Real Hypersurfaces with Quadratic Killing Normal Jacobi Operator in the Real Grassmannians of Rank Two

In this paper, first we introduce a new notion of (quadratic) Killing normal Jacobi operator (or cyclic parallel normal Jacobi operator) and its geometric meaning for real hypersurfaces in the real Grassmannians of rank two Qm(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}^{m}(\varepsilon )$$\end{document}, ε=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =\pm 1$$\end{document}, where Qm(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}^{m}(\varepsilon )$$\end{document} denotes the complex quadric Qm(ε)=Qm=SOm+2/SOmSO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb Q^{m}(\varepsilon )=Q^{m}=SO_{m+2}/SO_{m}SO_{2}$$\end{document} for ε=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =1$$\end{document} and Qm(ε)=Qm∗=SOm,20/SOmSO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}^{m}(\varepsilon )= Q^{m*}=SO_{m,2}^{0}/SO_{m}SO_{2}$$\end{document} for ε=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =-1$$\end{document}, respectively. Next, we give a non-existence theorem for Hopf real hypersurfaces satisfying quadratic Killing normal Jacobi operator in the real Grassmannians of rank two Qm(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}^{m}(\varepsilon )$$\end{document}.