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The geometry of almost Einstein $$(2,3,5)$$ distributions. (English) Zbl 1372.32033
Authors’ abstract: We analyze the classical problem of the existence of Einstein metrics in a given conformal structure for the class of conformal structures induced, following 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 one with 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.

##### MSC:
 32Q20 Kähler-Einstein manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32V05 CR structures, CR operators, and generalizations 53A30 Conformal differential geometry (MSC2010) 53B35 Local differential geometry of Hermitian and Kählerian structures
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