• Journal of the Korean Mathematical Society
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• v.45 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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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 $\hat{\nabla}^{(k)}$.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.489-501
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• 2016

In this paper, we study the rigidity under $K{\ddot{a}}hler$ deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $Gr_{\omega}$(n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

• 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 Korean Mathematical Society
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• v.52 no.5
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• pp.1109-1122
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• 2015

A Schubert variety in a rational homogeneous variety G/P is defined by the closure of an orbit of a Borel subgroup B of G. In general, Schubert varieties are singular, and it is an old problem to determine which Schubert varieties are smooth. In this paper, we classify all smooth Schubert varieties in the symplectic Grassmannians.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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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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• v.39 no.3_4
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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.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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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.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.13-23
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• 2015

We introduce a decomposition on a symplectic subspace determined by symplectic structure and study its properties. As a consequence, we give an elementary proof of the deformation of the Grassmannians of symplectic subspaces to the complex Grassmannians.