Nearly Kähler and nearly parallel -structures on spheres
In some other context, the question was raised how many nearly Kähler structures exist on the sphere 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 of the Laplacian acting on -forms. A similar result concerning nearly parallel -structures on the round sphere holds, too. An alternative proof by Riemannian Killing spinors is also indicated.