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