On moduli spaces for finite-order jets of linear connections

TL;DR

This paper characterizes the structure of moduli spaces for jets of linear connections as orbit spaces and proves that the only scalar differential invariants are constants, providing insights into their algebraic properties.

Contribution

It describes the ringed-space structure of these moduli spaces and establishes the triviality of scalar differential invariants for linear connections.

Findings

Moduli spaces are orbit spaces of linear representations of GL(n).

Scalar differential invariants of linear connections are constant.

Reproduces known expressions for Poincaré series of these moduli spaces.

Abstract

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group. Then, we use this fact to prove that the only (scalar) differential invariants associated to linear connections are constant functions, as well as to recover various expressions appearing in the literature regarding the Poincar\'{e} series of these moduli spaces.