Remarks on some integral formulas for $G_2$-structures

TL;DR

This paper clarifies and unifies various integral formulas and divergence equations related to $G_2$-structures on seven-dimensional manifolds, expressing intrinsic torsion components via exterior algebra.

Contribution

It provides a comprehensive and detailed comparison of existing integral formulas for $G_2$-structures, expressing intrinsic torsion components through exterior algebra.

Findings

All integral formulas and divergence equations are consistent with the presented framework.

Intrinsic torsion components are explicitly expressed using exterior algebra.

The work consolidates diverse results into a unified approach.

Abstract

For seven-dimensional Riemannian manifolds equipped with a $G_2$-structure, we show in a full detailed way that all integral formulas and divergence equations, given by diverse authors, are agree with the ones displayed here in terms of the intrinsic torsion of the $G_2$-structure. Likewise the components of such an intrinsic torsion is expressed by means of exterior algebra.