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 $P_{\pm }⟶M$ 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 ${\mathcal{H}}^4$ and ${\mathcal{H}}_{\mathbb{C}}^2$, respectively, the real and complex hyperbolic space. Only in the end we present the new ${\mathrm{G}}_2$ structures on principal bundles.

20 pages; final version, to appear in Annales Academiæ Scientiarum Fennicæ

References in corpus (2)

• Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
• The Srni lectures on non-integrable geometries with torsion