Cocalibrated G_2-manifolds with Ricci flat characteristic connection

Any 7-dimensional cocalibrated G_2-manifold admits a unique connection $\mathrm{\nabla }$ with skew symmetric torsion. We study these manifolds under the additional condition that the $\mathrm{\nabla }$-Ricci tensor vanishes. In particular, we describe their geometry in case of a maximal number of $\mathrm{\nabla }$-parallel vector fields.