Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition

Given a maximally non-integrable 2-distribution ${\mathcal D}$ on a 5-manifold $M$, it was discovered by P. Nurowski that one can naturally associate a conformal structure $[g]_{\mathcal D}$ of signature (2,3) on $M$. We show that those conformal structures $[g]_{\mathcal D}$ which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of $[g]_{\mathcal D}$ can be decomposed into a symmetry of ${\mathcal D}$ and an almost Einstein scale of $[g]_{\mathcal D}$.Comment: Misprints in Theorem B are correcte

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 $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, 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; K\"ahler-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 K\"ahler-Einstein or para-K\"ahler-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

New relations between G2G_2-geometries in dimensions 5 and 7

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure on a seven-dimensional bundle over a five-manifold endowed with a $(2,3,5)$-distribution. For a class of distributions the induced Lie contact structure is constructed explicitly and we determine its symmetries. We further study the relation between the canonical normal Cartan connections associated with the two structures. In particular, we show that the Cartan holonomy of the induced Lie contact structure reduces to $G_2$. Moreover, the curved orbit decomposition associated with a $\mathrm{G}_2$-reduced Lie contact structure on a seven-manifold is discussed. It is shown that in a neighbourhood of each point on the open curved orbit the structure descends to a $(2,3,5)$-distribution on a local leaf space, provided an additional curvature condition is satisfied. The closed orbit carries an induced $G_2$-contact structure.Comment: We changed abstract a bit, and correctly defined the $G_2$ contact structur

On Nurowski's conformal structure associated to a generic rank two distribution in dimension five

For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution. In the second part of the article, we relate this conformal class to the canonical Cartan connection associated to the distribution. This is used to prove that it coincides with the conformal class constructed by Nurowski.Comment: AMSLaTeX, 23 page

A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry

The twistor spinors of generic 2- and 3-distributions

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then characterized by their conformal holonomy and equivalently by the existence of a twistor spinor which satisfies a genericity condition. Moreover, we show that given such a twistor spinor we can decompose a conformal Killing field of the structure. We obtain explicit formulas relating conformal Killing fields, almost Einstein structures and twistor spinors.Comment: 26 page

Parabolic geometries determined by filtrations of the tangent bundle

summary:Summary: Let {\germ g} be a real semisimple $|k|$-graded Lie algebra such that the Lie algebra cohomology group H^1({\germ g}_-,{\germ g}) is contained in negative homogeneous degrees. We show that if we choose G= \operatorname{Aut}({\germ g}) and denote by $P$ the parabolic subgroup determined by the grading, there is an equivalence between regular, normal parabolic geometries of type $(G,P)$ and filtrations of the tangent bundle, such that each symbol algebra $\text{gr}(T_xM)$ is isomorphic to the graded Lie algebra {\germ g}_-. Examples of parabolic geometries determined by filtrations of the tangent bundle are discussed