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

Published in Symmetry, Integrability and Geometry: Methods and Applications, 2017, with K. Sagerschnig
(emis.de | arXiv:1201.2670)

Abstract

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures induced [by] 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 $\bf c$ 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 $\bf c$. Second, they are characterized by the existence of a holonomy reduction to $\operatorname{SU}(1,2)$, $\operatorname{SL}(3,R)$, or a particular semidirect product $\operatorname(2,R) \times Q_+$, 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 [w]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.