Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians

In this paper, we prove that there are no Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ such that the normal Jacobi operator is generalized $\mathfrak{F}$-recurrent, where $\mathfrak{F}=\operatorname{span}\{\xi,\xi_1,\xi_2,\xi_3\}$. We also prove that there are no Hopf real hypersurfaces in $G_2(\mathbb{C}^{m+2})$ such that the normal Jacobi operator is $\mathfrak{D}^\perp$-recurrent and the Hopf principal curvature is invariant along the Reeb flow, where $\mathfrak{D}^\perp=\operatorname{span}\{\xi_1,\xi_2,\xi_3\}$.