the smoothness of the total space. Results are proven in the much more general setting of submetries. These are metric generalizations of Riemannian submersions and isometric actions, which have recently appeared in different areas of geometry.