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 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 Kähler geometry and was indicated by Cvetič 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.
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Communicated by G.W. Gibbons
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Cleyton, R., Ivanov, S. On the Geometry of Closed G2-Structures. Commun. Math. Phys. 270, 53–67 (2007). https://doi.org/10.1007/s00220-006-0145-7
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DOI: https://doi.org/10.1007/s00220-006-0145-7