Desingularization of singular Riemannian foliation

TL;DR

This paper develops a method to desingularize singular Riemannian foliations on compact manifolds by successive blow-ups, resulting in a regular foliation and establishing a connection between their leaf spaces.

Contribution

It generalizes Molino's desingularization result and shows that the leaf space of a singular Riemannian foliation can be approximated by Riemannian orbifolds in the Gromov-Hausdorff sense.

Findings

Constructed a regular Riemannian foliation via blow-ups

Established a desingularization map projecting leaves

Proved leaf space is a Gromov-Hausdorff limit of orbifolds

Abstract

Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of the s.r.f are compact, then, for each small epsilon, the regular foliation can be chosen so that the desingularization map induces an epsilon-isometry between the leaf space of the regular Riemannian foliation and the leaf space of the s.r.f. This implies in particular that, the leaf space of the s.r.f is is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds.