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

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G 2 (ℂ m +2 ) with a commuting condition between the shape operator A and the structure tensors φ and φ 1 for M in G 2 (ℂ m +2 ). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ 1 induced by two structure tensors φ and φ 1 . That is, this commuting shape operator is given by φφ 1 A = A φφ 1 . Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G 2 (ℂ m +1 ) in G 2 (ℂ m +2 ).