Ruled Real Hypersurfaces in the Complex Quadric

Abstract

First we introduce the notions of $$\eta $$ -parallel and $$\eta $$ -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 $$\mathfrak {A}$$ -principal.