Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians with commuting 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 $S{U}_{2,m}\mathrm{/}S(U_2\cdot U_m)$, $m\ge 2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $S{U}_{2,m}\mathrm{/}S(U_2\cdot U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $S{U}_{2,m}\mathrm{/}S(U_2\cdot U_m)$ with commuting Ricci tensor. Each can be described as a tube over a totally geodesic $S{U}_{2,m-1}\mathrm{/}S(U_2\cdot U_{m-1})$ in $S{U}_{2,m}\mathrm{/}S(U_2\cdot U_m)$ or a horosphere whose center at infinity is singular.