Torsion-free $G_2$-structures with identical Riemannian metric

TL;DR

This paper investigates the topological and Lie group structures of the space of torsion-free $G_2$-structures that produce the same Riemannian metric on compact 7-manifolds, enhancing understanding of their moduli space.

Contribution

It characterizes the topological and Lie group structures of torsion-free $G_2$-structures sharing the same metric and describes their moduli space via covering spaces.

Findings

Identifies the topological structure of the space of $G_2$-structures with the same metric.

Determines the Lie group-theoretic structure of this space.

Describes the moduli space in specific cases using covering space theory.

Abstract

Based on a general formula due to R.Bryant, we work out the topological structure of the space of torsion-free $G_2$-structures generating the same associated Riemannian metric on a compact $7$-manifold. We also identify a corresponding Lie group-theoretic structure of the space. These observations are then used to describe the moduli space of torsion-free $G_2$-structures in certain cases - by way of covering spaces.