An Approach to Differential Invariants of $G$-Structures

TL;DR

This paper develops a framework for understanding scalar differential invariants of $G$-structures by representing them as functions on quotient spaces, providing bounds and explicit calculations for certain cases like conformal invariants.

Contribution

It introduces a method to identify scalar differential invariants of $G$-structures as functions on specific quotient spaces, generalizing previous approaches.

Findings

Scalar differential invariants are functions on quotient spaces of jet bundles.

A lower bound for the number of independent invariants is established.

Explicit calculation of invariants for conformal structures is provided.

Abstract

In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated bundle correspond to the real functions on the orbit space of the action of the jet group on the typical fiber, which enables to construct the said associated bundle. Scalar differential invariants are functions on bundles of jets of sections of associated bundles, which are invariant by the standard action of local diffeomorphisms. We show that these jet bundles are, in turn, associated bundles with reference frame bundles of higher order. By the previous result, this will allow to recognize scalar differential invariants as real functions over a quotient space, regardless of the base manifold. We apply the above to $G$-structures, and describe their scalar differential invariants as functions over certain quotient spaces. Each quotient space contains a dense differentiable manifold, whose dimension is "the number of functionally independent scalar differential invariants". We will obtain a lower bound of that number and calculate it for some cases, in particular for the scalar conformal differential invariants.