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

TL;DR

This paper explores the geometric structures arising from free 3-distributions, revealing connections to Fefferman constructions, holonomy reductions, and dual distributions, with extensions to higher dimensions.

Contribution

It introduces new holonomy reductions and dual distribution constructions for free 3-distributions, expanding understanding of their geometric and algebraic properties.

Findings

Existence of normal Fefferman constructions over CR and Lagrangian contact structures.

Holonomy reductions to SO(4,2), SO(3,3), and G_2' lead to specific geometric structures.

Extension of holonomy constructions to free n-distributions for n>3.

Abstract

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 paper concludes with some holonomy constructions for free $n$-distributions for $n>3$.