Complete Affine K$\ddot{a}$hler Manifolds

Abstract

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv 0,$ ($n\le 5$), then the universal covering manifold $\tilde{M}$ of $M$ is isometric to the Euclidean n-space $E^n$.