The Weitzenböck formula for the Fueter-Dirac operator

Abstract

We find Weitzenböck formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we give a different proof of rigidity for one of Lotay's associatives in the round $7$-sphere from those given by Kawai. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated $G_2$-structure obtained by Fernandez-Fino-Raffero.

34 pages; v3: Examples 3.14, 3.15 moficated and some further details

References in corpus (2)

• Associative Submanifolds of the 7-Sphere
• Locally conformal calibrated $G_2$-manifolds