• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1315-1327
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• 2011

Let M be a real hypersurface of a complex space form with almost contact metric structure (${\phi}$, ${\xi}$, ${\eta}$, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},\;{\xi}){\xi}$ is ${\xi}$-parallel. In particular, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when $R_{\xi}{\phi}S=R_{\xi}S{\phi}$ holds on M, where S denotes the Ricci tensor of type (1,1) on M.

• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.131-136
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• 1995

We give another detailed proof of the existence of a holomorphic structure on a smooth complex vector bundle over a complex manifold.

• 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.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.371-386
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• 1998

Let G be a compact Lie group and M a semialgebraic G space in some orthogonal representation space of G. We prove that if G is finite then M has an equivariant semialgebraic triangulation. Moreover this triangulation is unique. When G is not finite we show that M has a semialgebraic G CW complex structure, and this structure is unique. As a consequence compact semialgebraic G space has an equivariant simple homotopy type.

• 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.

• Journal of the Korea Safety Management & Science
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• v.14 no.2
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• pp.49-54
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• 2012

As all kind of industry has developed, metal structure and machine instrument use bolt, pin, rivet and welding for assembly and combination. For pin and hinge, dimension accuracy is crucial to keep the operation and safety of the structure and machine instrument. In case of complex machine, the hinge for cover open-loop system is one of the significant design elements. Most of the hinges are being imported and assembled sine they give high technology development cost for its unit cost position. The reason is that the localization of hinge is inadequate. As the demand increase and the necessity of localization grow, it is now more important than ever to develop low cost structure. By the low cost structure, a new technology could be obtained for electronic product and structural hinge since it would enable for complex machine hinge to be guaranteed, technologically. Open-loop hinge is the link type and designed for the structure to keep constant open-loop. And, the hinge is examined in design stability by finite element analysis method. In this paper, the operation result is presented when the hinge for complex machine open-loop is designed for link type structure.

• 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.

• 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.