If a Spin(7)-manifold $$N^8$$ admits a free $$S^1$$ action preserving the fundamental 4-form, then the quotient space $$M^7$$ is naturally endowed with a $$G_2$$ -structure. We derive equations relating the intrinsic torsion of the Spin(7)-structure to that of the $$G_2$$ -structure together with the additional data of a Higgs field and the curvature of the $$S^1$$ -bundle; this can be interpreted as a Gibbons–Hawking-type ansatz for Spin(7)-structures. In particular, we show that if N is a Spin(7)-manifold, then M cannot have holonomy contained in $$G_2$$ unless the N is in fact a Calabi–Yau fourfold and M is the product of a Calabi–Yau threefold and an interval. By inverting this construction, we give examples of SU(4) holonomy metrics starting from torsion-free SU(3)-structures. We also derive a new formula for the Ricci curvature of Spin(7)-structures in terms of the torsion forms. We then describe this $$S^1$$ -quotient construction in detail on the Bryant–Salamon Spin(7) metric on the spinor bundle of $$S^4$$ and on flat $$\mathbb {R}^8$$ .

中文翻译：

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$$S^1$$S1-Spin(7)-结构的商

如果一个 Spin(7)-流形 $$N^8$$ 承认一个自由的 $$S^1$$ 动作并保留了基本的 4-形式，那么商空间 $$M^7$$ 自然被赋予了一个 $ $G_2$$ -结构。我们推导出将 Spin(7) 结构的内在扭转与 $$G_2$$ 结构的内在扭转以及希格斯场的附加数据和 $$S^1$$ 束的曲率相关联的方程；这可以解释为 Spin(7) 结构的 Gibbons-Hawking 型 ansatz。特别地，我们证明，如果 N 是 Spin(7)-流形，则 M 不能包含在 $$G_2$$ 中的完整，除非 N 实际上是 Calabi-Yau 四重并且 M 是 Calabi-Yau 的乘积三倍和一个区间。通过反转这个结构，我们给出了从无扭转 SU(3) 结构开始的 SU(4) 完整度量的例子。我们还根据扭转形式推导出了 Spin(7) 结构的 Ricci 曲率的新公式。然后，我们在 $$S^4$$ 的旋量丛和平面 $$\mathbb {R}^8 上的 Bryant–Salamon Spin(7) 度量上详细描述了这个 $$S^1$$ -商构造$$ 。