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Geometry of symmetric spaces of rank one

一阶对称空间的几何

基本信息

批准号：
19540084
负责人：
MAEDA Sadahiro
金额：
$ 2.41万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2007
资助国家：
日本
起止时间：
2007 至 2009
项目状态：
已结题

项目摘要

It is not too much to say that Riemannian geometry has been developed with the investigation of geodesics. Among many smooth curves on a Riemannian manifold geometers have mainly studied geodesics. In this research, we propose to study some families of "nice" curves containing geodesics in order to investigate some other properties of Riemannian manifolds. It is known that on an arbitrary Riemannian symmetric space every geodesic is an orbit of some one-parameter subgroup of its isometry group. Noticing this fact, we say that a curve on a Riemannian manifold $M$ is a Killing helix if it is an orbit of some one-parameter subgroup of the isometry group of $M$, and we shed some light on the geometric study of them. Since they are integral curves of some Killing vector field, needless to say they are simple ; namely, they do not have self intersection points.Our program is to pick up some of the Killing helices in connection with some other geometric objects and study them or study other geometric objects by use of their properties. In this research we study Killing helices in connection with submanifolds. On many homogeneous submanifolds in a symmetric space of rank one, some kinds of geodesics are Killing helices if we consider them as curves on a symmetric space of rank one. This suggests to us that Killing helices are related to submanifolds when we study symmetric spaces of rank one.In the first half of this research, we obtain properties of Killing helices on a symmetric space of rank one which are obtained from the viewpoint of submanifolds. In the latter half, changing the point of view, we obtain properties of submanifolds obtained by making use of some properties of Killing helices on them.

可以毫不夸张地说，黎曼几何是随着测地线的研究而发展起来的。在黎曼流形上的许多光滑曲线中，几何学者主要研究测地线。在这项研究中，我们建议研究一些包含测地线的“NICE”曲线族，以研究黎曼流形的一些其他性质。众所周知，在任意黎曼对称空间上，每条测地线都是它的等距群的某个单参数子群的轨道。注意到这一事实，我们说黎曼流形上的一条曲线是一条Killing螺旋线，如果它是$M$的等距群的某个单参数子群的轨道，我们给出了一些关于它们的几何研究。由于它们是某些KILING向量场的积分曲线，因此不用说它们是简单的，即它们没有自交点。我们的程序是提取一些与其他几何对象相关的KILING螺旋线，并利用它们的性质来研究它们或研究其他几何对象。在这项研究中，我们研究了与子流形有关的杀伤螺旋。在一阶对称空间中的许多齐次子流形上，如果我们将其看作一阶对称空间上的曲线，则某些测地线是Killing螺旋。这表明，当我们研究一阶对称空间时，Killing螺旋与子流形有关。在本研究的前半部分，我们从子流形的角度得到了一阶对称空间上的Killing螺旋的性质。在后半部分，我们改变了观点，利用子流形上的Killing螺旋线的一些性质得到了它们的性质。