Abstract

In 1984, inspired by Penrose’s successful twistor program, 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 my talk I shall report on my joint work with Domenico Fiorenza (arXiv:2203.04233) on a unification and generalization of LeBrun’s, Rossi’s and Verbitsky’s construction to the case where a Riemannian manifold (M, g) has a VCP structure. I shall explain how to express the formal integrability of the CR-structure 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 I shall discuss some open problems.