Smooth G-manifolds as collections of fiber bundles

Abstract

This paper is about the general theory of differentiable actions of compact Lie groups. Let G be a compact Lie group acting smoothly on a manifold M. Both M and M/G have natural stratifications, and M/G inherits a “smooth structure” from M. The map M → M/G exhibits many of the properties of a smooth fiber bundle. For example, it is proved that a smooth G-manifold can be pulled back via a “weakly stratified” map of orbit spaces. Also, it is wellknown (and obvious) that a smooth G-manifold is determined by a certain collection of fiber bundles together with some attaching data. Several precise formulations of this observation are given. © 1978, University of California, Berkeley. All Rights Reserved.