Solvmanifolds with Integrable and Non-integrable G2 Structures

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G2 can admit non-integrable G2 structures of type R⊕S 2 0 (R )⊕R in the sense of Fernández and Gray. This relies on the construction of some G2 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.