Real hypersurfaces in the complex hyperbolic quadric with Reeb invariant Ricci tensor

Abstract

 We first give the notion of Reeb invariant Ricci tensor for real hypersurfaces $M$ in the complex quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$, which is defined by $\mathcal{L}_{\xi}\operatorname{Ric}=0$, where $\operatorname{Ric}$ denotes the Ricci tensor of $M$ in ${Q^m}^*$, and $\mathcal{L}_{\xi}$ the Lie derivative along the direction of the Reeb vector field $\xi=-JN$. Next we give a complete classification of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Reeb invariant Ricci tensor.