Smooth Banach structure on orbit spaces and leaf spaces

Author:

Daniel Beltita, Fernand Pelletier

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Functional Analysis (math.FA)

journal:

date:

2023-05-24 16:00:00

Abstract

We investigate the quotients of Banach manifolds with respect to free actions of pseudogroups of local diffeomorphisms. These quotient spaces are called H-manifolds since the corresponding simply transitive action of the pseudogroup on its orbits is regarded as a homogeneity condition. The importance of these structures stems from the fact that for every regular foliation without holonomy of a Banach manifold, the corresponding leaf space has the natural structure of an H-manifold. This is our main technical result, and one of its remarkable consequences is an infinite-dimensional version of Sophus Lie's third fundamental theorem, to the effect that every real Banach-Lie algebra can be integrated to an H-group, that is, a group object in the category of H-manifolds. In addition to these general results we discuss a wealth of examples of H-groups which are not Banach-Lie groups.

PDF: Smooth Banach structure on orbit spaces and leaf spaces.pdf