Commuting Jacobi operators on Real hypersurfaces of Type B in the complex quadric

Abstract

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar{R}}_N$ and the structure Jacobi operator~$R_{\xi }$ for Hopf real hypersurfaces in the complex quadric~$Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$, $m\ge 3$, which is defined by ${\bar{R}}_N{R}_{\xi }=R_{\xi }{\bar{R}}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak{A}$-principal singular normal vector field in the complex quadric~$Q^m$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^m$ with commuting Jacobi operators.

20 pages. arXiv admin note: text overlap with arXiv:1907.04661, arXiv:1605.05316