Smoothness of submetries

TL;DR

This paper proves that submetries between certain curved Riemannian manifolds must be smooth, resolving a longstanding conjecture, and extends the result to more general metric settings without curvature assumptions.

Contribution

It establishes the smoothness of submetries in non-negatively curved Riemannian manifolds, resolving a conjecture and generalizing to broader metric contexts.

Findings

Riemannian submersions between non-negatively curved manifolds are smooth.

Smoothness of the base follows from the total space's smoothness without curvature constraints.

Results apply to general submetries, broadening the scope of classical Riemannian geometry.

Abstract

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.