GEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS

This paper is the survey of joint works with K. Ogiue ([7]) and B.H. Kim, I.B. Kim ([5]). Geodesic spheres G(r) are fundamental examples of (real) hypersurfaces in a Riemannian manifold. In this paper, as an ambient space we take an n􀀀dimensional complex projective space CPn(c); n ≧ 2 of constant holomorphic sectional curvature c(> 0). By observing geodesics on G(r) in CPn(c) we characterize all G(r) (0 < r < =pc ) (Theorems 1 and 2)and some G(r) which are called Berger spheres (Theorem 3).