Orbifold-like and proper $\mathfrak g$-manifolds

TL;DR

This paper introduces orbifold-like $rak g$-manifolds, a class where the completion of infinitesimal group actions results in orbifold structures, enabling better geometric analysis of proper $rak g$-actions.

Contribution

It defines and studies orbifold-like $rak g$-manifolds, extending the theory of infinitesimal group actions to a well-behaved class with orbifold completions.

Findings

Orbifold-like $rak g$-manifolds have orbifold structures upon completion.

Proper $rak g$-actions on these manifolds exhibit properties similar to classical proper group actions.

The class allows for many geometric constructions and generalizations.

Abstract

In [4] and [5], we generalized the concept of completion of an infinitesimal group action $\zeta : {\mathfrak g} \to \mathfrak X (M)$ to an actual group action on a (non-compact) manifold $M$, originally introduced by R. Palais [9], and showed by examples that this completion may have quite pathological properties (much like the leaf space of a foliation). In the present paper, we introduce and investigate a tamer class of $\mathfrak g$-manifolds, called orbifold--like, for which the completion has an orbifold structure. This class of $\mathfrak g$-manifolds is reasonably well-behaved with respect to its local topological and smooth structure to allow for many geometric constructions to make sense. In particular, we investigate proper $\mathfrak g$-actions and generalize many of the usual properties of proper group actions to this more general setting.