Ruled Real Hypersurfaces in the Complex Quadric

First we introduce the notions of η -parallel and η -commuting shape operator for real hypersurfaces in the complex quadric Q^m = SO_m+2/SO_mSO_2 . Next we give a complete classification of real hypersurfaces in the complex quadric Q^m with such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex totally geodesic hyperplanes Q^m-1 in Q^m whose unit normal vector field in Q^m is 𝔄 -principal.