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

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