Deformations of nearly parallel G_2-structures

TL;DR

This paper characterizes infinitesimal deformations of nearly parallel G_2-structures using differential equations and eigenforms, identifying cases with and without deformations on specific manifolds.

Contribution

It provides a differential equation characterization of deformations and identifies specific manifolds with or without such deformations.

Findings

No deformations on squashed S^7 and SO(5)/SO(3)

Existence of deformations on Aloff-Wallach manifold N(1,1)

Deformations correspond to co-closed eigenforms with eigenvalue 8/21

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 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 SO(5)/SO(3), but that there are infinitesimal deformations on the Aloff-Wallach manifold N(1,1) = SU(3)/U(1).