Global Geometrical Approach to Contact Manifolds

接触流形的全局几何方法

• 批准号: 11440016
• 负责人: ITOH Mitsuhiro
• 金额: $ 3.52万
• 依托单位: UNIVERSITY OF TSUKUBA
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B).
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440016/
• 关键词: Contact manifold; CR twistor space; Almost CR structure; Self-dual Weyl conformal tensor; 佐々木多様体; 自己双対性; 調和形式; 接触カップ積

In this project we studied the following researches.1. Study of CR twistor space over a 5-dim contact metric manifold was developed. In analogy of 4-dim manifold it is shown that the CR twistor space admits an almost CR structure and it is verified that this almost CR structure is integrable under the curvature conditions on a given base contact metric 5-manifold, that the anti-self-dual Weyl conformal tensor vanishes and also the scalar curvature s=-4.2. 4-dimensional geometry can be applied to the contact subbundle of contact metric manifolds. By Tachibana's theorem and also by N.Tanaka's systematic theory on CR geometry harmonic k-forms over a compact Sasakian (2n+1)-manifold take values in the contact subbundle, when k<n+1. So the self-duality in Sasakian contact structure was defined like 4-dim manifold theory. Remark that Sasakian contact structure turns out to be nothing but a normal strongly pseudo convex CR structure, a main subject in CR geometry.3. Study of Legendrian surfaces minimally immersed in a Sasakian contact 5-manifold was proceeded in terms of Hopf differential, the cubic differential and also in terms of the second variation.

在本项目中，我们进行了以下研究：1.研究了5维切触度量流形上的CR扭量空间。通过与四维流形的类比，证明了CR扭量空间存在一个几乎CR结构，并证明了在给定的基切触度量5-流形上，在曲率条件下，这个几乎CR结构是可积的，反自对偶Weyl共形张量为零，标量曲率s=-4.2. 4-维几何可以应用于切触度量流形的切触子丛。根据Tachibana定理和N.Tanaka关于CR几何的系统理论，当k<n+1时，紧致Sasakian（2n+1）-流形上的调和k-形式取值于切触子丛。因此，Sasakian接触结构中的自对偶被定义为类似四维流形理论。注：Sasakian切触结构只是一个正规强伪凸CR结构，这是CR几何中的一个重要课题.利用Hopf微分、三次微分和二阶变分研究了极小浸入Sasakian切触5-流形的Legendrian曲面.

项目成果

Mitsuhiro Itoh: "Weyl Manifolds and the Morse Functional"Proceedings 4th Intern. Workshop of Diff.Geometry (Korea). 4巻. 9-17 (2000)

Mitsuhiro Itoh：“Weyl 流形和莫尔斯泛函”论文集第四届 Diff.Geometry 实习生（韩国）第 4 卷 9-17（2000 年）。

M,Itoh and T,Satou: "Self-Dual METRICS on 4-dimensional Circle Bundles"Nihonkai Mathematical Jornal. 10巻1号. 71-86 (1999)

M,Itoh 和 T,Satou：“4 维圆束的自对偶度量”《日本海数学杂志》第 10 卷，第 1. 71-86 (1999)

Mitsthiro Itoh: "Global Geometry of Sasavian Manifolds"Proceedings of 4th International Workshop of Differential Geometry. 掲載予定. (2000)

Mitsthiro Itoh：“Sasavian 流形的全局几何”第四届国际微分几何研讨会论文集（2000 年）。

Mitsuhiro Itoh: "Weyl Manifolds and the Morse Functional"Proceedings 4th Intern.Workshop of Diffi Geometry (Korea). 4巻. 9-17 (2000)

Mitsuhiro Itoh：“Weyl 流形和莫尔斯泛函”论文集第 4 期实习生。Diffi 几何研讨会（韩国）第 4 卷 9-17（2000 年）。

Mitsuhiro, Itoh: "MInimally immersed Legendrian surfaces in Sasakian 5-manifolds"Kodai Math.Journal. 23. 358-375 (2000)

Mitsuhiro, Itoh：“佐佐木 5 流形中的最小沉浸式传奇曲面”Kodai Math.Journal。