Kahler manifolds of quasi-constant holomorphic sectional curvatures

The Kahler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kahler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kahler metrics into Kahler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kahler metrics is shown to be exactly the class of Kahler metrics whose potential function is only a function of the distance from the origin in C(n). Finally we show that any rotational even dimensional hypersurface carries locally a natural Kahler structure which is of quasi-constant holomorphic sectional curvatures.