Persistence of Freeness for Lie Pseudogroup Actions

TL;DR

This paper proves that the local and global freeness of a Lie pseudogroup's action on jet spaces persists under prolongation, supporting the construction of complete moving frames and invariant analysis.

Contribution

It establishes the persistence of freeness in Lie pseudogroup actions on jet spaces, foundational for invariant object construction.

Findings

Freeness of pseudogroup actions persists under prolongation.

Supports the construction of complete moving frames.

Facilitates analysis of invariants in jet spaces.

Abstract

The action of a Lie pseudogroup $G$ on a smooth manifold $M$ induces a prolonged pseudogroup action on the jet spaces $J^n$ of submanifolds of $M$. We prove in this paper that both the local and global freeness of the action of $G$ on $J^n$ persist under prolongation in the jet order $n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the pseudogroup action on $J^\infty$.