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The intrinsic torsion of \(SU(3)\) and \(G_2\) structures. (English) Zbl 1024.53018

Gil-Medrano, Olga (ed.) et al., Differential geometry. Proceedings of the international conference held in honour of the 60th birthday of A. M. Naveira, Valencia, Spain, July 8-14, 2001. Singapore: World Scientific. 115-133 (2002).
The purpose of this paper is to present the relationship between the component of the intrinsic torsion of an \(SU(3)\)-structure on a 6-manifold and a \(G_2\)-structure on a 7-manifold (for the cases: \(SU(3)\subset SO(6)\), \(G_2\subset SO(7))\). The fact that \(SU(3)\) is a maximal subgroup of \(G_2\) gives a direct relationship between the two structures. A 6-manifold \(M\) with a \(SU(3) \)-structure is equipped with a Riemannian metric \(g\), an orthogonal almost-complex structure \(J\) and a 2-form \(\omega\). An \(SU(3)\)-structure is determined by a real 3-form \(\psi_+\). Let \(\tau_1\) be the intrinsic torsion of the \(SU(3)\)-structure and \(W_k\) the components of \(\tau_1\). The following theorem is proved: “The five components of \(\tau_1\) are determined by \(d\omega\), \(d\psi_+\), \(d \psi_-\), \((\psi_-=J \psi_+)\), in the following manner: \(W_1\leftrightarrow (d \omega)^{30}\), \(W_2\leftrightarrow ((d\psi_+)^{11}_0\), \((d\psi_-)^{11}_0)\), \(W_3 \leftrightarrow (d\omega)^{21}_0\), \(W_4\leftrightarrow \omega\wedge d\omega\), \(W_5 \leftrightarrow (d\psi_\pm)^{31}\).”
Let \(P\) be a 7-manifold with an \(SU(3)\)-structure. The authors investigate some properties of a \(G_2\)-structure determined on \(P\) by a 3-form \(\varphi=\omega \wedge\alpha +d\psi_+\). They show that the torsion \(\tau_2\) is determined by \(\tau_1\). In addition, certain examples are discussed. The examples illustrate the type of \(SU(3)\)-structure that can arise as a reduction of a metric with holonomy \(G_2\).
For the entire collection see [Zbl 0995.00012].

MSC:

53C10 \(G\)-structures
53B21 Methods of local Riemannian geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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