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

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)SU2,m/S(U2⋅Um), m≥2m≥2 from the equation of Gauss. Next we derive a new formula for the Ricci tensor S of M   in SU2,m/S(U2⋅Um)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)SU2,m/S(U2⋅Um) with Reeb invariant Ricci tensor, that is, LξS=0LξS=0. Each can be described as a tube over a totally geodesic SU2,m−1/S(U2⋅Um−1)SU2,m−1/S(U2⋅Um−1) in SU2,m/S(U2⋅Um)SU2,m/S(U2⋅Um) or a horosphere whose center at infinity is singular.