Self-duality and associated parallel or cocalibrated ${\mathrm{G}}_2$ structures

Abstract

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ 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 cons\-truction of complete $\mathrm{G}_2$-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 ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$, respectively, the real and complex hyperbolic space. Only in the end we present the new $\mathrm{G}_2$ structures on principal bundles.