Equifocality of a singular riemannian foliation

TL;DR

This paper proves that regular leaves in a singular Riemannian foliation are equifocal, allowing reconstruction of the foliation via parallel submanifolds, and shows the endpoint map's properties under certain conditions.

Contribution

It generalizes previous results by establishing equifocality and endpoint map properties for singular Riemannian foliations without sections.

Findings

Regular leaves are equifocal.

The foliation can be reconstructed from parallel submanifolds.

The endpoint map is a covering map under trivial holonomy.

Abstract

A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previous results of the authors on singular riemannian foliations with sections.