Harmonic G2-structures on almost Abelian Lie groups

Abstract

We consider left-invariant $G_2$-structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket A of the corresponding Lie algebra. In those terms, we establish the algebraic condition on A for each of the possible 16-torsion classes of a $G_2$-structure. In particular, we show that four of those torsion classes are not admissible, since ${\tau }_3=0$ implies ${\tau }_0=0$. Finally, we use the above results to provide the algebraic criteria on A, satisfying the harmonic condition $\mathrm{div}T=0$, and this allows to identify which torsion classes are harmonic.