Real hypersurfaces in the complex quadric with Reeb parallel shape operator

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric Q m = SO m+2 /SO m SO 2 . Next, we give a complete proof of non-existence of real hypersurfaces in Q m = SO m+2 /SO m SO 2 with shape operator of Codazzi type. Motivated by this result we have given a complete classification of real hypersurfaces in Q m = SO m+2 /SO m SO 2 with Reeb parallel shape operator.