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

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 Grassmannian 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.