On characterizations of Hopf hypersurfaces in a nonflat complex space form with commuting operators

Abstract

Let $M$ be a real hypersurface in a complex space form $\mn$, $c \neq 0$. In this paper we prove that if $\rx\li = \li\rx$ holds on $M$, then $M$ is a Hopf hypersurface, where $\rx$ and $\li$ denote the structure Jacobi operator and the induced operator from the Lie derivative with respect to the structure vector field $\xi$, respectively. We characterize such Hopf hypersurfaces of $\mn$.