Real hypersurfaces in a complex space form with a condition on the structure Jacobi operator

Abstract In this paper we classify the real hypersurfaces in a non-flat complex space form with its structure Jacobi operator R ξ satisfying (∇X R ξ)ξ = 0, for all vector fields X in the maximal holomorphic distribution D. With this result, we prove the non-existence of real hypersurfaces with D-parallel as well as D-recurrent structure Jacobi operator in complex projective and hyperbolic spaces. We can also prove the non-existence of real hypersurfaces with recurrent structure Jacobi operator in a non-flat complex space form as a corollary.