Authors: Richard Cleyton Stefan Ivanov
Publish Date: 2006/12/05
Volume: 270, Issue: 1, Pages: 53-67
Abstract
We give an answer to a question posed in physics by Cvetič et al 9 and recently in mathematics by Bryant 3 namely we show that a compact 7dimensional manifold equipped with a G 2structure 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 Kähler geometry and was indicated by Cvetič et al in 9 The result was generalized by Bryant to closed G 2structures with too tightly pinched Ricci tensor We extend it in another direction proving that a compact G 2manifold with closed fundamental form and divergencefree Weyl tensor is a G 2manifold with parallel fundamental form We introduce a second symmetric Riccitype tensor and show that Einstein conditions applied to the two Ricci tensors on a closed G 2structure again imply that the induced metric has holonomy group contained in G 2
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