Harmonic $G_2$-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 $divT=0$, and this allows to identify which torsion classes are harmonic.

15 pages, v3: Additional reference [17] to the bibliography, the same version as v2

References in corpus (2)

• Harmonic G-structures
• Weyl-Einstein structures on conformal solvmanifolds

Cited by in corpus (1)

• Applications of the quaternionic Jordan form to hypercomplex geometry