• 대한수학회지
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• 제57권6호
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• pp.1435-1449
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• 2020

In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

• 대한수학회보
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• 제45권2호
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• pp.321-338
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• 2008

Let $\bar{M}$ be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures $\{L^{\tau}\}_{{\tau}{\in}I}$, $0{\in}I$, which extend smoothly up to bM, the boundary of M, and assume that there is ${\lambda}{\in}C^{\infty}$(bM) which is strictly subharmonic with respect to the structure $L^0|_{bM}$ in any direction where the Levi-form vanishes on bM. We obtain explicit estimates for the $\bar{\partial}$-Neumann problem in Sobolev spaces both in space and parameter variables. Also we get a similar result when $\bar{M}$ is strongly pseudoconvex.

• 대한수학회보
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• 제46권2호
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• pp.255-261
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• 2009

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

• 한국수학교육학회지시리즈D:수학교육연구
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• 제20권1호
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• pp.39-49
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• 2016

The History of Mathematics (HOM) was integrated in teaching the Cartesian Coordinate Plane (CCP) to Grade Seven learners of Moonwalk National High School using Lesson Study. After the lesson was taught, there were three valuable issues emerged: (1) HOM is a Springboard and/or a Medium of Motivation in Teaching CCP; (2) The History of CCP Opened a Wider Perspective about Its Real-life Application in the Modern World (3) Integration of History Developed a Sense of Purpose and an Appreciation of Mathematics Among Learners. Feedbacks solicited from the learners showed that they have understanding of the importance of studying Mathematics after they learned the life and contributions of Rene Descartes to Mathematics. Hence, integration of history plays a vital role in developing positive attitudes among learners towards Math.

• 대한수학회지
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• 제36권5호
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• pp.959-1008
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• 1999

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

• 대한수학회보
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• 제47권5호
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• pp.1089-1103
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• 2010

In this paper, we study lightlike submanifolds of a semi-Riemannian manifold admitting a semi-symmetric non-metric connection. We obtain a necessary and a sufficient condition for integrability of the screen distribution. Then we give the conditions under which the Ricci tensor of a lightlike submanifold with a semi-symmetric non-metric connection is symmetric. Finally, we show that the Ricci tensor of a lightlike submanifold of semi-Riemannian space form is not parallel with respect to the semi-symmetric non-metric connection.

• 대한수학회지
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• 제40권3호
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• pp.461-486
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• 2003

It is known that the CR geometries of Levi non-degen-erate hypersurfaces in $\C^n$ and of the elliptic or hyperbolic CR submanifolds of codimension two in $\C^4$ share many common features. In this paper, a special class of normalized coordinates is introduced for any CR manifold M which is one of the above three kinds and it is shown that the explicit expression in these coordinates of an isotropy automorphism $f{\in}Aut(M)_o {\subset}Aut(M),\;o{\in}M$, is equal to the expression of a corresponding element of the automorphism group of the homogeneous model. As an application of this property, an extension theorem for CR maps is obtained.

• 대한수학회지
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• 제50권3호
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• pp.479-491
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• 2013

Let M be a smooth real hypersurface in complex space of dimension $n$, $n{\geq}3$, and assume that the Levi-form at $z_0$ on M has at least $(q+1)$-positive eigenvalues, $1{\leq}q{\leq}n-2$. We estimate solutions of the local $\bar{\partial}$-closed extension problem near $z_0$ for $(p,q)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

• 한국의류학회지
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• 제20권3호
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• pp.493-501
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• 1996

An understanding of consumers' evaluative processes and the consumer variables that influence cross-cultural evaluations of foreign products is essential to effectively penetrate international markets. This study compared consumers in two developing countries, Korea and Mexico, regarding their product evaluations and purchase intentions for US Levi's jeans. Both Korean and Mexican consumers are generally quality and price conscious. Korean consumers appeared to be more value-conscious than Mexican consumers. Mexican consumers exhibited high level of brand and prestige conscious behavior. Findings in this study support that culture bound consumer characteristics vary from country to country resulting in differences in perception of quality, value and purchase intentions for a particular product. The perception of product quality differs from perception of value and subsequent willingness to buy. For a comprehensive understanding of consumer purchase behavior, the impact of value-conscious attitude on percetion of quality and on purchase intentions should be examined.

• 호남수학학술지
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• 제42권3호
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• pp.521-536
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• 2020

The object of the present paper is to study of Almost Quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons on an Lorentzian concircular structure manifolds briefly say (LCS)_n-manifolds under infinitesimal CL-transformations and obtained sufficient conditions for such solitons to be expanding, steady and shrinking. Also we obtained a necessary and sufficient condition of an almost quasi-Yamabe soliton with respect to the CL-connection to be an almost quasi-Yamabe soliton on (LCS)_n-manifolds with respect to Levi-Civita connection. Finally, we construct an example of steady almost quasi-Yamabe soliton on 3-dimensional (LCS)_n-manifolds.