Geodesic
Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres
Archivum mathematicum, Tome 42 (2006) no. 5, pp. 241-243.

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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.
Classification : 53C15, 53C29
Keywords: nearly Kähler structures; nearly parallel $\mathrm {G}_2$-structures 
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     author = {Friedrich, Thomas},
     title = {Nearly {K\"ahler} and nearly parallel $G\sb 2$-structures on spheres},
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Friedrich, Thomas. Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres. Archivum mathematicum, Tome 42 (2006) no. 5, pp. 241-243. http://geodesic.mathdoc.fr/item/ARM_2006__42_5_a10/