Stratified vector fields on orbit spaces

TL;DR

This paper develops a correspondence between vector fields on differentiable stacks and stratified vector fields on orbit spaces, extending classical theorems to proper Lie groupoids.

Contribution

It introduces a Morita stratification-based framework linking stacky and stratified vector fields, leading to generalized Gauss and Palais' theorems.

Findings

Established a one-to-one correspondence between stack and stratified vector fields.

Derived a stacky version of the generalized Gauss lemma.

Proved a smooth version of Palais' covering isotopy theorem for proper Lie groupoids.

Abstract

Using Morita type stratifications, we establish a one-to-one correspondence between geometric vector fields on a separated differentiable stack and stratified vector fields on its orbit space. This correspondence enables us to derive a stacky version of the generalized Gauss lemma and to prove a smooth version of Palais' covering isotopy theorem for a class of proper Lie groupoids, thereby extending the classical result for proper Lie group actions.