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nearly Kähler structures; nearly parallel $\mathrm {G}_2$-structures
In some other context, the question was raised how many nearly Kähler structures exist on the sphere $\mathbb {S}^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda = 12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel $\mathrm {G}_2$-structures on the round sphere $\mathbb {S}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.
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