We first show that there exist no real hypersurfaces $M^{2n-1}$ which are Kenmotsu manifolds with respect to the almost contact metric structure $(\phi, \xi, \eta, g)$ on $M$ induced from the K$\ddot{a}$hler structure of a complex $n(\geqq 2)$-dimensional nonflat complex space form $\widetilde{M}_n(c)$. Next, weakening this condition, we classify normal real hypersurfaces $M^{2n-1}$ in $\widetilde{M}_n(c)$ and give some necessary and sufficient conditions for a real hypersurface $M$ to be normal from the viewpoint of submanifold theory.