On characterizations of real hypersurface in complex space form with Codazzi type structure Lie operator

In this paper, we prove that if $(nabla _{X} L_{xi })Y= (nabla _{Y} L_{xi })X$ holds on $M$ , then $M$ is a Hopf hypersurface, where $L_xi $ denote the induced operator from the Lie derivative with respect to the structure vector field $xi $ . We characterize such Hopf hypersurfaces of $M_n(c)$ .