Self-duality and associated parallel or cocalibrated G_2 structures

Abstract

We prove the existence of a large family of naturally defined G2 structures on certain compact principal SO(3)-bundles P+ and P− associated with any given oriented Riemannian 4-manifold M . A nice surprise is that such structures, though never calibrated, are always cocalibrated. As we start our study with a recast of the Bryant-Salamon contruction of G2 holonomy on the vector bundle of anti-selfdual 2-forms on M , we then discover incomplete examples of that restricted holonomy on disk bundles over H 4 and H 2 C , respectively, the real and complex hyperbolic space. Also, the existence of a G2 metric on Λ 2 + T * M for any K3 surface M is shown.