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).

我们已经观察到平滑歧管上的几何结构和拓扑结构之间的沟幅。 H.Weyl从仪表理论的角度引入了保形结构的概念。他发现了一个不变结构，现在称为Weyl Sonformal曲率张量。差异几何形状的基本结果是，当M^n在局部在局部等于平坦的欧几里得空间时，n维riemannian歧管m^n（n> 3）的Weyl串正曲线张量就消失了。沿着这条线，但只有riemannian几何形状，我们研究了与其他几何结构相结合坚固的不变性。更确切地说，作为几何结构和（奇数）维歧管的连形结构，我们将Bochner曲率张量的定义和Cr-manifolds和Cr-manifolds分别定义为CR-Manifolds，我们将Bochner曲率张量的定义分别定义为Bochner曲率张量。我们将定义与给定的几何结构的形式等效，然后在其上构建一个形式不变（张量）。当不变的消失时，我们观察到了哪种新（或经典）几何形状（G，X）出现了什么样的。同样，我们获得了一个分类理论，即这种消失不变张量的歧管可以相对于模型空间（G，x）均匀。

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: "Transformation groups on Heisenberg geometry," to appear in Kumamoto J.of Math.(1996)

Yoshinobu Kumamoto：“海森堡几何上的变换群”，出现在 Kumamoto J.of Math. (1996)

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。