Bulletin of the
Korean Mathematical Society BKMS

Let $M$ be a real hypersurface with almost contact metric structure $(\phi, \xi, \eta, g)$ in a nonflat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ commutes with both the structure tensor $\phi$ and the Ricc tensor $S$, then $M$ is a Hopf hypersurface in $M_n(c)$ provided that the mean curvature of $M$ is constant or $g(S\xi, \xi)$ is constant.

Keywords: structure Jacobi operator, Ricci tensor, Hopf hypersurface, nonflat complex space form

MSC numbers: Primary 53C40; Secondary 53C15