An integral formula for $G_2$-structures

TL;DR

This paper derives an integral formula connecting curvature and quadratic invariants of the intrinsic torsion for $G_2$-structures, providing insights into their geometric properties.

Contribution

The paper applies a general integral formula to $G_2$-structures, relating curvature and intrinsic torsion invariants, and interprets special cases.

Findings

Derived an integral formula linking curvature and quadratic invariants.

Provided geometric interpretations in special cases.

Enhanced understanding of $G_2$-structure properties.

Abstract

We apply an integral formula obtained by the author for a general $G$--structure to the case of $G=G_2$. We derive an integral formula relating curvatures and some quadratic invariants of the endomorphism induced by the intrinsic torsion. We interpret the formula in certain special cases.