Nearly Kaehler and nearly parallel G$_2$-structures on spheres

Thomas Friedrich

Institut fuer Mathematik, Humboldt-Universitaet zu Berlin,
Sitz: WBC Adlershof, D-10099 Berlin, Germany


In some other context, the question was raised how many nearly K\"ahler 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 G$_2$-structures on the round sphere ${\mathbb S}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

AMSclassification. Primary 53C25; Secondary 81T30.

Keywords. Nearly K\"ahler structures, nearly parallel G$_2$-structures.