• 대한수학회지
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• 제44권4호
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• pp.941-947
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• 2007

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of nonpositive scalar curvature. Let ${\mu}0$ be the least eigenvalue of the Laplacian acting on $L^2-functions$ on M. We show that if $Ric^M{\ge}-{\mu}0$ at all $x{\in}M$ and either $Ric^M>-{\mu}0$ at some point x0 or Vol(M) is infinite, then every harmonic morphism ${\phi}:M{\to}N$ of finite energy is constant.

• 대한설비공학회지:설비저널
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• 제17권4호
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• pp.426-434
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• 1988

Series expansion is applied to solve the laminar boundary layer equations for the problem of natural convection from vertical cylinder with uniform surface heat flux. The series in terms of transverse curvature parameter ${\xi}$ is extended to five terms and is well converged by applying the Shanks transform twice. In case of natural convection from a vertical cylinder heated with uniform surface heat flux, it is possible to consider the vertical cylinder as vertical plate under the condition of D/L${\geq}$A/$(Gr_L^*)^{1/5}$, where A is in the range of 5.7~55.2. Also, mean Nusselt number ${\overline{Nu_L}}$ can be represented as $C_1(Ra_L^*)^{1/5}$, where $C_1$ is a constant which depends on Pr and is in the range of 0.5~0.8.

• 대한수학회논문집
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• 제36권4호
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• pp.817-827
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• 2021

In this paper, we prove that a diffeomorphism f on a normal almost contact 3-manifold M is a CRL-transformation if and only if M is an α-Sasakian manifold. Moreover, we show that a CR-loxodrome in an α-Sasakian 3-manifold is a pseudo-Hermitian magnetic curve with a strength $q={\tilde{r}}{\eta}({\gamma}^{\prime})=(r+{\alpha}-t){\eta}({\gamma}^{\prime})$ for constant 𝜂(𝛄'). A non-geodesic CR-loxodrome is a non-Legendre slant helix. Next, we prove that let M be an α-Sasakian 3-manifold such that (∇_YS)X = 0 for vector fields Y to be orthogonal to ξ, then the Ricci tensor 𝜌 satisfies 𝜌 = 2α²g. Moreover, using the CRL-transformation $\tilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\tilde{R}$, the pseudo-Ricci tensor $\tilde{\rho}$ and the torsion tensor field $\tilde{T}^t(\tilde{S}X,Y)$.

• 대한수학회보
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• 제56권5호
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• pp.1219-1233
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• 2019

Let $E{\rightarrow}M$ be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection $D^E$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when $D^E$ is flat. We study also the Einstein property on E proving, among other results, that if $k{\geq}2$ and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.

• 대한수학회보
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• 제56권5호
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• pp.1315-1325
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• 2019

The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.

• 천문학회보
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• 제44권1호
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• pp.56.1-56.1
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• 2019

The concordance LCDM model has been very successful at reproducing a wide range of observations. However, the nature of its main components, such as dark energy, dark matter, and inflation, are still unkown. Therefore, it is of prime importance to question the underlying hypotheses of the model and tests there prediction. While most constraints have been obtained assuming a LCDM universe, model-independent approaches, which do not make assumptions regarding the model, are a powerful approach. To falsify the LCDM model, I applied model-indepedent methods to the latests available data to test different aspects of the concordance model, such as the FLRW metric, the curvature, dark energy as the cosmological constant, and gravity as general relativity. The Universe is consistent with flat-LCDM with GR. However, at z>1, tensions start to appear, and more data are required.

• 충청수학회지
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• 제34권1호
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• pp.1-15
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• 2021

In the recent papers, S'anchez-Reyes [Appl. Math. Model. 40 (2016), 1676-1682] described the method for finding a minimal surface through a geodesic, and Li et al. [Appl. Math. Model. 37 (2013), 6415-6424] studied the approximation of minimal surfaces with a geodesic from Dirichlet function. In the present article, we consider an isoparametric surface generated by Frenet frame of a curve introduced by Wang et al. [Comput. Aided Des. 36 (2004), 447-459], and give the necessary and sufficient condition to satisfy both geodesic of the curve and minimality of the surface. From this, we construct minimal surfaces in terms of constant curvature and torsion of the curve. As a result, we present a new approach for constructions of the minimal surfaces from a prescribed closed geodesic and unclosed geodesic, and show some new examples of minimal surfaces with a circle and a helix as a geodesic. Our approach can be used in design of minimal surfaces from geodesics.

• 대한수학회보
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• 제59권6호
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• pp.1567-1594
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• 2022

In this article, we first introduce the notion of commuting Ricci tensor and pseudo-anti commuting Ricci tensor for Hopf hypersurfaces in the homogeneous nearly Kähler 𝕊³ × 𝕊³ and prove that the mean curvature of hypersurface is constant under certain assumptions. Next, we prove the nonexistence of Ricci soliton on Hopf hypersurface with potential Reeb vector field, which improves a result of Hu et al. on the nonexistence of Einstein Hopf hypersurfaces in the homogeneous nearly Kähler 𝕊³ × 𝕊³.

• 호남수학학술지
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• 제45권2호
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• pp.325-339
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• 2023

The object of the present work is to study a generalized W₃ recurrent manifold. We obtain a necessary and sufficient condition for the scalar curvature to be constant in such a manifold. Also, sufficient condition for generalized W₃ recurrent manifold to be special quasi-Einstein manifold are given. Ricci symmetric and decomposable generalized W₃ recurrent manifold are studied. Finally, the existence of such a manifold is ensured by a non-trivial example.

• 대한수학회논문집
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• 제38권4호
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• pp.1271-1280
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• 2023

The object of the present paper is to introduce a type of non-flat Riemannian manifold, called a weakly cyclic generalized B-symmetric manifold (W CGBS)_n. We obtain a sufficient condition for a weakly cyclic generalized B-symmetric manifold to be a generalized quasi Einstein manifold. Next we consider conformally flat weakly cyclic generalized B-symmetric manifolds. Then we study Einstein (W CGBS)_n (n > 2). Finally, it is shown that the semi-symmetry and Weyl semi-symmetry are equivalent in such a manifold.