Structure Lie operator on real hypersurfaces of complex two-plane Grassmannians

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, the structure Lie operator is parallel if and only if $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb {C}^{m+1})$ in $G_2(\mathbb {C}^{m+2})$.