Real hypersurfaces with commuting Jacobi operator in the complex quadric

In this paper, first we introduce a new notion of commuting normal Jacobi operator ${\bar R}_N{\phi}={\phi}{\bar R}_N$ or commuting structure Jacobi operator $R_{\xi}{\phi}={\phi}R_{\xi}$ for real hypersurfaces in the complex quadrics $Q^m=SO_{m+2}/SO_mSO_2$. Next, we give a complete classification for real hypersurfaces in $Q^{m}$ satisfying commuting normal Jacobi operator or structure Jacobi operator, respectively.