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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.

给出了单重六主曲率等参超曲面齐性的一个新的简单证明。对此，我们研究了球面中退化高斯映射的子流形，并构造了许多满足Ferus不等式中等式的例子。所有等参超曲面的焦点子流形和所有退化高斯映射的等参超曲面都是严格子流形。已知一个严格子流形（锥）的法丛的正则嵌入给出一个特殊的拉格朗日子流形。我们证明了R ^n中的真完备严格子流形与维数不大于其半维的CW复形具有相同的同伦类型。从体积最小化法丛的观点来看，这是有趣的。作为S^3中常平均曲率曲面的Gauss映射分裂的反问题，我们得到了从一对调和映射到S^2中得到常平均曲率曲面的充要条件. Kaneyuki研究了广义共形变换群的轨道.内山研究了一类非线性常微分方程的奇性。Yokoyama完成了扩展DS图的理论。Tamaru对某些非紧型对称空间的各向同性轨道进行了分类。Aiyama给出了许多曲面的表示公式，特别是C^2中的Langarnian曲面。Ishikawa给出了空间曲线可展曲面的拓扑分类，并研究了切触流形的奇异性。Udagawa利用椭圆函数描述了从环面到Grassmannian流形的调和映射。他还研究了圆埃尔米特对称空间。Umehara研究了曲线的奇异性、极小曲面的端点，并结合它们的Gauss映射，研究了常曲率曲面的一个内在性质1. Kimura构造了子流形上圆丛和球丛的许多齐次例子.