The geometry of almost Einstein \((2,3,5)\) distributions. (English) Zbl 1372.32033

Sagerschnig, Katja; Willse, Travis

SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 004, 56 p. (2017).

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.

Reviewer: Zbigniew Olszak (Wrocław)

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

Keywords:

Einstein metrics; conformal strutures; \((2,3,5)\) distributions; Sasaki-Einstein geometry; Kähler-Einstein geometry

References:

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