Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

Abstract

In this paper, we have considered new commuting conditions, that is, (R ξ ϕ)S = S(R ξ ϕ) (resp. \((\bar{R}_{N}\phi )S = S(\bar{R}_{N}\phi )\)) between the Jacobi operators R ξ (resp. \(\bar{R}_{N}\)), the structure tensor field ϕ and the Ricci tensor S for real hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\). With such a condition we give a complete classification of Hopf hypersurfaces M in \(G_{2}(\mathbb{C}^{m+2})\).