  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.
