Mutual Invariance between Geometric Structures and Toplogical Structures on Manifolds

流形上几何结构与拓扑结构的互不变性

• 批准号: 06640161
• 负责人: KAMISHIMA Yoshinobu
• 金额: $ 1.28万
• 依托单位: Kumamoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06640161/
• 关键词: Weyl Curvature tensor; Symplectic structure; Pseoudo-Hermiyian structure; Uniformization; Geometric structure; Vanishing curvature; CR-structure; Conformal atructure; Contactization; CR-structure; Contactization; pseudo-Hermitian構造; Contact fo

We have observed the ralation between geometric and topological structures on smooth manifolds. H.Weyl has introduced the notion of conformal structure from the viewpoint of the Gauge theory. And he found an invariant on conformal structure, which is now called Weyl Conformal Curvature Tensor. It is the fundamental result in differential geometry that the Weyl conformal curvature tensor of an n-dimensional Riemannian manifold M^n (n>3) vanishes if and only if M^n is locally conformally equivalent to the flat euclidean space. Along this line but only Riemannian geometry, we have examined the invariance of conformal sturusture to other geometric structures. More precisely, as a geometric structure and a conformal invariant to even (resp. odd) dimensional manifolds, we brougth into focus Kahler manifolds for which the Bochner curvature tensor has been defined and CR-manifolds for which the Chern-Moser-Webster curvature tensor has been defined respectively. We shall define a conformal equivalence to the given geometric structure, and then construct a conformal invariant (tensor) on it.When that invariant vanishes, we observed what kind of new (or classical) geometry (G,X) comes out. Similtaneously, we have obtained a classification theorerm that such a manifold with vanishing invariant tensor can be uniformized with respect to the model space (G,X).

我们观察了光滑流形上几何结构与拓扑结构之间的关系。H.Weyl从规范理论的观点引入了共形结构的概念。他发现了一个关于共形结构的不变量，现在被称为外尔共形曲率张量。微分几何中的基本结果是n维黎曼流形M^n（n>3）的Weyl共形曲率张量为零当且仅当M^n局部共形等价于平坦欧氏空间。沿着这条线，但只限于黎曼几何，我们已经研究了共形结构对其他几何结构的不变性。更确切地说，作为一个几何结构和共形不变量，甚至（分别）。奇）维流形，分别引入了定义了Bochner曲率张量的焦点Kahler流形和定义了Chern-Moser-韦伯斯特曲率张量的CR-流形.我们将定义一个与给定几何结构的共形等价，然后在其上构造一个共形不变量（张量），当该不变量消失时，我们观察到什么样的新（或经典）几何（G，X）出现。同样，我们得到了一个分类定理，证明了这样一个不变张量为零的流形可以关于模型空间（G，X）一致化。

项目成果

神島芳宣: "Uniformization of Kahler manifolds with vanishing Bochner tensor" Acta Mathematica. 172. 299-308 (1994)

Yoshinobu Kamishima：“卡勒流形与消失的博赫纳张量的均匀化”Acta Mathematica 172. 299-308 (1994)。

Y.Kamishima: "Topolgy of CR-manifolds and Kahler manifolds, Symplectic Geometry and Related Topics, Proceedings, Kyowon University in Korea, Jeong Seog Ryu (ed.)" Proceedings of Workshops in Pure Math. (to appear in). (1996)

Y.Kamishima：“CR 流形和卡勒流形的拓扑学、辛几何和相关主题，韩国教园大学论文集，Jeong Seog Ryu（编辑）”纯数学研讨会论文集。

神島芳宣: "Pseudo-Hermitian structure on manifolds from Riemannian geometry" Differential Geometry and Related Topics Proceedings of Workshops in Pure Math. Part III. 13. 165-213 (1994)

Yoshinobu Kamishima：“黎曼几何流形上的伪埃尔米特结构”纯数学研讨会的微分几何和相关主题论文集第三部分。13. 165-213 (1994)

神島芳宣: "to appear in Kumamoto J. of Math." Transformation groups on Heisenbery geometry,(1996)

Yoshinobu Kumamoto：“出现在 Kumamoto J. of Math 上的 Heisenbery 几何变换群”，（1996 年）

Y.Kamishima: "Standard pseudo-Hermitian structure and Seifert fibration on CR manifold" Annals of Global Analysis and Geometry. 12. 261-289 (1994)

Y.Kamishima：“CR 流形上的标准伪厄米特结构和 Seifert 纤维化”全局分析与几何年鉴。