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
Keywords: nearly Kähler structures; nearly parallel $\mathrm {G}_2$-structures
@article{ARM_2006__42_5_a10,
author = {Friedrich, Thomas},
title = {Nearly {K\"ahler} and nearly parallel $G\sb 2$-structures on spheres},
journal = {Archivum mathematicum},
pages = {241--243},
publisher = {mathdoc},
volume = {42},
number = {5},
year = {2006},
mrnumber = {2322410},
zbl = {1164.53353},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2006__42_5_a10/}
}
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/