• 대한수학회논문집
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• 제36권1호
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• pp.165-171
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• 2021

The purpose of this paper is to investigate the geometry of complete gradient Yamabe soliton (Mn, g, f, λ) with constant scalar curvature admitting a non-homothetic conformal vector field V leaving the potential vector field invariant. We show that in such manifolds the potential function f is constant and the scalar curvature of g is determined by its soliton scalar. Considering the locally conformally flat case and conformal vector field V, without constant scalar curvature assumption, we show that g has constant curvature and determines the potential function f explicitly.

• 대한수학회논문집
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• 제25권2호
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• pp.245-249
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• 2010

We show that on a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ (dim $M^{2n}\;=\;2n\;{\geq}\;4$), $M^{2n}$ is K$\ddot{a}$hler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of $M^{2n}$ everywhere. As a consequence, if a compact locally conformal K$\ddot{a}$hler manifold $M^{2n}$ is both conformally flat and scalar flat, then $M^{2n}$ is K$\ddot{a}$hler. In contrast with the compact case, we show that there exists a locally conformal K$\ddot{a}$hler manifold with k equal to s, which is not K$\ddot{a}$hler.

• 대한수학회논문집
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• 제13권3호
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• pp.539-544
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• 1998

Let (M, g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvature S, which is close to O. With conditions on a conformal invariant and scalar curvature of (M, g), we show that there exists a conformal metric (equation omitted), near g, whose scalar curvature (equation omitted) = 0 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_{i}$ with ∪$K_{i}$ = M.

• Kyungpook Mathematical Journal
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• 제51권1호
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• pp.109-124
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• 2011

The main objective of the paper is to provide a full classification of quasi-conformally recurrent Riemannian manifolds with harmonic quasi-conformal curvature tensor. Among others it is shown that a quasi-conformally recurrent manifold with harmonic quasi-conformal curvature tensor is any one of the following: (i) quasi-conformally symmetric, (ii) conformally flat, (iii) manifold of constant curvature, (iv) vanishing scalar curvature, (v) Ricci recurrent.

• 대한수학회논문집
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• 제12권1호
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• pp.101-108
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• 1997

We let (M,g) be a noncompact complete Riemannina manifold of dimension $n \geq 3$ with finite volume and positive scalar curvature. We show the existence of a conformal metric with constant positive scalar curvature on (M,g) by gluing solutions of Yamabe equation on each compact subsets $K_i$ with $\cup K_i = M$ .

• 호남수학학술지
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• 제33권1호
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• pp.121-127
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• 2011

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

• 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.

• 대한수학회보
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• 제35권2호
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• pp.195-201
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• 1998

We let (M,g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvatue S, which is close to -1. We show the existence of a conformal metric $\bar{g}$, near to g, whose scalar curvature $\bar{S}$ = -1 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_i$ with ${\bigcup}K_i$ = M.

• 대한수학회논문집
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• 제37권3호
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• pp.865-879
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• 2022

Let (M^m, g) be an m-dimensional Riemannian manifold. In this paper, we introduce a new class of metric on (M^m, g), obtained by a non-conformal deformation of the metric g. First we investigate the Levi-Civita connection of this metric. Secondly we characterize the Riemannian curvature, the sectional curvature and the scalar curvature. In the last section we characterizes some class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when (M^m, g) is an Euclidean space.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제4권1호
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• pp.27-33
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• 1997

Let (M = B$\times$f F, g) be an ($n \geq3$ )-dimensional differential manifold with Riemannian metric g. We solve the following elliptic nonlinear partial differential equation (equation omitted). where $\Delta_{g}$ is the Laplacian in the $\Delta$g-metric and ($h(\chi)$) is the scalar curvature of g and ($H(\chi)$) is a function on M.