Riemannian foliations and quasifolds

TL;DR

This paper extends the understanding of leaf spaces of Riemannian foliations by showing they are diffeological quasifolds under mild conditions, generalizing orbifold structures to include more complex quotient spaces.

Contribution

It proves that the leaf space of a Killing Riemannian foliation is a diffeological quasifold and relates the holonomy groupoid to affine actions of countable groups.

Findings

Leaf space is a diffeological quasifold modeled by quotients of Euclidean space.

Holonomy groupoid is Morita equivalent to an affine action groupoid.

Results generalize orbifold structures to quasifolds.

Abstract

It is well known that, by the Reeb stability theorem, the leaf space of a Riemannian foliation with compact leaves is an orbifold. We prove that, under mild completeness conditions, the leaf space of a Killing Riemannian foliation is a diffeological quasifold: as a diffeological space, it is locally modelled by quotients of Cartesian space by countable groups acting affinely. Furthermore, we prove that the holonomy groupoid of the foliation is, locally, Morita equivalent to the action groupoid of a countable group acting affinely on Cartesian space.