• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.895-920
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• 2021

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_ξ and its derivatives ∇R_ξ under the Levi-Civita connection ∇ of M. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ∇_ξR_ξ = 0, in the complex quadric Q^m for m ≥ 3. In addition, we also consider a new notion of 𝒞-parallel structure Jacobi operator of M and give a nonexistence theorem for Hopf real hypersurfaces with 𝒞-parallel structure Jacobi operator in Q^m, for m ≥ 3.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.473-486
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• 2021

We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the k-th generalized Tanaka-Webster connection.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.309-339
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• 2024

Let M be a real hypersurface in the complex hyperbolic quadric Q^m*, m ≥ 3. The Riemannian curvature tensor field R of M allows us to define a symmetric Jacobi operator with respect to the Reeb vector field ξ, which is called the structure Jacobi operator R_ξ = R( · , ξ)ξ ∈ End(TM). On the other hand, in [20], 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_ξ 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.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.747-761
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• 2010

It is known that there are no real hypersurfaces with parallel structure Jacobi operator $R_{\xi}$ (cf.[16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator $R_{\xi}$ which are weaker than ${\nabla}R_{\xi}$ = 0. Under further condition $S\phi={\phi}S$ for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.525-535
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• 2019

In relation to the generalized Tanaka-Webster connection, we consider a new notion of parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and prove the non-existence of real hypersurfaces in $G_2({\mathbb{C}}^{m+2})$ with generalized Tanaka-Webster parallel structure Jacobi operator.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.211-219
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• 2009

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies a certain cyclic condition.

• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.255-278
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• 2022

In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface M in complex hyperbolic two-plane Grassmannians SU_2,m/S (U₂·U_m). Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians SU_2,m/S (U₂·U_m) with generalized Killing structure Jacobi operator.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.849-861
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• 2023

In this paper, we introduce the notion of semi-symmetric structure Jacobi operator for Hopf real hypersufaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. Next we prove that there does not exist any Hopf real hypersurface in the complex quadric Q^m = SO_m+2/SO_mSO₂ 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.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.573-589
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• 2006

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ in a non flat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}$ is ${\xi}$-parallel and the Ricci tensor S commutes with the structure operator $\phi$, then a real hypersurface in $M_n(c)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_n(c)$.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.447-461
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• 2009

In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.