• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1101-1114
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• 1996

In Riemannian geometry, it is one of the important things to study the topological or geometric invariants of manifolds since they characterize the topology of manifolds.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.863-879
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• 2019

In this paper we study the Myers' theorem for orientable Riemannian hypersurfaces under some restrictions on the mean curvature, the second-order mean curvature, or the divergence of the shape operator and give some estimates for the diameter of such hypersurfaces.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.273-285
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• 2005

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

• International Journal of Naval Architecture and Ocean Engineering
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• v.4 no.3
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• pp.291-301
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• 2012

A scalar metric for the assessment of hull surface producibility was known to be useful in estimating the complexity of a hull form of ships or large offshore structures by looking at their shape. However, it could not serve as a comprehensive measuring tool due to its lack of important components of the hull form such as longitudinals, stiffeners, and web frames attached to the hull surface. To have a complete metric for cost estimation, these structural members must be included. In this paper, major inner structural members are considered by measuring the complexity of their geometric shape. The final scalar metric thus consists of the classes containing inner members with various curvature magnitudes as well as the classes containing curved plates with single and double curvature distribution. Those two distinct metrics are merged into a complete scalar metric that accounts for the total cost estimation of complex structural bodies.

• Kyungpook Mathematical Journal
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• v.45 no.1
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• pp.55-65
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• 2005

We find the expression of the curvature tensor field for a manifold with is endowed with an almost contact structure satisfying the condition (1.7). By using this condition we obtain some properties of the Ricci tensor and scalar curvature (d. Theorem 3.2 and Proposition 3.2).

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.537-557
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• 2021

In this paper, we investigate the curvature behavior of complete noncompact gradient expanding Ricci solitons with nonnegative Ricci curvature. For such a soliton in dimension four, it is shown that the Riemann curvature tensor and its covariant derivatives are bounded. Moreover, the Ricci curvature is controlled by the scalar curvature. In higher dimensions, we prove that the Riemann curvature tensor grows at most polynomially in the distance function.

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.897-906
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• 2024

On a Hermitian manifold, the Chern connection can induce a metric connection on the background Riemannian manifold. We call the sectional curvature of the metric connection induced by the Chern connection the Chern sectional curvature of this Hermitian manifold. First, we derive expression of the Chern sectional curvature in local complex coordinates. As an application, we find that a Hermitian metric is Kähler if the Riemann sectional curvature and the Chern sectional curvature coincide. As subsequent results, Ricci curvature and scalar curvature of the metric connection induced by the Chern connection are obtained.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.547-556
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• 2011

In this paper, we estimated the Ricci curvature and the scalar curvature on SU(3)/T (k, l) under the condition (k, l) ${\in}\mathbb{R}^2$ (${\mid}k{\mid}+{\mid}l{\mid}{\neq}0$), where the four isotropy irreducible representations in SU(3)/T (k, l) are, not necessarily, mutually equivalent or inequivalent.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.775-779
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• 2005

On a compact n-dimensional manifold $M^n$, a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satifies the critical point equation (CPE), given by $Z_g\;=\;s_g^{1\ast}(f)$. It has been conjectured that a solution (g, f) of CPE is Einstein. The purpose of the present paper is to prove that every compact stable minimal hypersurface is in a certain hypersurface of $M^n$ under an assumption that Ker($s_g^{1\ast}{\neq}0$).

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.787-799
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• 2013

In this paper, we compute the sectional curvatures and the scalar curvature of the Siegel-Jacobi space $\mathb{H}_1{\times}\mathb{C}$ of degree 1 and index 1 explicitly.