• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.287-311
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• 2023

In this paper, we introduce the notion of cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex hyperbolic quadric Q^m* = SO⁰_2,m/SO₂SO_m. We give a classifiction of when real hypersurfaces in the complex hyperbolic quadric Q^m* having 𝔄-principal or 𝔄-isotropic unit normal vector fields have cyclic structure Jacobi semi-symmetric tensor.

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.683-699
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• 2017

We introduce the notion of Killing shape operator for real hypersurfaces in the complex hyperbolic quadric $Q^{m*}=SO_{m,2}/SO_mSO_2$. The Killing shape operator implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^{m*}=SO_{m,2}/SO_mSO_2$ with Killing shape operator.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.551-570
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• 2020

We introduce the notion of Lie invariant normal Jacobi operators for real hypersurfaces in the complex hyperbolic quadric Q^m∗ = SO^o_m,2/SO_mSO₂. The invariant normal Jacobi operator implies that the unit normal vector field N becomes 𝕬-principal or 𝕬-isotropic. Then in each case, we give a complete classification of real hypersurfaces in Q^m∗ = SO^o_m,2/SO_mSO₂ with Lie invariant normal Jacobi operators.

• Honam Mathematical Journal
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• v.28 no.1
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• pp.83-111
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• 2006

After introducing Tropical Algebraic Varieties, we give a polyhedral description of tropical hypersurfaces. Using TOPCOM and GAP, we show that there exist 59 types of two dimensional tropical quadric surfaces. We also show a criterion for a quadric hypersurface to be non-degenerate in terms of a tropical rank.

• 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.

• East Asian mathematical journal
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• v.33 no.5
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• pp.571-585
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• 2017

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigen vectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we showed that a 2-type surface M in $E^3$ satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle},{\rangle}$ is the usual inner product in $E^3$, then M is an open part of a circular cylinder. Also we showed that if a quadric hypersurface M in a Euclidean space satisfies ${\langle}{\Delta}x,x{\rangle}=c$ for a constant c, then it is one of a minimal quadric hypersurface, a genaralized cone, a hypersphere, and a spherical cylinder.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.429-437
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• 1994

Let $M^n$ be a connected hypersurface in Euclidean (n + 1)-space $E^{n+1}$, and let $G : G^n \longrightarrow S^n(1) \subset E^{n+1}$ be its Gauss map.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.2
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• pp.181-188
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• 2021

Let X be a nondegenerate reduced closed subscheme in ℙⁿ. Assume that π_q : X → Y = π_q(X) ⊂ ℙ^n-1 is a generic projection from the center q ∈ Sec(X) \ X where Sec(X) = ℙⁿ. Let Z be the singular locus of the projection π_q(X) ⊂ ℙ^n-1. Suppose that I_X has the almost minimal presentation, which is of the form R(-3)^β_2,1 ⊕ R(-4) → R(-2)^β_1,1 → I_X → 0. In this paper, we prove the followings: (a) Z is either a linear space or a quadric hypersurface in a linear subspace; (b) $H^1({\mathcal{I}_X(k)})=H^1({\mathcal{I}_Y(k)})$ for all k ∈ ℤ; (c) reg(Y) ≤ max{reg(X), 4}; (d) Y is cut out by at most quartic hypersurfaces.