Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions

TL;DR

This paper explores the geometric structures arising from free n-distributions, focusing on holonomy reductions, special connections, and Fefferman constructions, especially in free 3-distributions with applications to CR and Lagrangian contact structures.

Contribution

It introduces new holonomy reduction results and constructs explicit examples, particularly in free 3-distributions, linking them to CR, Lagrangian contact, and dual distributions.

Findings

Holonomy reductions imply special geometric structures.

Explicit examples of holonomy reductions in simple cases.

Construction of dual distributions via holonomy reduction to G_2'.

Abstract

This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are 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 $G_2'$.