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Studies on Curvatures of Submanifolds

子流形曲率的研究

基本信息

批准号：
10640061
负责人：
KITAGAWA Yoshihisa
金额：
$ 2.11万
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：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640061/
关键词：
submanifold flat torus isometric deformation

项目摘要

In this research, using a method established by Y.Kitagawa, we studied flat tori in the unit 3-sphere SィイD13ィエD1, and obtained many interesting results. We summarize the results as follows.(1) Isometric deformations of flat tori in SィイD13ィエD1 with nonconstant mean curvature : In this research, we studied isometric deformations of an isometric immersion f of a flat torus M into SィイD13ィエD1, and proved that if the mean curvature of f is not constant, then the immersion f admits a nontrivial isometric deformation preserving the total mean curvature.(2) Isometric deformations of flat tori in SィイD13ィエD1 with constant mean curvature : It is easy to see that if f is an isometric immersion of a flat torus M into SィイD13ィエD1 with constant mean curvature, then f is a covering map onto a Clifford torus in SィイD13ィエD1. In this research, we classified the covering maps which admit no isometric deformation.(3) Periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1 : In 1988, it was shown that if M is a 2-dimensional flat torus isometrically immersed in SィイD13ィエD1, then every asymptotic curve of M is periodic. In this research, we studied periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1, and proved that there exists a 3-dimensional flat torus isometrically embedded in SィイD15ィエD1 all of whose asymptotic curves have no period.

在这项研究中，我们利用Y.Kitakawa建立的方法，研究了单位3球体SィイD13ィエD1中的扁平环面，并获得了许多有趣的结果。我们将结果总结如下。（1）具有非恒定平均曲率的SィイD13ィエD1中平坦环面的等距变形：在本研究中，我们研究了平坦环面M的等距浸入f到SィイD13ィエD1中的等距变形，并证明如果f的平均曲率不恒定，则浸入f承认非平凡等距变形 (2) SィイD13ィエD1 中具有恒定平均曲率的平坦环面的等距变形：很容易看出，如果 f 是平坦环面 M 到具有恒定平均曲率的 SィイD13ィエD1 中的等距浸入，则 f 是 SィイD13ィエD1 中 Clifford 环面的覆盖贴图。在本研究中，我们对不允许等距变形的覆盖图进行了分类。(3)等距浸入SィイD12n-1ィエD1中的n维平面环面渐近曲线的周期性：1988年，人们证明，如果M是等距浸入SィイD13ィエD1中的二维平面环面，则M的每条渐近曲线为周期性的。本研究研究了等距嵌入SィイD12n-1ィエD1中的n维平面环面渐近曲线的周期性，并证明了等距嵌入SィイD15ィエD1中存在3维平面环面，其渐近曲线均无周期。