Journal of the
Korean Mathematical Society JKMS

Let $M$ be a real hypersurface in the complex hyperbolic quadric~${Q^{m}}^{*}$, $m \geq 3$. The Riemannian curvature tensor field~$R$ of~$M$ allows us to define a symmetric Jacobi operator with respect to the Reeb vector field~$\xi$, which is called the structure Jacobi operator~$R_{\xi} = R(\, \cdot \, , \xi) \xi \in \text{End}(TM)$. On the other hand, in~\cite{Semm03}, Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator~$R_{\xi}$ for a real hypersurface~$M$ in the complex hyperbolic quadric~${Q^{m}}^{*}$. Furthermore, we give a complete classification of Hopf real hypersurfaces in ${Q^{m}}^{*}$ with such a property.

Keywords: Complex hyperbolic quadric, Hopf real hypersurface, Killing structure Jacobi operator, cyclic parallel structure Jacobi operator, $\mathfrak A$-isotropic vector field, $\mathfrak A$-principal vector field, singular vector field

MSC numbers: Primary 53C40, 53C15

Supported by: The first author was supported by grant Proj.~No.~NRF-2022-R1A2C-100456411, the second author by NRF-2022-R1I1A1A-01055993, and the third by NRF-2018-R1D1A1B-05040381 & NRF-2021-R1C1C-2009847 from National Research Foundation of Korea.