Solving Local Equivalence Problems with the Equivariant Moving Frame Method

TL;DR

This paper extends the equivariant moving frame method to handle local equivalence problems for submanifolds where the Lie pseudo-group action is not free, providing an alternative to Cartan's method.

Contribution

It introduces modifications to the equivariant moving frame construction to apply in non-free action scenarios, enhancing its applicability to local equivalence problems.

Findings

Extended moving frame method to non-free actions

Applied framework to submanifold equivalence problems

Provided an alternative to Cartan's method

Abstract

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudo-group does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan's equivalence method based on the theory of G-structures.