Length spectrum of geodesic spheres in a non-flat complex space form

Abstract

We investigate the distribution of length of closed geodesics on geodesic spheres and tubes around complex hyperplane in a non-flat complex space form. The feature of the length spectrum of a geodesic sphere of radius $r$ in a complex projective space of holomorphic sectional curvature 4 is quite different according as ${\tan }^{2}r$ is rational or irrational. Each length spectrum is simple when ${\tan }^{2}r$ is irrationaj but when ${\tan }^{2}r$ is rationaj it is not necessarily simple and moreover the multiplicity is not uniformly bounded.