The Geometry of Almost Einstein (2,3,5) Distributions

TL;DR

This paper characterizes conformal structures induced by (2,3,5) distributions that admit almost Einstein scales, linking them to various geometries and constructing new examples with positive and negative Einstein constants.

Contribution

It provides a novel characterization of almost Einstein (2,3,5) distributions via holonomy reductions and links to multiple classical geometries, including new construction methods.

Findings

Characterization of conformal structures with almost Einstein scales via holonomy reductions.

Partitioning of manifolds into submanifolds with diverse geometric structures.

Construction of new nonflat almost Einstein (2,3,5) structures with positive and negative Einstein constants.

Abstract

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures $\mathbf{c}$ that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce $\mathbf{c}$. Second, they are characterized by the existence of a holonomy reduction to $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; K\"ahler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional K\"ahler-Einstein or para-K\"ahler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative.