Real hypersurfaces of complex quadric in terms of star-Ricci tensor

Abstract

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\ge 3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}{P}^{\frac{m}{2}}\subset Q^m,m\ge 4$.

13 pages, accepted by Tokyo Journal of Mathematics