Real hypersurfaces in a complex space form with non-commuting operators

Abstract

Let M be a real hypersurface with almost contact metric structure (ϕ,g,ξ,η) in a complex space form Mn(c), c≠0. In this paper we prove that if RξϕA+AϕRξ=0 holds on M, then M is a Hopf hypersurface in Mn(c), where A denotes the shape operator, ϕ the structure tensor and Rξ the Jacobi operator with respect to the structure vector field ξ. We characterize such Hopf hypersurfaces of Mn(c).