Purely coclosed G${}_2$-structures on 2-step nilpotent Lie groups

Abstract

We consider left-invariant (purely) coclosed G${}_2$-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G${}_2$-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G${}_2$-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G${}_2$-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

28 pages

References in corpus (1)

• The moduli space of left-invariant metrics of a class of six-dimensional nilpotent Lie groups

Cited by in corpus (2)

• Moduli spaces of (co)closed ${\mathrm{G}}_2$-structures on nilmanifolds
• Families of solutions of the heterotic G${}_2$ system