Differentiable spaces that are subcartesian

TL;DR

This paper proves that the orbit space of a proper Lie group action on a smooth manifold has a rich differentiable structure, including differential forms and vector fields, satisfying Smith's de Rham theorem.

Contribution

It establishes that such orbit spaces are subcartesian differentiable spaces with a reflexive differential structure, extending the understanding of their geometric properties.

Findings

Orbit space has a differentiable space structure in Smith's sense.

The orbit space admits an exterior algebra of differential forms.

The space supports vector fields and their flows.

Abstract

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfies Smith's version of de Rham's theorem. Because the orbit space is a locally closed subcartesian space, it has vector fields and their flows.