THE RICCI TENSOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

Abstract

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