On Nurowski's conformal structure associated to a generic rank two distribution in dimension five

TL;DR

This paper constructs an intrinsic conformal structure for generic rank two distributions in five dimensions and relates it to Nurowski's canonical conformal structure via Cartan connections.

Contribution

It introduces a generalized contact form and explicitly constructs a conformal class of metrics, linking it to Nurowski's conformal structure through Cartan geometry.

Findings

The conformal class is intrinsic to the distribution.

The constructed metric's conformal class matches Nurowski's.

The approach uses generalized contact forms and Cartan connections.

Abstract

For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution. In the second part of the article, we relate this conformal class to the canonical Cartan connection associated to the distribution. This is used to prove that it coincides with the conformal class constructed by Nurowski.