Ruled Real Hypersurfaces in he Complex Quadric

Abstract

First we introduce the notions of $\eta$-parallel and $\eta$-commuting shape operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_m SO_2$. Next we give a complete classification of real hypersurfaces in the complex quadric $Q^m$ with such king of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurfaces foliated by complex totally geodesic hyperplanes $Q^{m-1}$ in $Q^m$ whose unit normal vector field in $Q^m$ is $\mathfrak{A}$-principal.