• Bulletin of the Korean Mathematical Society
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• v.34 no.3
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• pp.403-409
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• 1997

Let X be a closed almost complex 4-manifold with $b_2^+(X) > 1$, and have its canonical line bundle as a basic class. Then the pseudo-holomorphic 2-dimensional submanifolds in X with nonnegative self-intersection minimize genus in their homology classes.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.51-59
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• 2005

Let (M, $g$) be a compact oriented self-dual 4-dimensional Einstein manifold with positive sectional curvature. Then we show that, up to rescaling and isometry, (M, $g$) is $S^4$ or $\mathbb{C}\mathbb{P}_2$, with their cannonical metrics.

• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.61-66
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• 1984

It is well-known that the most interesting non-integrable almost Hermitian manifold are the nearly Kaehlerian manifolds ([2] and [3]), and that there exists a complex but not a Kaehlerian structure on Riemannian product manifolds of two normal contact manifolds [4]. The purpose of the present paper is to study nearly Kaehlerian product manifolds of two almost contact metric manifolds and investigate the geometrical structures of these manifolds. Unless otherwise stated, we shall always assume that manifolds and quantities are differentiable of class $C^{\infty}$. In Paragraph 1, we give brief discussions of almost contact metric manifolds and their Riemannian product manifolds. In paragraph 2, we investigate the perfect conditions for Riemannian product manifolds of two almost contact metric manifolds to be nearly Kaehlerian and the non-existence of a nearly Kaehlerian product manifold of contact metric manifolds. Paragraph 3 will be devoted to a proof of the following; A conformally flat compact nearly Kaehlerian product manifold of two almost contact metric manifolds is isomatric to a Riemannian product manifold of a complex projective space and a flat Kaehlerian manifold..

• Journal of Advanced Marine Engineering and Technology
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• v.37 no.7
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• pp.694-700
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• 2013

The design of the exhaust manifold for the pulse converters of a 4 strokes high power medium-speed diesel engine is presented in terms of fatigue analysis. The said system undergoes thermal expansion due to high temperature of exhaust gas and is exposed to intrinsic vibration of the internal combustion engine. Moreover, the exhaust pulse generates pressure pulsating along the runner inside manifold. Under such circumstances, the design and construction of exhaust manifold must be carried out in a way to prevent early failure due to fracture. To validate the design concept, a test rig was developed to simulate the combination of thermal and vibrational movements, simultaneously. Experimental results showed that a certain sense of reliability can be achieved by considering a field factor obtained from the results of engine bench tests.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.817-827
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• 2021

In this paper, we prove that a diffeomorphism f on a normal almost contact 3-manifold M is a CRL-transformation if and only if M is an α-Sasakian manifold. Moreover, we show that a CR-loxodrome in an α-Sasakian 3-manifold is a pseudo-Hermitian magnetic curve with a strength $q={\tilde{r}}{\eta}({\gamma}^{\prime})=(r+{\alpha}-t){\eta}({\gamma}^{\prime})$ for constant 𝜂(𝛄'). A non-geodesic CR-loxodrome is a non-Legendre slant helix. Next, we prove that let M be an α-Sasakian 3-manifold such that (∇_YS)X = 0 for vector fields Y to be orthogonal to ξ, then the Ricci tensor 𝜌 satisfies 𝜌 = 2α²g. Moreover, using the CRL-transformation $\tilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\tilde{R}$, the pseudo-Ricci tensor $\tilde{\rho}$ and the torsion tensor field $\tilde{T}^t(\tilde{S}X,Y)$.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1279-1287
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• 2019

The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient ${\rho}$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient ${\rho}$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost ${\eta}$-Ricci soliton.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.639-651
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• 2020

Consider a gradient Einstein-type metric in the setting of K-contact manifolds and (κ, µ)-contact manifolds. First, it is proved that, if a complete K-contact manifold admits a gradient Einstein-type metric, then M is compact, Einstein, Sasakian and isometric to the unit sphere 𝕊²ⁿ⁺¹. Next, it is proved that, if a non-Sasakian (κ, µ)-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension 3, and for higher dimension, M is locally isometric to the product of a Euclidean space 𝔼ⁿ⁺¹ and a sphere 𝕊ⁿ(4) of constant curvature +4.

• Honam Mathematical Journal
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• v.41 no.4
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• pp.669-682
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• 2019

The objective of this paper is to introduce and study Einstein-like para-Kenmotsu manifolds. For a para-Kenmotsu manifold to be Einstein-like, a necessary and sufficient condition in terms of its curvature tensor is obtained. We also obtain the scalar curvature of an Einstein-like para-Kenmotsu manifold. A necessary and sufficient condition for an almost para-contact metric hypersurface of a locally product Riemannian manifold to be para-Kenmotsu is derived and it is shown that the para-Kenmotsu hypersurface of a locally product Riemannian manifold of almost constant curvature is always Einstein.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1101-1114
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• 2016

Let ($M^{2n+1}$, ${\phi}$, ${\xi}$, ${\eta}$, g) be a (k, ${\mu}$)'-almost Kenmotsu manifold with k < -1 which admits a gradient Ricci almost soliton (g, f, ${\lambda}$), where ${\lambda}$ is the soliton function and f is the potential function. In this paper, it is proved that ${\lambda}$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton ${\mathbb{H}}^{n+1}(-4){\times}{\mathbb{R}}^n$, and the soliton is expanding with ${\lambda}=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature -1 or the potential vector field is pointwise colinear with the Reeb vector field.