Solvmanifolds with integrable and non-integrable [formula omitted] structures

Abstract

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G 2 can admit non-integrable G 2 structures of type R ⊕ S 0 2 ( R 7 ) ⊕ R 7 in the sense of Fernández and Gray. This relies on the construction of some G 2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symmetric torsion that has parallel spinors as well. The family turns out to be a deformation of the Levi-Civita connection. This is in contrast with the case of compact scalar-flat Riemannian spin manifolds, where any metric connection with closed torsion admitting parallel spinors has to be torsion-free.