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

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 $[g{]}_{\mathcal{D}}$.

Misprints in Theorem B are corrected

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• New relations between $G_2$-geometries in dimensions 5 and 7
• Pure spinors, intrinsic torsion and curvature in odd dimensions
• Analytic torsion of generic rank two distributions in dimension five
• A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors
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• Parallel tractor extension and ambient metrics of holonomy split G_2
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