Geometry of symmetric spaces of rank one

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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.