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

Abstract

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um), m ≥ 2 from the equation of Gauss. Next we derive a new formula for the Ricci tensor of M in SU2,m/S(U2 ⋅ Um). Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um) with commuting Ricci tensor. Each can be described as a tube over a totally geodesic SU2,m-1/S(U2 ⋅ Um-1) in SU2,m/S(U2 ⋅ Um) or a horosphere whose center at infinity is singular.