Singular Riemannian Foliations: Exceptional Leaves; Tautness

TL;DR

This paper studies singular Riemannian foliations, establishing global tubular neighborhoods, describing the foliation as a stratification, and proving tautness of leaves under certain geometric conditions.

Contribution

It provides a global geometric and topological framework for singular Riemannian foliations, including the existence of tubular neighborhoods and the tautness of leaves.

Findings

Existence of global tubular neighborhoods for singular Riemannian foliations.

Foliation can be described as a stratification by leaf types.

Leaves are taut when the foliation has no horizontal conjugate points.

Abstract

For a singular Riemannian foliation whose leaves are properly embedded, we show in the first part of this article the existence of global tubular neighbourhoods, and we develop a global description of the foliation as stratification by types of leaves. The second part deals with the further restriction to a foliation without horizontal conjugate points, introduced by Lytchak and Thorbergsson, which in the special case of an isometric group action equals the concept of variationally completeness. Therefrom, we deduce a global geometric description -- the focal points of the leaves are exactly the singular points -- as well as a topological one: tautness of the leaves.