Core reduction for singular Riemannian foliations in positive curvature

TL;DR

This paper investigates the geometric structure of singular Riemannian foliations with positive curvature, showing that the existence of a pre-section implies the leaf space has a boundary, especially in polar foliations.

Contribution

It establishes a link between positive curvature, the existence of pre-sections, and boundary properties of leaf spaces in singular Riemannian foliations.

Findings

Leaf space has boundary when a pre-section exists in positively curved foliations.

Polar foliations in positively curved manifolds have leaf spaces with nonempty boundary.

Abstract

We show that for a smooth manifold equipped with a singular Riemannian foliation, if the foliated metric has positive sectional curvature, and there exists a pre-section, that is a proper submanifold retaining all the transverse geometric information of the foliation, then the leaf space has boundary. In particular, we see that polar foliations of positively curved manifolds have leaf spaces with nonempty boundary.