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.
@article{108030,
author = {Thomas Friedrich},
title = {Nearly K\"ahler and nearly parallel $G\sb 2$-structures on spheres},
journal = {Archivum Mathematicum},
volume = {042},
year = {2006},
pages = {241-243},
zbl = {1164.53353},
mrnumber = {2322410},
language = {en},
url = {http://dml.mathdoc.fr/item/108030}
}
Friedrich, Thomas. Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres. Archivum Mathematicum, Tome 042 (2006) pp. 241-243. http://gdmltest.u-ga.fr/item/108030/
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