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.

可以说黎曼几何是随着测地线的研究而发展起来的。在众多光滑曲线中，黎曼流形上的几何学家主要研究的是测地线。在这项研究中，我们建议研究一些包含测地线的“好”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，每个测地线都是其等距群的某个单参数子群的轨道。注意到这一事实，如果黎曼流形 $M$ 上的一条曲线是 $M$ 等距群的某个单参数子群的轨道，我们就说它是杀伤螺旋，并且我们对他们。由于它们是某种 Killing 矢量场的积分曲线，所以不用说它们很简单；也就是说，它们没有自交点。我们的程序是拾取一些与其他几何对象相关的杀伤螺旋并研究它们或利用它们的属性来研究其他几何对象。在这项研究中，我们研究了与子流形相关的杀伤螺旋。在一阶对称空间中的许多齐次子流形上，如果我们将某些类型的测地线视为一阶对称空间上的曲线，那么它们就是杀死螺旋。这告诉我们，当我们研究一阶对称空间时，杀伤螺旋与子流形相关。在本研究的前半部分，我们从子流形的角度获得了一阶对称空间上杀伤螺旋的性质。后半部分，换个角度，我们得到了利用子流形上的杀伤螺旋的一些性质得到的子流形的性质。