J. Korean Math. Soc. 2021; 58(4): 895-920
Online first article December 23, 2020 Printed July 1, 2021
https://doi.org/10.4134/JKMS.j200311
Copyright © The Korean Mathematical Society.
Hyunjin Lee, Young Jin Suh
Kyungpook National University; Kyungpook National University
In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface $M$ in the complex quadric~$Q^{m}$ from the equation of Gauss and some important formulas for the structure Jacobi operator ~$R_{\xi}$ and its derivatives $\nabla R_{\xi}$ under the Levi-Civita connection $\nabla$ of $M$. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi}=0$, in the complex quadric $Q^{m}$ for $m \geq 3$. In addition, we also consider a new notion of $\mathcal C$-parallel structure Jacobi operator of $M$ and give a nonexistence theorem for Hopf real hypersurfaces with $\mathcal C$-parallel structure Jacobi operator in $Q^{m}$, for $m \geq 3$.
Keywords: Reeb parallel structure Jacobi operator, $\mathcal C$-parallel structure Jacobi operator, singular normal vector field, K\"{a}hler structure, complex conjugation, complex quadric
MSC numbers: 53C40, 53C55
Supported by: The first author was supported by grant Proj. No. NRF-2019-R1I1A1A-01050300 and the second author by grant Proj. No. NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea.
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