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

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 $\mathrm{SU}(1, 2)$, $\mathrm{SL}(3, {\mathbb R})$, or a particular semidirect product $\mathrm{SL}(2, {\mathbb R}) \ltimes Q_+$, 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 $5$; Kähler–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 Kähler–Einstein or para-Kähler–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.