• 대한수학회지
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• 제45권4호
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• pp.1113-1133
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• 2008

In this paper we give some non-existence theorems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$ with Lie ${\xi}$-parallel normal Jacobi operator $\bar{R}_N$ and another geometric conditions.

• 대한수학회보
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• 제52권1호
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• pp.57-68
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• 2015

In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, $m{\geq}3$, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection $\hat{\nabla}^{(k)}$.

• 대한수학회보
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• 제46권3호
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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.

• 대한수학회보
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• 제54권3호
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• pp.975-992
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• 2017

In this article, we consider a real hypersurface of complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$, $m{\geq}3$, admitting commuting ${\ast}$-Ricci and pseudo anti-commuting ${\ast}$-Ricci tensor, respectively. As the applications, we prove that there do not exist ${\ast}$-Einstein metrics on Hopf hypersurfaces as well as ${\ast}$-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

• 대한수학회보
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• 제45권3호
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• pp.495-507
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• 2008

In this paper we give a complete classification of real hyper-surfaces in complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$ with commuting structure Jacobi operator $R_{\xi}$ and another geometric condition.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.327-338
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• 2021

In this paper, we have introduced a new notion of recurrent structure Jacobi of real hypersurfaces in complex hyperbolic two-plane Grassmannians G^*₂(ℂ^m+2). Next, we show a non-existence property of real hypersurfaces in G^*₂(ℂ^m+2) satisfying such a curvature condition.

• 대한수학회보
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• 제54권5호
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• pp.1527-1575
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• 2017

In a spirit of Givental's constructions Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested Landau-Ginzburg models for smooth Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections in complex tori equipped with special functions called superpotentials. We provide a particular algorithm for constructing birational isomorphisms of these models for complete intersections in Grassmannians of planes with complex tori. In this case the superpotentials are given by Laurent polynomials. We study Givental's integrals for Landau-Ginzburg models suggested by Batyrev, Ciocan-Fontanine, Kim, and van Straten and show that they are periods for pencils of fibers of maps provided by Laurent polynomials we obtain. The algorithm we provide after minor modifications can be applied in a more general context.

• 대한수학회지
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• 제44권1호
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• pp.211-235
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• 2007

In this paper, first we introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grass-mannians $G_2(\mathbb{C}^{m+2})$ from the equation of Gauss and derive a new formula for the Ricci tensor of M in $G_2(\mathbb{C}^{m+2})$. Next we prove that there do not exist any Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ with parallel and commuting Ricci tensor. Finally we show that there do not exist any Einstein Hopf hypersurfaces in $G_2(\mathbb{C}^{m+2})$.

• 대한수학회지
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• 제59권2호
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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.

• Kyungpook Mathematical Journal
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• 제59권3호
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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.