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In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \\cite{DWW05} and \\cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a K\\\"{a}hler-Einstein manifold. 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Klaus Kr\\\"{o}ncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \\cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \\cite{DWW05} and \\cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a K\\\"{a}hler-Einstein manifold. 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