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

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.

arXiv comment16 pages