A note on the relation between joint and differential invariants

TL;DR

This paper explores the relationship between joint and differential invariants under Lie group actions, providing simplified versions of known results and a method to derive differential invariants from joint invariants.

Contribution

It introduces a general method to compute differential invariants from joint invariants and clarifies their relationship, simplifying existing theories.

Findings

Simplified versions of Lie's finiteness theorem for joint invariants

A method to derive differential invariants from joint invariants

Enhanced understanding of the relation between joint and differential invariants

Abstract

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.