Purely coclosed G$_{\mathbf2}$-structures on 2-step nilpotent Lie groups

Abstract

We consider left-invariant (purely) coclosed G2-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 G2-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 G2-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G2-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

Families of solutions of the heterotic G$_2$ system

“…It would also be interesting to construct solutions involving Sasaki or Sasaki-Einstein manifolds, or more general Aloff-Wallach spaces N p,q making use of the connections of [61]. Another option would be to generalize the solutions of [37,39] to other nilmanifolds using the detailed study of [75]. Finally, one could attempt to exploit the existence of almost contact metric 3-structures on manifolds with a G 2 -structure to try to generalize the example of section 5.1 of [41].…”