Real hypersurfaces in the complex quadric with Reeb invariant shape operator

First we introduce the notion of Reeb invariant shape operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2Qm=SOm+2/SOmSO2. Next we give a complete classification of real hypersurfaces in Qm=SOm+2/SOmSO2Qm=SOm+2/SOmSO2 with invariant shape operator.