BERGER SPHERES: THEIR REDEFINITION AND RELATED EXAMPLES

Abstract

This expository paper consists of two parts. In the first half, using geometric properties of geodesic spheres with sufficiently big radii in a complex projective space, we redefine Berger spheres (cf. [18]). We study such Berger spheres from the viewpoints of contact geometry, submanifold geometry and length spectral geometry (see Theorems 1 and 2). In the latter half (cf. [17]), considering geodesic spheres of sufficiently small radii in a complex hyperbolic space, we present examples of non-Berger spheres related to the redefinition of Berger spheres (see Theorem 3). We finally characterize such non-Berger spheres considered as real hypersurfaces isometrically immersed into a complex hyperbolic space (see Theorem 4).