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 projec-tive space of holomorphic sectional curvature 4 is quite different according as tan²r is rational or irrational. Each length spectrum is simple when tan²r is irrational, but when tan²r is rational, it is not necessarily simple and moreover the multiplicity is not uniformly bounded.