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

TL;DR

This paper introduces the star-Ricci tensor for real hypersurfaces in complex quadrics, proving non-existence of certain Hopf hypersurfaces with specific tensor properties and characterizing star-Ricci solitons.

Contribution

It defines star-Ricci tensors in complex quadric hypersurfaces and characterizes hypersurfaces with star-Ricci solitons as tubes around complex projective spaces.

Findings

No Hopf hypersurfaces with commuting or parallel star-Ricci tensor exist in $Q^m$, $m extgreater=3$.

Star-Ricci solitons correspond to tubes around totally geodesic $ ext{C}P^{m/2}$ in $Q^m$, $m extgreater=4$.

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 $Q^m,m\geq3$, 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 $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.