Deformations of nearly $G_2$-structures

TL;DR

This paper investigates the second order obstructions to deforming nearly $G_2$ structures on compact manifolds, demonstrating rigidity for a specific structure on the Aloff-Wallach space.

Contribution

It identifies the second order obstruction to deformation of nearly $G_2$ structures and proves rigidity for the structure on $N(1,1)$, extending previous work.

Findings

Second order obstruction to deformation described

Rigidity established for the nearly $G_2$ structure on $N(1,1)$

Builds on prior work by Alexandrov and Semmelmann

Abstract

We describe the second order obstruction to deformation for nearly $G_2$ structures on compact manifolds. Building on work of B.Alexandrov and U.Semmelmann this allows proving rigidity under deformation for the proper nearly $G_2$ structure on the Aloff-Wallach space $N(1,1)$.