Parabolic Geometries, CR–Tractors, and the Fefferman Construction

An Analog of the Fefferman Construction

The Fefferman construction associates to a manifold carrying a CR–structure a conformal structure on a sphere bundle over the manifold. There are some analogs to this construction, with one giving a Lie contact structure, a refinement of the contact bundle on the bundle of rays in the cotangent bundle of a manifold with a conformal metric. Since these structures are parabolic geometries, these ...

Free 3-distributions: holonomy, Fefferman constructions and dual distributions

This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4, 2) and SO(3, 3), respectively. There is also a fascinating construction of a ‘dual’ distribution when the holonomy reduces to G′2. The p...

Free CR Distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1, n)/P with a suitable parabolic subgroup P . We s...

Parabolic Geometries , Cr { Tractors , And

Two Constructions with Parabolic Geometries

This is an expanded version of a series of lectures delivered at the 25th Winter School “Geometry and Physics” in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the equivalence between parabolic geometries and underlying geometric structures. The second part of the paper is devoted to constructions which relate parabolic geometr...