Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2021; 58(4): 895-920

Published online July 1, 2021 https://doi.org/10.4134/JKMS.j200311

Copyright © The Korean Mathematical Society.

A new classification of real hypersurfaces with Reeb parallel structure Jacobi operator in the complex quadric

Hyunjin Lee, Young Jin Suh

Kyungpook National University; Kyungpook National University

Abstract

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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