On the Inner Automorphisms of a Singular Foliation

TL;DR

This paper proves that the time-one flow of an element in a singular foliation always acts as an automorphism, highlighting a fundamental property of these geometric structures.

Contribution

It provides an alternative proof that the exponential of a singular foliation element is an automorphism, a non-trivial fact in the theory.

Findings

Time-one flow of a singular foliation element is an automorphism

Alternative proof of a key property in singular foliation theory

Highlights the structure-preserving nature of flows in singular foliations

Abstract

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give an alternative proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation (i.e. its exponential) is an automorphism of the singular foliation.