In this note, using the spinorial description of $SU(3)$ and $G_2$-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique section induced by a $G$-structure in consideration.

中文翻译：

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谐波 SU(3)- 和 $$G_2$$- 结构通过 Spinors

在这篇笔记中，我们利用其他作者最近获得的 $SU(3)$ 和 $G_2$-结构的旋向描述，给出了上述结构调和性的充分必要条件。我们描述了在适当的齐次空间上获得的结果。这里，和谐性是指由考虑到的 $G$ 结构引起的独特部分的和谐性。