Notes on the redefinition of Berger-spheres

Studying geometric properties of geodesic spheres of sufficiently small radii in a complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c$ $(< 0)$, we present examples of Riemannian manifolds which are diffeomorphic to Euclidean spheres and are not so called Berger spheres. These manifolds are closely related to the redefinition of Berger spheres given in [9]. We next characterize these manifolds considered as real hypersurfaces in $\mathbb{C}H^n(c)$ from the viewpoint of submanifold geometry.