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 well-known (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.
