CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM USED BY THE ζ-PARALLEL STRUCTURE JACOBI OPERATOR

Let M be a real hypersurface of a complex space form with almost contact metric structure $({\phi},{\xi},{\eta},g)$ . In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},{\xi}){\xi}$ is ${\xi}$ -parallel. In particular, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex: projective space $P_n{\mathbb{C}}$ or a complex hyperbolic space $H_n{\mathbb{C}}$ when $g({\nabla}_{\xi}{\xi},{\nabla}_{\xi}{\xi})$ is constant and not equal to -c/24 on M, where c is a constant holomorphic sectional curvature of a complex space form.