RICCI TENSOR OF REAL HYPERSURFACES

Let M be a real hypersurface of a complex space form M-n(c), c not equal 0, and suppose that the structure vector field xi is an eigen vector field of the Ricci tensor S, which satisfies S xi = beta xi where beta is a function. We show that if (del(X) S)Y is proportional to xi for any vector fields X and Y orthogonal to xi, then M is a Hopf hypersurface, and if it is perpendicular to xi, then M is a ruled real hypersurface.