Deformation theory of nearly $G_2$ manifolds

Abstract

$\def\G{\mathrm{G}_2}$We study the deformation theory of nearly $\G$ manifolds. These are seven-dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $\G$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $\G$ structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $\G$ manifolds.