New examples of G2-structures with divergence-free torsion

Abstract

Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group G2 have spurred extensive research in geometric flows of G2-structures defined on seven-dimensional manifolds in recent years. Among many possible geometric flows, the so-called isometric flow has the distinctive feature of preserving the underlying metric induced by that G2-structure, so it can be used to evolve a G2-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 G2-structures in such a way that its critical points are precisely G2-structures with divergence-free torsion. In this article we study three large families of pairwise non-equivalent non-closed left-invariant G2- structures defined on simply connected solvable Lie groups previously studied in [17] and compute the divergence of their full torsion tensor, obtaining that it is identically zero in all cases.