On the geometry of closed G2-structure

Details On the geometry of closed G2-structure On the geometry of closed G2-structure

2003-06-25

arxiv.org/abs/math/0306362v3

Commun.Math.Phys.270:53-67,2007

Mathematics

Differential Geometry

14 pages, the Einstein condition in the assumptions of the Main theorem is generalized to the assumption that the Weyl tensor

Scientific paper

10.1007/s00220-006-0145-7

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.

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