Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold

Abstract

We study homogeneous instantons on the seven dimensional Stiefel manifold V in the context of $G_2$ and Sasakian geometry. According to the reductive decomposition of V we provide an explicit description of all invariant $G_2$ and Sasakian structures. In particular, we characterise the invariant $G_2$- structures inducing a Sasakian metric, among which the well known nearly parallel $G_2$-structure (Sasaki- Einstein) is included. As a consequence, we classify the invariant connections on homogeneous principal bundles over V with gauge group U(1) and SO(3), satisfying either the $G_2$ or the Sasakian instanton condition. In addition, we study infinitesimal deformations of $G_2$-instantons on coclosed $G_2$-manifolds using a spinorial approach. By means of a Weitzenb\"ock-type formula with torsion, we obtain curvature obstructions to the existence of non-trivial infinitesimal deformations and prove rigidity results for certain homogeneous $G_2$-instantons.