The Weitzenbock formula for the Fueter–Dirac operator

We find a Weitzenböck formula for the Fueter–Dirac operator which controls 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 find a different proof of rigidity for one of Lotay’s associatives in the round $7$-sphere from those given by Kawai [14, 15]. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric [11]. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated $G_2$-structure obtained by Fernández–Fino–Raffero [9].

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Received 15 September 2017

Accepted 23 May 2019

Published 22 July 2022