On Characterizations of Hopf Hypersurfaces in a Nonflat Complex Space Form with Anti-commuting Operators

Abstract

Let M be a real hypersurface in a complex space form Mn(c), \({c \neq 0}\). In this paper we prove that if \({R_{\xi}(\phi A - A\phi) + (\phi A - A\phi)R_{\xi} = 0}\) holds on M, then M is a Hopf hypersurface, where \({R_{\xi}}\) is the structure Jacobi operator, A is the shape operator of M in Mn(c) and \({\phi}\) is the tangential projection of the complex structure of Mn(c). We characterize such Hopf hypersurfaces of Mn(c).