Harmonic \(G_2\)-structures on almost Abelian Lie groups

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 \(div T=0\), and this allows to identify which torsion classes are harmonic.