Kaehler Manifolds of Quasi-Constant Holomorphic Sectional Curvatures

Abstract

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler 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 Kaehler metrics into Kaehler 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 Kaehler metrics is shown to be exactly the class of Kaehler metrics whose potential function is only a function of the distance from the origin in complex Euclidean space. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kaehler structure, which is of quasi-constant holomorphic sectional curvatures.