## On the geometry of closed $$G_{2}$$-structures.(English)Zbl 1122.53026

Let $$e_1,\dots, e_7$$ be an oriented orthonormal basis of $$\mathbb R^7$$ and denote $$e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p}$$ by $$e_{i_i}\cdots e_{i_p}$$. The subgroup of $$\text{GL}(7)$$ that fixes the three-form
$\omega= e_{124}+ e_{235}+ e_{346}+ e_{457}+ e_{561}+ e_{672}+ e_{713}$
is the exceptional Lie group $$G_2$$ and is a compact, connected, simply-connected, simple Lie subgroup of $$\text{SO}(7)$$ of $$\dim 14$$. A $$G_2$$-structure on a 7-manifold is a reduction of the structure group of the tangent bundle to $$G_2$$, and $$\omega$$ is called the fundamental form of the $$G_2$$-manifold $$M$$. When $$\omega$$ is parallel, it is known that the holonomy group is contained in $$G_2$$. Compact examples of such parallel $$G_2$$-manifolds were obtained by D. Joyce [J. Differ. Geom 43, No. 2, 291–328 (1996; Zbl 0861.53022)] which are all called Joyce spaces.
In this paper the authors study the geometry of closed $$G_2$$-structures and present as main theorem the following result: A compact and closed $$G_2$$-manifold with harmonic Weyl tensor is a Joyce space. With this theorem associated is the following corollary: If $$M$$ as above admits a harmonic curvature or is an Einstein space, then it is a Joyce space.

### MSC:

 53C29 Issues of holonomy in differential geometry 53C10 $$G$$-structures

Zbl 0861.53022
Full Text:

### References:

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