论文标题
Cr-Twistor空间以$ g_2 $ - 和$ spin（7）$ - 结构
CR-twistor spaces over manifolds with $G_2$- and $Spin(7)$-structures
论文作者
论文摘要
1984年，勒布伦（Lebrun）在任意的riemannian 3曼福尔德（Riemannian 3-Manifold）上建造了一个Cr-Twistor空间，并证明了CR结构是正式的。罗西（Rossi）在1985年以$ m $二维的利曼尼亚歧管（M $ M $）的形式概括了这种扭曲的建筑，并具有$（M-1）$ - 折叠矢量跨产品（VCP）。在2011年，Verbitsky概括了勒布朗（Lebrun）的扭曲空间的建设，达到了$ 7 $ -Manifolds，并具有$ G_2 $ - 结构。在本文中，我们统一并概括了勒布朗，罗西（Rossi's）和韦比茨基（Verbitsky）的CR-Twistor空间的构建，其中Riemannian歧管$（M，G）$具有VCP结构。我们表明，CR结构的形式可集成性是根据扭曲空间上的扭转张量表示的，这是$（M，G）$的Grassmanian Bundle。 If the VCP structure on $(M,g)$ is generated by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion tensor vanishes if and only if $(M, g)$ has constant curvature, and the horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or $Spin(7)$-manifold.最后，我们讨论一些开放问题。
In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for $m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmanian bundle over $(M, g)$. If the VCP structure on $(M,g)$ is generated by a $G_2$- or $Spin(7)$-structure, then the vertical component of the torsion tensor vanishes if and only if $(M, g)$ has constant curvature, and the horizontal component vanishes if and only if $(M,g)$ is a torsion-free $G_2$ or $Spin(7)$-manifold. Finally we discuss some open problems.
