## 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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