We find aWeitzenb?ck formula for the Fueter-Dirac operator which controls infinitesimal deformations of an associative submanifold in a 7–manifold with a G2–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 G2-structure obtained by Fernández- Fino-Raffero [9].
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