Let P"(~) denote the complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4 and suppose m >2. It is well-known that there does not exist a totally umbilical real hypersurface M of P"(C) (see Tashiro and Tachibana [7]). Moreover, as a consequence of Codazzi equation, there does not exist a real hypersurface M of pro(C) with parallel second fundamental form. To get this, Y. Maeda [4] estimated the norm of the derivative of the second fundamental form. He showed