Nearly hypo structures and compact nearly Kähler 6‐manifolds with conical singularities
Nearly hypo structures and compact nearly Kähler 6‐manifolds with conical singularities
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DOI：
10.1112/jlms/jdn044
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发表时间：
2006-02
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影响因子：
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通讯作者：
Marisa Fern'andez;S. Ivanov;V. Muñoz;L. Ugarte
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作者：
Marisa Fern'andez;S. Ivanov;V. Muñoz;L. Ugarte
我们证明了6维近似Kähler流形M6的任何全测地超曲面N5是Sasaki-Einstein流形，因此它具有Conti和Salamon意义下的hypo结构[Trans. Amer. Math. Soc. 359（2007）5319-5343]。我们证明了任何Sasaki-Einstein 5-流形在sin-cone N5 × π上定义了一个近似Kähler结构，并且当N5是紧的时，在N5 × [0，π]上定义了一个具有圆锥奇点的紧近似Kähler结构，从而提供了锥N5 × [0，π]上的Calabi-Yau结构和sin-cone N5 × [0，π]上的近似Kähler结构之间的联系。定义了近次结构的概念，并由此构造了N5 × N上的近Kähler结构.我们将双亚结构刻画为亚结构和近亚结构的交集，并对第一Betti数不为零的5维李代数上的双亚结构进行了分类.引入了近似Kähler结构概念的一个扩展，我们称之为近似半平坦SU（3）-结构，并由此推广了Bilal和Metzger [Nuclear Phys. B 663（2003）343-364]给出的M6 × π上近似平行G2-结构的构造。对于N5 = S5 <$S6和N5 = S2 × S3 <$S3 × S3，我们明确地描述了一个Sasaki-Einstein亚结构以及相应的N5 × π和N5 × [0，π]上的近似Kähler结构，以及N5 × π 2和（N5 × [0，π]）× [0，π]上的近似平行G2-结构.
We prove that any totally geodesic hypersurface N5 of a 6‐dimensional nearly Kähler manifold M6 is a Sasaki–Einstein manifold, and so it has a hypo structure in the sense of Conti and Salamon [Trans. Amer. Math. Soc. 359 (2007) 5319–5343]. We show that any Sasaki–Einstein 5‐manifold defines a nearly Kähler structure on the sin‐cone N5 × ℝ, and a compact nearly Kähler structure with conical singularities on N5 × [0, π] when N5 is compact, thus providing a link between the Calabi–Yau structure on the cone N5 × [0, π] and the nearly Kähler structure on the sin‐cone N5 × [0, π]. We define the notion of nearly hypo structure, which leads to a general construction of nearly Kähler structure on N5 × ℝ. We characterize double hypo structure as the intersection of hypo and nearly hypo structures and classify double hypo structures on 5‐dimensional Lie algebras with non‐zero first Betti number. An extension of the concept of nearly Kähler structure is introduced, which we refer to as nearly half‐flat SU(3)‐structure, and which leads us to generalize the construction of nearly parallel G2‐structures on M6 × ℝ given by Bilal and Metzger [Nuclear Phys. B 663 (2003) 343–364]. For N5 = S5 ⊂ S6 and for N5 = S2 × S3 ⊂ S3 × S3, we describe explicitly a Sasaki–Einstein hypo structure as well as the corresponding nearly Kähler structures on N5 × ℝ and N5 × [0, π], and the nearly parallel G2‐structures on N5 × ℝ2 and (N5 × [0, π]) × [0, π].