• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1595-1603
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• 2022

We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.67-83
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• 2014

The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and $n_2$ of the curve, [1]. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifying curves in $\mathbb{R}^3_1$. Moreover, by the aid of these characterizations, we have investigated semi real quaternionic rectifying curves in semi-quaternionic space $\mathbb{Q}_v$.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.523-535
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• 2015

We study surfaces in Euclidean space which are obtained as the sum of two curves or that are graphs of the product of two functions. We consider the problem of finding all these surfaces with constant Gauss curvature. We extend the results to non-degenerate surfaces in Lorentz-Minkowski space.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.999-1006
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• 1997

We proved that if a fibred Riemannian space $\tilde{M}$ with cosymplectic structure is conformally flat, then $\tilde{M}$ is the locally product manifold of locally Euclidean spaces, that is locally Euclidean. Moreover, we investigated the fibred Riemannian space with cosymplectic structure when the Riemannian metric $\tilde{g}$ on $\tilde{M}$ is Einstein.

• Honam Mathematical Journal
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• v.44 no.2
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• pp.219-230
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• 2022

We examine the bi-rotational hypersurface x = x(u, v, w) with the third Laplace-Beltrami operator in the four dimensional Euclidean space 𝔼⁴. Giving the i-th curvatures of the hypersurface x, we obtain the third Laplace-Beltrami operator of the bi-rotational hypersurface satisfying ∆^IIIx =𝒜x for some 4 × 4 matrix 𝒜.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.593-611
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• 2016

This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

• Journal of the Korean Mathematical Society
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• v.41 no.2
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• pp.379-396
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• 2004

In this article, we study rotation surfaces in the 4-dimensional pseudo-Euclidean space E$_2$$^4$. Also, we obtain the complete classification theorems for the flat rotation surfaces with finite type Gauss map, pointwise 1-type Gauss map and an equation in terms of the mean curvature vector. In fact, we characterize the flat rotation surfaces of finite type immersion with the Gauss map and the mean curvature vector field, namely the Gauss map of finite type, pointwise 1-type Gauss map and some algebraic equations in terms of the Gauss map and the mean curvature vector field related to the Laplacian of the surfaces with respect to the induced metric.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.419-431
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• 2022

Mannheim introduced the concept of a pair of curves, called as Mannheim partner curves, in 1878. Until now, Mannheim partner curves have been studied widely in the literature. In this study, we take into account of this concept according to Positional Adapted Frame (PAF) for the particles moving in the 3-dimensional Euclidean space. We introduce a new type special trajectory pairs which are called Mannheim partner P-trajectories in the Euclidean 3-space. The relationships between the PAF elements of this pair are investigated. Also, the relations between the Serret-Frenet basis vectors of Mannheim partner P-trajectories are given. Afterwards, we obtain the necessary conditions for one of these trajectories to be an osculating curve and for other to be a rectifying curve. Moreover, we provide an example including an illustrative figure.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.310-324
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• 2022

In this paper, we define timelike curves in R⁴₂ and characterize such curves in terms of Frenet frame. Also, we examine the timelike helices of R⁴₂, taking into account their curvatures. In addition, we study timelike slant helices, timelike B₁-slant helices, timelike B₂-slant helices in four dimensional semi-Euclidean space, R⁴₂. And then we obtain an approximate solution for the timelike B₁ slant helix with Taylor matrix collocation method.

• Journal of Astronomy and Space Sciences
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• v.19 no.1
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• pp.1-6
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• 2002

Recently, one interesting possibility is proposed that a magnetar can be a progenitor of short and hard gamma-ray bursts (GRBs). If this is true, one may expect that the short and hard GRBs, at least some of GRBs in this class, are distributed in the Euclidean space and that the angular position of these GRBs is correlated with galaxy clusters. Even though it is reported that the correlation is statistically marginal, the observed value of < $V/V_{max}$ > deviates from the Euclidean value. The latter fact is often used as evidence against a local extragalactic origin for short GRB class. We demonstrate that GRB sample of which the value of < $V/V_{max}$ > deviates from the Euclidean value can be spatially confined within the low value of z. We select very short bursts (TgO < 0.3 sec) from the BATSE 4B catalog. The value of < $V/V_{max}$ > of the short bursts is 0.4459. Considering a conic-beam and a cylindrical beam for the luminosity function, we deduce the corresponding spatial distribution of the GRB sources. We also calculate the fraction of bursts whose redshifts are larger than a certain redshift z', i.e. f>z'. We find that GRBs may be distributed near to us, despite the non-Euclidean value of < $V/V_{max}$ >. A broad and uniform beam pattern seems compatible with the magnetar model in that the magnetar model requires a small $z_{max}$.