On the geometry of closed G2-structure
Abstract
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Goldberg conjecture in almost Kahler geometry. The result was generalized by R.L.Bryant to closed G2-structures with too tightly pinched Ricci tensor. We extend it in another direction proving that a compact G2-manifold with closed fundamental form and divergence-free Weyl tensor is a G2-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 G2-structure again imply that the induced metric has holonomy group contained in G2.
Keywords
Cite
@article{arxiv.math/0306362,
title = {On the geometry of closed G2-structure},
author = {Richard Cleyton and Stefan Ivanov},
journal= {arXiv preprint arXiv:math/0306362},
year = {2008}
}
Comments
14 pages, the Einstein condition in the assumptions of the Main theorem is generalized to the assumption that the Weyl tensor is divergence-free, clarity improved, typos corrected