Geometry of the flat tori in the 3-sphere and its higher dimensional generalization

3-球面平面环面的几何形状及其高维推广

基本信息

批准号：
12640059
负责人：
KITAGAWA Yoshihisa
金额：
$ 2.3万
依托单位：
Utsunomiya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640059/
关键词：
differential geometry submanifold flat torus isometric deformation mean curvature 3-sphere meromorphic mapping dynamical system

项目摘要

In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings and dynamical systems. The main results of this research are summarized as follows.(1) Studies on isometric deformations of flat tori in the S-sphere.In this research, Y. Kitagawa studied isometric deformations of flat tori isometrically immersed in the 3-sphere S^3 with constant mean curvature. As a result, he obtained a classification of the flat tori isometrically immersed in S^3 which admit no isometric deformation.(2) Studies on algebraic dependence of meromorphic mappings.In this research, Y. Aihara proved some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. Moreover, applying these criteria, he obtained unicity theorems for meromorphic mappings, and gave conditions under which two holomorphic mappings from X into a smooth elliptic curve E are algebraically related.(3) Studies on vector fields with topological stability.In this research, K. Sakai (with K. Moriyasu and N. Sumi) gave a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it was proved that the C^1 interior of the set of all topologically stable C^1 vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.

在这项研究中，我们研究了在三个球体，mer态映射和动力学系统中的平托里的几何形状。这项研究的主要结果总结如下。（1）关于S-Sphere中Flat Tori等轴测变形的研究。在这项研究中，Y. Kitagawa研究了平面液的平面Tori的等轴测变形，并浸入了3个球体的S^3中，均具有恒定的平均曲率。结果，他获得了固定在s^3中的扁平圆花生的分类。代数歧管。 Moreover, applying these criteria, he obtained unicity theorems for meromorphic mappings, and gave conditions under which two holomorphic mappings from X into a smooth elliptic curve E are algebraically related.(3) Studies on vector fields with topological stability.In this research, K. Sakai (with K. Moriyasu and N. Sumi) gave a characterization of the structurally stable vector fields by利用拓扑稳定性的概念。更确切地说，证明所有拓扑稳定的C^1矢量场的集合的C^1内部与满足公理A的所有向量场的集合和强横向横向条件相吻合。