On Real Representations of Kaehlerian Manifolds

Abstract

This chapter presents the study on real representations of Kaehlerian manifolds, especially those of Kaehlerian manifolds of constant holomorphic sectional curvature. The chapter states the characteristic properties of real analytic manifolds that can represent complex analytic manifolds. The chapter discusses affine connections that do not change the collineation defined by ϕi, in every tangent space of the manifold, then Hermitian and Kaehlerian metrics in such manifolds is discussed. The chapter discusses the curvature in a Kaehlerian mailifold and examines the case in which the so-called “holomorphic sectional curvature” is constant. If it is assume that the so-called axiom of planes holds not for all the planes but only for the holomorphir planes, then the manifold will be of constant, holomorphic curvature. The study the distance between two consecutive conjugate points on a geodesic in a Kaehlerian manifold of positive constant holomorphic curvature is presented.