On the geometry of closed G2-Structures

We give an answer to a question posed in physics by Cvetic et al. [9] and recently in mathematics by Bryant [3], namely we show that a compact 7-dimensional manifold equipped with a G(2)-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G(2). This could be considered to be a G(2) analogue of the Goldberg conjecture in almost Kahler geometry and was indicated by Cvetic et al. in [9]. The result was generalized by Bryant to closed G(2)-structures with too tightly pinched Ricci tensor. We extend it in another direction proving that a compact G(2)-manifold with closed fundamental form and divergence-free Weyl tensor is a G(2)-manifold with parallel fundamental form. We introduce a second symmetric Ricci-type tensor and show that Einstein conditions applied to the two Ricci tensors on a closed G(2)-structure again imply that the induced metric has holonomy group contained in G(2).