New examples of G$_2$-structures with divergence-free torsion

TL;DR

This paper constructs new examples of G₂-structures on solvable Lie groups with divergence-free torsion, contributing to the understanding of geometric flows that evolve G₂-structures towards minimal torsion configurations.

Contribution

It provides explicit examples of non-closed G₂-structures with divergence-free torsion on solvable Lie groups, expanding the known classes of such structures.

Findings

Divergence of torsion tensor is zero in all studied examples.

Constructed three large families of non-closed G₂-structures.

Analyzed properties of G₂-structures under the isometric flow.

Abstract

Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called \textit{isometric flow} has the distinctive feature of preserving the underlying metric induced by that $\mathrm{G}_2$-structure, so it can be used to evolve a $\mathrm{G}_2$-structure to one with the smallest possible torsion in a given metric class. This flow is built upon the divergence of the full torsion tensor of the flowing $\mathrm{G}_2$-structures in such a way that its critical points are precisely $\mathrm{G}_2$-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant $\mathrm{G}_2$-structures defined on simply connected solvable Lie groups previously studied in \cite{KL} and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.