The Weitzenbock formula for the Fueter-Dirac operator (Q2157341)

Let \((M^7,\phi)\) be a smooth manifold with a \(G_2\)-structure and let \(G:=\mathrm{Stab}(\phi) \subset \mathrm{Aut} (M)\) be the group of global authomorphisms preserving \(\phi\). An associative submanifold \(Y^3\) in \(M\) is called rigid if all infinitesimal deformations of \(Y\) are given by the action of \(G\) on \(Y\). In this paper the authors find a Weitzenböck formula for the Fueter-Dirac operator in order to control infinitesimal deformations of an associative submanifold \(Y^3\) of \((M^7,\phi)\), they obtain rigidity under some positivity conditions. As an application they find alternative proofs of some known cases. For example they obtain the rigidity of \(S^3\) as an associative submanifold of \(S^7\) with the \(G_2\)-structure defined by Lotay. Moreover they obtain the rigidity of \(S^3=S^3\times\{0\}\) in \(S^3\times\mathbb R^4\) with the structure defined by Bryant and Salamon. Also the authors provide a new example of a rigid associative submanifold in a compact locally conformal calibrated \(G_2\)-manifold studied by \textit{M. Fernández} et al. [Ann. Mat. Pura Appl. (4) 195, No. 5, 1721--1736 (2016; Zbl 1357.53033)].