Bulletin of the
Korean Mathematical Society BKMS

In this paper, we introduce the notion of {\it semi-symmetric structure Jacobi operator } for Hopf real hypersufaces in the complex quad\-ric $Q^m = SO_{m+2}/SO_mSO_2$. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ with semi-symmetric structure Jacobi operator. As a corollary, we also get a non-existence property of Hopf real hypersurfaces in the complex quadric $Q^m$ with either symmetric (parallel), or recurrent structure Jacobi operator.

Keywords: Semi-symmetric structure Jacobi operator, complex quadric, Hopf hypersurface, $\mathfrak{A}$-isotropic or $\mathfrak{A}$-principal vector fields

MSC numbers: Primary 53C40; Secondary 53C15

Supported by: The first author was supported by grant Proj. No. NRF-2021-R1F1A1064192. The second author was supported by grant by Proj. No. NRF-2020-R1G1A1A-01003570. The third author was supported by grant Proj. No. NRF-2020-R1A2C1A-01101518.