Bulletin of the
Korean Mathematical Society BKMS

Let $M$ be a real hypersurface with almost contact metric structure $(\phi, g, \xi, \eta)$ in a complex space form $M_n(c)$, $c \neq 0$. In this paper we prove that if $R_\xi \mathcal L_\xi g=0$ holds on $M$, then $M$ is a Hopf hypersurface in $M_n(c)$, where $R_\xi$ and $\mathcal L_x$ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field $\xi$ respectively. We characterize such Hopf hypersurfaces of $M_n(c)$.

Keywords: real hypersurface, structure Jacobi operator, Hopf hypersurface

MSC numbers: Primary 53C40; Secondary 53C15