Self-duality and associated parallel or cocalibrated G2 structures

Abstract

We find a remarkable family of G2 structures defined on certain principal SO(3)-bundles P± → M associated with any given oriented Riemannian 4-manifold M. Such structures are always cocalibrated. The study starts with a recast of the Singer–Thorpe equations of 4-dimensional geometry. These are applied to the Bryant–Salamon construction of complete G2-holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on M. We then discover new examples of that special holonomy on disk bundles over H4 and HC2, respectively, the real and complex hyperbolic space. Only in the end we present the new G2 structures on principal bundles.