Real hypersurfaces in complex two-plane Grassmannians with commuting restricted Jacobi operators

Abstract

In this paper, we have considered a new commuting condition, that is, $(R_{\xi }\varphi )S=S(R_{\xi }\varphi )$ \big(resp. $(\Bar{R}_Nϕ) S = S (\Bar{R}_Nϕ$)\big) between the restricted Jacobi operator~$R_{\xi }\varphi$ (resp. $\Bar{R}_Nϕ$), and the Ricci tensor $S$ for real hypersurfaces $M$ in $G_2({\mathbb{C}}^{m+2})$. In terms of this condition we give a complete classification for Hopf hypersurfaces $M$ in $G_2({\mathbb{C}}^{m+2})$.