Journal of the
Korean Mathematical Society JKMS

In this paper, first we introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex two-plane Grassmannians $G_2({\Bbb C}^{m+2})$ from the equation of Gauss and derive a new formula for the Ricci tensor of $M$ in $G_2({\Bbb C}^{m+2})$. Next we prove that there do not exist any Hopf real hypersurfaces in complex two-plane Grassmannians $G_2({\Bbb C}^{m+2})$ with parallel and commuting Ricci tensor. Finally we show that there do not exist any Einstein Hopf hypersurfaces in $G_2({\Bbb C}^{m+2})$.

Keywords: real hypersurfaces, complex two-plane Grassmannians, parallel Ricci tensor, commuting Ricci tensor, Einstein

MSC numbers: Primary 53C40; Secondary 53C15