# On the stratification by orbit types II

## TL;DR

This paper improves the understanding of stratifications by orbit types under Lie group actions, establishing smooth local triviality and strengthening previous conditions like Whitney (b) and Verdier.

## Contribution

It advances the theory by proving smooth local triviality of the stratification, building on prior results about Whitney (b) and Verdier conditions.

## Key findings

- Stratification satisfies Whitney (b) condition.
- Stratification satisfies strong Verdier condition.
- Establishes smooth local triviality of the stratification.

## Abstract

When we have a proper action of a Lie group on a manifold, it is well known that we get a stratification by orbit types and it is known that this stratification satisfies the Whitney (b) condition. In a previous article we have seen that the stratification satisfies the strong Verdier condition. In this article we improve this result and obtain smooth local triviality.