Real Hypersurfaces with Killing Shape Operator in the Complex Quadric

Abstract

We introduce the notion of Killing shape operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_m SO_2$. The Killing shape operator condition implies that the unit normal vector field $N$ becomes $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in $Q^m = SO_{m+2}/SO_m SO_2$ with Killing shape operator.