Stratified Subcartesian Spaces

TL;DR

This paper demonstrates that under certain conditions, the orbit structure of vector fields on a subcartesian space forms a Whitney A stratification, and relates orbit type stratification to vector field orbits in quotient spaces.

Contribution

It establishes conditions under which orbit families induce Whitney A stratifications and connects orbit type stratification with vector field orbits in quotient spaces.

Findings

Orbit families define Whitney A stratifications under local finiteness and closedness.

Orbit type stratification of M/G corresponds to vector field orbits.

Provides a framework linking vector fields, stratifications, and group actions.

Abstract

We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a Lie group G on a smooth manifold M is given by orbits of the family of all vector fields on M/G.