Nearly Kaehler and nearly parallel G_2-structures on spheres

Thomas Friedrich

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arXiv · math/0509146

Nearly Kaehler and nearly parallel G_2-structures on spheres

Thomas Friedrich

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Abstract

In some other context, the question was raised how many nearly Kähler structures exist on the sphere $§^6$ 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 $λ= 12$ of the Laplacian acting on 2-forms. A similar result concerning nearly parallel $\G_2$-structures on the round sphere $§^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

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Thomas Friedrich. 2005-09-07. Nearly Kaehler and nearly parallel G_2-structures on spheres. https://arxiv.org/abs/math/0509146

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