Kaehler Metrics Generated by Functions of the Time-Like Distance in the Flat Kaehler-Lorentz Space

Abstract

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kaehler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kaehler structures. We prove that these rotational hypersurfaces carry Kaehler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kaehler metrics are Bochner-Kaehler (especially of constant holomorphic sectional curvatures) are also described.