Weyl structures for parabolic geometries

TL;DR

This paper introduces Weyl structures and preferred connections within the broad framework of parabolic geometries, extending classical notions and providing new descriptions of Cartan bundles and connections.

Contribution

It generalizes the concepts of Weyl structures, scales, and Rho-tensors to all parabolic geometries, offering a unified approach and new characterizations.

Findings

Extended the notion of Weyl structures to all parabolic geometries

Provided a new description of Cartan bundles and connections

Characterized classes of scales, Weyl connections, and Rho-tensors

Abstract

Motivated by the rich geometry of conformal Riemannian manifolds and by the recent development of geometries modeled on homogeneous spaces $G/P$ with $G$ semisimple and $P$ parabolic, Weyl structures and preferred connections are introduced in this general framework. In particular, we extend the notions of scales, closed and exact Weyl connections, and Rho--tensors, we characterize the classes of such objects, and we use the results to give a new description of the Cartan bundles and connections for all parabolic geometries.