• 제어로봇시스템학회:학술대회논문집
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• 2004.08a
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• pp.1623-1628
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• 2004

In the 3-dimensional cyclic Lotka-Volterra equations, we show the solution on the invariant hyperplane. In addition, we show the existence of the invariant hyperplane by the center manifold theorem under the some conditions. With this result, we can lead the hyperplane of the n-dimensional cyclic Lotka-Volterra equaions. In other section, we study the 3- or 4-dimensional Hamiltonian Lotka-Volterra equations which satisfy the Jacobi identity. We analyze the solution of the Hamiltonian Lotka- Volterra equations with the functions called the split Liapunov functions by [4], [5] since they provide the Liapunov functions for each region separated by the invariant hyperplane. In the cyclic Lotka-Volterra equations, the role of the Liapunov functions is the same in the odd and even dimension. However, in the Hamiltonian Lotka-Volterra equations, we can show the difference of the role of the Liapunov function between the odd and the even dimension by the numerical calculation. In this paper, we regard the invariant hyperplane as the important item to analyze the motion of Lotka-Volterra equations and occur the chaotic orbit. Furtheremore, an example of the asymptoticaly stable and stable solution of the 3-dimensional cyclic Lotka-Volterra equations, 3- and 4-dimensional Hamiltonian equations are shown.

• Honam Mathematical Journal
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• v.43 no.4
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• pp.655-665
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• 2021

In the present paper, we study the geometric properties of the invariant submanifold of an almost Kenmotsu structure whose Riemannian curvature tensor has (𝜅, 𝜇, 𝜈)-nullity distribution. In this connection, the necessary and sufficient conditions are investigated for an invariant submanifold of an almost Kenmotsu (𝜅, 𝜇, 𝜈)-space to be totally geodesic under the behavior of functions 𝜅, 𝜇, and 𝜈.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.635-645
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• 2002

Two examples of $\varepsilon$-famed manifolds are constructed. It is proved that an $\varepsilon$-framed structure on a manifold is not unique. Automorphism groups of r-framed manifolds are studied. Lastly we prove that a connected Lie group G admits a left invariant normal $\varepsilon$-framed structure if and only if the Lie algebra of all left invariant vector fields on G is an $\varepsilon$-framed Lie algebra.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.255-261
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• 2009

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

• 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.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.435-445
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• 2005

The purpose of this paper is to study a Kenmotsu manifold which is derived from the almost contact Riemannian manifold with some special conditions. In general, we have some relations about semi-symmetric, Ricci semi-symmetric or Weyl semisymmetric conditions in Riemannian manifolds. In this paper, we partially classify the Kenmotsu manifold and consider the manifold admitting a transformation which keeps Riemannian curvature tensor and Ricci tensor invariant.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.741-751
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• 2001

In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1｜{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.793-805
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• 2017

In this paper, we prove that the Ricci tensor of a three-dimensional almost Kenmotsu manifold satisfying ${\nabla}_{\xi}h=0$, $h{\neq}0$, is ${\eta}$-parallel if and only if the manifold is locally isometric to either the Riemannian product $\mathbb{H}^2(-4){\times}\mathbb{R}$ or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.691-701
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• 2006

In this paper, we construct a new exotic smooth 4-manifold X which is homeomorphic, but not diffeomorphic, to ${\mathbb{C}}\mathbb{P}^2{\sharp}13\overline{\mathbb{C}\mathbb{P}}^2$. Moreover the manifold X has vanishing Seiberg-Witten invariants for all $Spin^c$-structures of X and has no symplectic structure.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.549-554
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• 2014

We present a smooth simply connected closed eight dimensional manifold with distinct symplectic deformation equivalence classes [[${\omega}_i$]], i = 1, 2 such that the symplectic Z invariant, which is defined in terms of the scalar curvatures of almost K$\ddot{a}$hler metrics in [5], satisfies $Z(M,[[{\omega}_1]])={\infty}$ and $Z(M,[[{\omega}_2]])$ < 0.