Glasnik Matematicki, Vol. 59, No. 2 (2024), 461-478. \( \)

REAL HYPERSURFACES WITH SEMI-PARALLEL NORMAL JACOBI OPERATOR IN THE REAL GRASSMANNIANS OF RANK TWO

Hyunjin Lee and Young Jin Suh

Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea
e-mail:lhjibis@hanmail.net

Department of Mathematics & RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail:yjsuh@knu.ac.kr

Abstract.   In this paper, we introduce the notion of a semi-parallel normal Jacobi operator for a real hypersurface in the real Grassmannian of rank two, denoted by \(\mathbb Q^{m}(\varepsilon)\), where \(\varepsilon=\pm 1\). Here, \(\mathbb Q^{m}(\varepsilon)\) represents the complex quadric \(\mathbb Q^{m}(1)=SO_{m+2}/SO_{m}SO_{2}\) for \(\varepsilon=1\) and \(\mathbb Q^{m}(-1)=SO_{m,2}^{0}/SO_{m}SO_{2}\) for \(\varepsilon =-1\), respectively. In general, the notion of semi-parallel is weaker than the notion of parallel normal Jacobi operator. In this paper we prove that the unit normal vector field of a Hopf real hypersurface in \({\mathbb Q^{m}(\varepsilon)}\), \(m \geq 3\), with semi-parallel normal Jacobi operator is singular. Moreover, the singularity of the normal vector field gives a nonexistence result for Hopf real hypersurfaces in \(\mathbb Q^{m}(\varepsilon)\), \(m \geq 3\), admitting a semi-parallel normal Jacobi operator.

2020 Mathematics Subject Classification.   53C40, 53C55

Key words and phrases.   Semi-parallelism, normal Jacobi operator, \(\mathfrak A\)-isotropic, \(\mathfrak A\)-principal, real hypersurfaces, real Grassmannian of rank two, complex quadric, complex hyperbolic quadric

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https://doi.org/10.3336/gm.59.2.09

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