Weyl connections and the local sphere theorem for quaternionic contact structures

TL;DR

This paper applies Weyl structures in parabolic geometries to compute the Weyl connection for quaternionic contact structures, leading to new tensorial formulas and insights into their conformal properties and flatness obstructions.

Contribution

It introduces a method to compute the Weyl connection for qc structures using parabolic geometry, providing new tensor formulas and clarifying their geometric significance.

Findings

Derived a tensorial formula for the qc Weyl curvature tensor.

Showed the vanishing of this tensor characterizes local flatness of qc structures.

Connected the tensor's properties to conformal invariance and geometric obstructions.

Abstract

We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carath\'eodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold. In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern-Moser tensor in CR geometry. This tensor agrees with the formula derived via independent methods by S. Ivanov and D. Vasillev. However, as a result of our derivation of this tensor, its fundamental properties -- conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure -- follow as easy corollaries from the general parabolic theory.