  In this paper we prove that for a complete, connected and oriented K\"{a}ler
affine manifold $(M,G)$ of dimension $n,$ if it is K\"ahler affine Ricci flat
or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then
the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the
Euclidean n-space $E^{n}$.
