Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ 1 A = A φ 1 φ between the shape operator A and the structure tensors φ and φ 1 for real hypersurfaces in G 2 (ℂ m +2 ). Suprisingly, real hypersurfaces of type ( A ), that is, a tube over a totally geodesic G 2 (ℂ m +1 ) in complex two plane Grassmannians G 2 (ℂ m +2 ) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G 2 (ℂ m +2 ) satisfying the commuting condition. Finally we get a characterization of Type ( A ) in terms of such commuting condition φφ 1 A = A φ 1 φ .