Sectional curvatures of geodesic spheres in a complex hyperbolic space

Abstract

We characterize geodesic spheres with sufficiently small radii in a complex hyperbolic space of constant holomorphic sectional curvature c(<0) by using their geometric three properties. These properties are based on their contact forms, geodesics and shape operators. These geodesic spheres are the only examples of hypersurfaces of type (A) which are of nonnegative sectional curvature in this ambient space. Moreover, in particular, when −1 ≤ c < 0, the class of these geodesic spheres has just one example of Sasakian space forms.