Weyl connections and the local sphere theorem for quaternionic contact structures
arXiv:1003.1850
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-Carathéodory 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.
17 pages; references corrected, acknowledgement added