Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 004, 56 pages      arXiv:1606.01069

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

Katja Sagerschnig a and Travis Willse b
a) Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
b) Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received July 26, 2016, in final form January 13, 2017; Published online January 19, 2017

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 ${\rm SU}(1, 2)$, ${\rm SL}(3, {\mathbb R})$, or a particular semidirect product ${\rm SL}(2, {\mathbb R}) \ltimes Q_+$, 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.

Key words: $(2,3,5)$ distribution; almost Einstein; conformal geometry; conformal Killing field; CR structure; curved orbit decomposition; Fefferman construction; ${\rm G}_2$; holonomy reduction; Kähler-Einstein; Sasaki-Einstein; second-order ordinary differential equation.

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