Nearly Kaehler and nearly parallel G_2-structures on spheres

Abstract

In some other context, the question was raised how many nearly K\"ahler structures exist on the sphere $\S^6$ 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 $\G_2$-structures on the round sphere $\S^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.