The *-Ricci Tensor of Real Hypersurfaces in Symmetric Spaces of Rank One or Two

Abstract

Complex projective and hyperbolic spaces, i.e. non-flat complex space forms, are symmetric spaces of rank one. Complex two-plane Grassmannians are symmetric spaces of rank two. Let M be a real hypersurface in a symmetric space of rank one or two. Many geometers, such as Berndt, Jeong, Kim, Ortega, Pérez, Santos, Suh, Takagi and others have studied real hypersurfaces in above spaces in terms of their operators and tensor fields. This paper will be divided into two parts. Firstly, results concerning real hypersurfaces in non-flat complex space forms in terms of their^∗-Ricci tensor, S ^∗, which in case of real hypersurfaces was first studied by Hamada (Real hypersurfaces of complex space forms in terms of Ricci *-tensor. Tokyo J. Math. 25, 473–483 (2002)), will be presented. More precisely, it will be answered if there exist or not real hypersurfaces, whose^∗-Ricci tensor is parallel, semi-parallel, i.e. R ⋅ S ^∗ = 0, or pseudo-parallel, i.e. $R(X,Y ) \cdot S^{{\ast}} = L\{(X \wedge Y ) \cdot S^{{\ast}}\}$ with L ≠ 0 (Kaimakamis and Panagiotidou, Parallel^∗-Ricci tensor of real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. Taiwan. J. Math., to appear, DOI 10.11650/tjm.18.2014.4271; Kaimakamis and Panagiotidou, Conditions of parallelism of^∗-Ricci tensor of real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. Preprint). Secondly, the formula of^∗-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians will be provided (Panagiotidou, The^∗-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, work in progress).