on characterizations of hypersurfaces in a sasakian space form

Let m be a real hypersurface in a sasakian space form m(c). in this paper, we prove that if rulu = luru holds on m, then m is a hopf hypersurface, where ru and lu denote the jacobi operator with respect to the structure vector field and the operator induced by the lie derivative along the unit normal vector field u, respectively. furthermore, we provide a complete characterization of such hopf hypersurfaces in m(c).