Geometry of symmetric spaces of rank one

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。在本研究中，我们提出研究一些包含测地线的“漂亮”曲线族，以探讨黎曼流形的一些其他性质。已知在任意黎曼对称空间上，每条测地线都是其等距群的单参数子群的轨道。注意到这个事实，我们说黎曼流形$M$上的曲线如果是$M$等距群的某个单参数子群的轨道，那么它就是一个杀戮螺旋，并且我们对它们的几何研究有所启发。因为它们是某个杀戮向量场的积分曲线，所以不用说它们是简单的；也就是说，它们没有自交点。我们的计划是选择一些与其他几何物体相关的杀戮螺旋，并研究它们，或者通过使用它们的属性来研究其他几何物体。在本研究中，我们研究了与子流形相关的消旋。在一阶对称空间中的许多齐次子流形上，如果把某些测地线看作一阶对称空间上的曲线，它们就是杀戮螺旋。这表明当我们研究1阶对称空间时，杀戮螺旋与子流形有关。在前半部分的研究中，我们从子流形的角度得到了对称1阶空间上的杀戮螺旋的性质。后半部分，换一种观点，利用子流形上的杀螺旋的一些性质，得到子流形的性质。

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.