CR-twistor spaces over manifolds with $G_2$- and $Spin(7)$-structures

Abstract

In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for $m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmanian bundle over $(M, g)$. If the VCP structure on $(M,g)$ is generated by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion tensor vanishes if and only if $(M, g)$ has constant curvature, and the horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or $Spin(7)$-manifold. Finally we discuss some open problems.