Bundles of Weyl structures and invariant calculus for parabolic geometries

TL;DR

This paper develops a universal calculus for constructing affine invariants of Weyl structures in parabolic geometries, advancing the understanding of invariant differential operators in these geometric settings.

Contribution

It introduces a systematic method to derive all affine invariants of Weyl connections based solely on tensorial deformations, broadening the toolkit for parabolic geometry analysis.

Findings

Provides a universal calculus for invariant construction.

Offers a procedure to generate all affine invariants of Weyl connections.

Enhances understanding of invariant differential operators in parabolic geometries.

Abstract

For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general tools were presented for the entire class of parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces $G/P$ with $P$ a parabolic subgroup in a semi-simple Lie group $G$. Similarly to conformal Riemannian and projective structures, all these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms $\Upsilon$. They correspond to reductions of $P$ to its reductive Levi factor, and they are called the Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In this article, we describe a universal calculus which provides an important first step to determine such invariants. We present a natural procedure how to construct all affine invariants of Weyl connections, which depend only tensorially on the deformations $\Upsilon$.