Differential systems and submanifolds theory

微分系统和子流形理论

基本信息

批准号：
12640087
负责人：
MIYAOKA Reiko
金额：
$ 1.86万
依托单位：
Sophia 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-12640087/
关键词：
Gauss mapping Lagrangian submanifolds Austere submanifolds Isoparametric hypersurfaces Shape operators Completely integrable systems Surfaces of constant mean curvature Harmonic maps 主曲率重複度退化スペシャルラグランジアン部分多様体

项目摘要

We gave a new simple proof of the homogeneity of isoparametric hypersurfaces with six principal curvatures with single multiplicity. Concerning this, we investigate submanifolds in the spheres with degenrate Gauss mapping, and constructed many examples satisfying the equality in the Ferus inequality. All focal submanifolds of isoparametric hypersurfaces and all isoparametric hypersurfaces with degenerate Gauss mapping are austere submanifolds. It is known that the canonical embedding of the normal bundle of a (cone of) austere submanifold gives a special Lagrangian submanifold. We have shown that a proper complete austere submanifolds in R^n have the same homotopy type as a CW complex of dimension not larger than half dimension of the submanifold. This is interesting from the view point of volume minimizing normal bundles. As an inverse problem of splitting Gauss maps of surfaces of constant mean curvature in S^3, we get necessary and sufficient conditions to get surfaces of constant mean curvature from a pair of harmonic maps intoS^2.Kaneyuki studied orbit of generalized conformal transformation groups. Uchiyama investigated singularities of a non-linear ordinary differential eqquations. Yokoyama completed the theory of extended DS diagram. Tamaru classified isotropy orbits of certain symmetric spaces of non-compact type. Aiyama gave many kind of reperesentation formulas for surfaces, especially Langarnian surfaces in C^2.Ishikawa gave a topological classification of developable surfaces of space curves and studied singularities of contact manifolds. Udagawa described harmonic maps from tori into Grassmannian manifolds by using elliptic functions. He also studied circles in Hermitian symmetric spaces. Umehara studied singularities of curves, ends of minimal surfaces and concerning with Gauss map of these, studied an intrinsic properties of surface with constant curvature 1.Kimura constructied many homogeneous examples of circle and sphere bundles over submanifolds.

我们给出了一个新的简单证明，证明具有六个主曲线的等距性超曲面的同质性具有单个多样性。关于这一点，我们研究了带有脱胶高斯映射球体中的子延伸，并构建了许多满足Ferus不平等平等的例子。等横侧曲面的所有局灶性亚频量和所有具有简化高斯映射的等型超曲面的均为submanifolds。众所周知，Austere Submanifold的（锥体锥）的正常束的规范嵌入给出了特殊的Lagrangian Submanifold。我们已经表明，R^n中的正确完整的朴素submanifolds具有与尺寸不大的CW复合物相同的同型类型。从最小化正常束的体积的角度来看，这很有趣。作为在s^3中分裂恒定平均曲率表面的高斯图的一个反问题，我们得到了必要和足够的条件，可以从一对谐波映射的恒定平均曲率表面获得一对谐波图的恒定表面^2.Kaneyuki研究了广义共形转换组的轨道。 Uchiyama调查了非线性普通差分式的奇异性。横山完成了扩展DS图的理论。 tamaru分类的非紧缩类型对称空间的各向同性轨道。 Aiyama为表面提供了许多类型的再生公式，尤其是C^的Langarnian表面。乌达瓦（Udagawa）通过使用椭圆函数将谐波图描述为从托里（Tori）到格拉曼尼亚（Grassmannian）歧管。他还在隐性对称空间学习了圆圈。 Umehara研究了曲线的奇异性，最小表面的末端以及与这些曲线的高图相关的，研究了表面的固有特性，其稳定曲率的表面1.Kimura 1. Kimura构建了许多圆形和球体捆绑的均匀实例。