• 대한수학회보
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• 제48권1호
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• pp.129-140
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• 2011

The aim of this paper is to study the structure of the fundamental group of a closed oriented Riemannian manifold with positive scalar curvature. To be more precise, let M be a closed oriented Riemannian manifold of dimension n (4 $\leq$ n $\leq$ 7) with positive scalar curvature and non-trivial first Betti number, and let be $\alpha$ non-trivial codimension one homology class in $H_{n-1}$(M;$\mathbb{R}$). Then it is known as in [8] that there exists a closed embedded hypersurface $N_{\alpha}$ of M representing $\alpha$ of minimum volume, compared with all other closed hypersurfaces in the homology class. Our main result is to show that the fundamental group ${\pi}_1(N_{\alpha})$ is always virtually free. In particular, this gives rise to a new obstruction to the existence of a metric of positive scalar curvature.

• 대한수학회지
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• 제56권2호
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• pp.329-352
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• 2019

In this paper, we study conformally flat (${\alpha}$, ${\beta}$)-metrics in the form $F={\alpha}(1+{\sum_{j=1}^{m}}\;a_j({\frac{\beta}{\alpha}})^j)$ with $m{\geq}2$, where ${\alpha}$ is a Riemannian metric and ${\beta}$ is a 1-form on a smooth manifold M. We prove that if such conformally flat (${\alpha}$, ${\beta}$)-metric F is of weakly isotropic scalar curvature, then it must has zero scalar curvature. Moreover, if $a_{m-1}a_m{\neq}0$, then such metric is either locally Minkowskian or Riemannian.

• 호남수학학술지
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• 제33권4호
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• pp.487-493
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• 2011

The Fisher information matrix plays a significant role in statistical inference in connection with estimation and properties of variance of estimators. In this paper, we define the parameter space of the t-manifold using its Fisher's matrix and characterize the t-manifold from the viewpoint of information geometry. The ${\alpha}$-scalar curvatures to the t-manifold are calculated.

• Journal of the Korean Data and Information Science Society
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• 제21권3호
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• pp.575-580
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• 2010

The Fisher information matrix plays a significant role in statistical inference in connection with estimation and properties of variance of estimators. In this paper, the parameter space of the t-distribution using its Fisher's matrix is de ned. The ${\alpha}$-scalar curvatures to parameter space are calculated.

• 대한수학회보
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• 제51권1호
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• pp.115-128
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• 2014

In this paper, we study the second approximate Matsumoto metric F = ${\alpha}+{\beta}+{\beta}^2/{\alpha}+{\beta}^3/{\alpha}^2$ on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

• 호남수학학술지
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• 제32권1호
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• pp.85-89
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• 2010

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations on space-times M = $({\alpha},{\infty}){\times}_fN$ with prescribed scalar curvature functions.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제6권2호
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• pp.95-101
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• 1999

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future complete Lorentzian metrics on $M{\;}={\;}[\alpha,\infty){\times}_f{\;}N$ with specific scalar curvatures.

• 통합자연과학논문집
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• 제5권1호
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• pp.42-45
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• 2012

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations on Riemannian warped product manifold $M=({\alpha},\;{\infty}){\times}_fN$ with prescribed scalar curvature functions.

• 대한수학회보
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• 제55권4호
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• pp.1231-1240
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• 2018

In this paper, we characterize the conformal transformations between two almost regular (${\alpha},{\beta}$)-metrics. Suppose that F is a non-Riemannian (${\alpha},{\beta}$)-metric and is conformally related to ${\widetilde{F}}$, that is, ${\widetilde{F}}=e^{{\kappa}(x)}F$, where ${\kappa}:={\kappa}(x)$ is a scalar function on the manifold. We obtain the necessary and sufficient conditions of the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature. Further, when both F and ${\widetilde{F}}$ are regular, the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature must be a homothety.