Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator

Abstract

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\). Next, we give a complete proof of non-existence of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with shape operator of Codazzi type. Motivated by this result, we give a complete classification of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with Reeb parallel shape operator.