Distribution of length spectrum of circles on a complex hyperbolic space

It is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being con cerned with circles, we also know that two closed circles in a real space f orm are congruent if and only if they have the same length. In this paper w e study how prime periods of circles on a complex hyperbolic space are dist ributed on a real line and show that even if two circles have the same leng th and the same geodesic curvature they are not necessarily congruent each other.