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

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.

18 pages