Deformations of nearly parallel $G_2$-structures

Abstract

We study the infinitesimal deformations of a proper nearly parallel $G_2$-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $\Lambda^3_{27}$-eigenforms of the Laplace operator for the eigenvalue $8\mathrm{scal} /21$. We give a similar description for the space of infinitesimal Einstein deformations of a fixed nearly parallel $G_2$-structure. Moreover we show that there are no deformations on the squashed $S^7$ and on $\mathrm{SO}(5)/\mathrm{SO}(3)$, but that there are infinitesimal deformations on the Aloff-Wallach manifold $N(1, 1) = \mathrm{SU}(3)/U(1)$.