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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

Citations:

Zbl 0861.53022
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References:

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