Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor

Abstract

In this paper, first we introduce the full expression for the Ricci tensor of a real hypersurface M in complex two-plane Grassmannians G 2 ( C m + 2 ) from the equation of Gauss. Next we prove that a Hopf hypersurface in complex two-plane Grassmannians G 2 ( C m + 2 ) with commuting Ricci tensor is locally congruent to a tube of radius r over a totally geodesic G 2 ( C m + 1 ) . Finally it can be verified that there do not exist any Hopf Einstein hypersurfaces in G 2 ( C m + 2 ) .