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

Abstract

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 $${\mathbb {Q}}^{m}(\varepsilon )$$ , $$\varepsilon =\pm 1$$ , where $${\mathbb {Q}}^{m}(\varepsilon )$$ denotes the complex quadric $$\mathbb Q^{m}(\varepsilon )=Q^{m}=SO_{m+2}/SO_{m}SO_{2}$$ for $$\varepsilon =1$$ and $${\mathbb {Q}}^{m}(\varepsilon )= Q^{m*}=SO_{m,2}^{0}/SO_{m}SO_{2}$$ for $$\varepsilon =-1$$ , 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 $${\mathbb {Q}}^{m}(\varepsilon )$$ .