Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(\varphi ,\xi ,\eta ,g)$. 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 ${\mathrm{\nabla }}_{\xi }{R}_{\xi }=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_{\xi }\varphi S=S\varphi R_{\xi }$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.