• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.191-203
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• 2021

Let M be an almost Kenmotsu manifold of dimension 2n + 1 having non-vanishing ��-sectional curvature such that trℓ > -2n - 2. We prove that any second order parallel tensor on M is a constant multiple of the associated metric tensor and obtained some consequences of this. Vector fields keeping curvature tensor invariant are characterized on M.

• Kyungpook Mathematical Journal
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• 제62권4호
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• pp.751-767
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• 2022

This paper deals with the Paneitz-Branson operator in compact Riemannian manifolds with negative Yamabe invariant. We start off by providing a new criterion for the positivity of the Paneitz-Branson operator when the Yamabe invariant of the manifold is negative. Another result stated in this paper is about the existence of a metric on a manifold of dimension 5 such that the Paneitz-Branson operator has multiple negative eigenvalues. Finally, we provide new inequalities related to the upper bound of the mean value of the Q-curvature.

• 대한수학회보
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• 제54권6호
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• pp.2001-2011
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• 2017

In 1997, H. Li [12] proposed a conjecture: if $M^n(n{\geqslant}3)$ is a complete spacelike hypersurface in de Sitter space $S^{n+1}_1(1)$ with constant normalized scalar curvature R satisfying $\frac{n-2}{n}{\leqslant}R{\leqslant}1$, then is $M^n$ totally umbilical? Recently, F. E. C. Camargo et al. ([5]) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^n$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^n$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ satisfies $K(M^n)$ > 0, then $M^n$ is totally umbilical, i.e., Theorem 2.

• Journal of Welding and Joining
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• 제22권3호
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• pp.45-49
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• 2004

Generally, it is used the compensation spring to compensate the inaccuracy of screen image induced by thermal deformation in CRT monitor. Its mechanism is hi-metallic system made of heterogeneous metals and these springs are bonded by laser welding. But laser welding induces a non-uniform temperature distribution, and residual stress is yielded locally by these temperature deviation. Therefore, this study measures the curvature constant to assess functions of the compensation spring of shadow mask with respect to increment temperature and estimates the effect of residual stress on the performance of tri-metal used to compensation spring.

• 대한수학회논문집
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• 제37권3호
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• pp.825-837
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• 2022

We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric P-connection. At first, it is proven that if the metric of such a manifold is a gradient m-quasi-Einstein metric, then either the gradient of the potential function 𝜓 is collinear with the vector field P or, λ = -(m + 2) and the manifold is of constant sectional curvature -1, provided P𝜓 ≠ m. Next, it is shown that if the metric of the manifold under consideration is a gradient 𝜌-Einstein soliton, then the gradient of the potential function is collinear with the vector field P. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric P-connection is a gradient 𝜔-Ricci soliton, then the manifold is of constant sectional curvature -1 and λ + 𝜇 = -2. Finally, we consider an example to verify our results.

• 대한수학회보
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• 제60권1호
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• pp.171-184
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• 2023

In this paper, we show that a complete translating soliton Σ^m in ℝⁿ for the mean curvature flow is stable with respect to weighted volume functional if Σ satisfies that the L^m norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of Σ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial f-harmonic 1-form of L²_f on Σ. With the additional assumption that Σ is contained in an upper half-space with respect to the translating direction then it has only one end.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.303-311
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• 1998

We study the conformally flat unit vector bundle $E_1$ of constant scalar curvature for the bundle ${\pi}:E^{n+2}{\rightarrow}M^n$ over an Einstein manifold M.

• 대한수학회논문집
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• 제27권4호
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• pp.771-780
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• 2012

We study biminimal curves in 2-dimensional Riemannian manifolds of constant curvature.

• 대한수학회지
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• 제47권5호
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• pp.967-983
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• 2010

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-{\kappa}^2$. Using the cone total curvature TC($\Gamma$) of a graph $\Gamma$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma\;\subset\;M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3, this density estimate implies that if $TC(\Gamma)$ < $3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})$, then the only possible singularities of a piecewise smooth (M, 0, $\delta$)-minimizing set $\Sigma$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.

• 호남수학학술지
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• 제4권1호
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• pp.75-84
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• 1982

Let M be a complete and orientable hypersurface of an odd-dimensional sphere $S^{2n+1}$ with quasi-integrable $(f,\;g,\;u,\;{\nu},\;{\lambda})$ -structure. The purpose of the present paper is to prove the following two theorems. (I) If the scalar curvature of M is constant and the function $\lambda$ is not locally constant, then M is a great sphere $S^{2n}$(1) or a product of two spheres with the same dimension $S^{n}(1/\sqrt{2}){\times}S^{n}(1/\sqrt{2})$. (II) Suppose that the sectional curvature of the section $\gamma(u,\;{\nu})$ spanned by u and $\nu$ is constant on M and M is compact. If the second fundamental tensor H of M is positive semi-definite and satisfies trace $$^{t}HH{\leq_-}{2n}$$, then M is a great sphere $S^{2n}$ (1) or a product of two spheres $S^{n}{\times}S^{n}$ or $S^{p}{\times}S^{2n-p}$, p being odd.