Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition

TL;DR

This paper characterizes conformal structures derived from generic rank 2 distributions on 5-manifolds using conformal Killing 2-forms and decomposes their Killing fields into symmetries and Einstein scales.

Contribution

It provides a characterization of conformal structures associated with rank 2 distributions via normal conformal Killing 2-forms and describes the decomposition of conformal Killing fields.

Findings

Characterization of conformal structures via decomposable conformal Killing 2-forms.

Decomposition of conformal Killing fields into symmetries and Einstein scales.

Connection between distribution symmetries and conformal Killing fields.

Abstract

Given a maximally non-integrable 2-distribution ${\mathcal D}$ on a 5-manifold $M$, it was discovered by P. Nurowski that one can naturally associate a conformal structure $[g]_{\mathcal D}$ of signature (2,3) on $M$. We show that those conformal structures $[g]_{\mathcal D}$ which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of $[g]_{\mathcal D}$ can be decomposed into a symmetry of ${\mathcal D}$ and an almost Einstein scale of $[g]_{\mathcal D}$.