• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.551-560
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• 2009

A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated $Poincar{\acute{e}}$ Recurrence Theorem is pointed out.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.251-263
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• 2012

A CW complex B is said to be I-trivial if there does not exist a $\mathbb{Z}_2$-map from $S^{i-1}$ to S(${\alpha}$) for any vector bundle ${\alpha}$ over B a any integer i with i > dim ${\alpha}$. In this paper, we consider the question of determining whether $\Sigma^k\mathbb{R}P^n$ is I-trivial or not, and to this question we give complete answers when k $\neq$ 1, 3, 8 and partial answers when k = 1, 3, 8. A CW complex B is I-trivial if it is "W-trivial", that is, if for every vector bundle over B, all the Stiefel-Whitney classes vanish. We find, as a result, that $\Sigma^k\mathbb{R}P^n$ is a counterexample to the converse of th statement when k = 2, 4 or 8 and n $\geq$ 2k.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.289-297
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• 2002

It was shown by L. Polterovich ([3]) that if L is a totally real submanifold of a symplectic manifold $(M,\omega)$ and L is parallelizable then L is normal. So we try to find an answer to the question of whether there is a compatible almost complex structure J on the symplectic vector bundle $TM$\mid$_{L}$ such that $TL{\cap}JTL=0$ assuming L is normal and parallelizable. Although we could not reach an answer, we observed that the claim holds at the vector space level. And related to the question, we showed that for a symplectic vector bundle $(M,\omega)$ of rank 2n and $E=E_1{\bigoplus}E_2$, where $E=E_1,E_2$are Lagrangian subbundles of E, there is an almost complex structure J on E compatible with ${\omega}$ and $JE_1=E_2$. And finally we provide a necessary and sufficient condition for a given embedding into a symplectic manifold to be normal.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1219-1233
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• 2019

Let $E{\rightarrow}M$ be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection $D^E$. R. Albuquerque constructed a general class of (two-weights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when $D^E$ is flat. We study also the Einstein property on E proving, among other results, that if $k{\geq}2$ and the base manifold is Einstein with positive constant scalar curvature, then there is a 1-parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat.

• Honam Mathematical Journal
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• v.25 no.1
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• pp.163-172
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• 2003

In this paper we study the property of harmonic vector fields. We call a vector fields ${\xi}$ harmonic if it is a harmonic map from the manifold into its tangent bundle with the Sasaki metric. We show that the characteristic polynomial of operator $A={\nabla}{\xi}\;in\;S^{2n+1}\;is\;(x^2+1)^n$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.319-335
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• 2011

We classify equivariant topoligical complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1443-1482
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• 2008

In this paper we define a Myller configuration in a Finsler space and use some special configurations to obtain results about Finsler subspaces. Let $F^{n}$ = (M,F) be a Finsler space, with M a real, differentiable manifold of dimension n. Using the pull back bundle $({\pi}^{*}TM,\tilde{\pi},\widetilde{TM})$ of the tangent bundle $(TM,{\pi},M)$ by the mapping $\tilde{\pi}={\pi}/TM$ and the Cartan Finsler connection of a Finsler space, we obtain an orthonormal frame of sections of ${\pi}^{*}TM$ along a regular curve in $\widetilde{TM}$ and a system of invariants, geometrically associated to the Myller configuration. The fundamental equations are written in a very simple form and we prove a fundamental theorem. Important lines in a Finsler subspace are defined like special lines in a Myller configuration, geometrically associated to the subspace: auto parallels, lines of curvature, asymptotes. Torse forming vector fields with respect to the Cartan Finsler connection are characterized by means of the invariants of the Frenet frame of a versor field along a curve, and the new notion of torse forming vector fields in the sense of Myller is introduced. The particular cases of concurrence and parallelism in the sense of Myller are completely studied, for vector fields from the distribution $T^m$ of the Myller configuration and also from the normal distribution $T^p$.

• Journal of the Korean Mathematical Society
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• v.27 no.1
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• pp.19-24
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• 1990

• East Asian mathematical journal
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• v.33 no.5
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• pp.533-542
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• 2017

We study the geometry of indefinite trans-Sasakian manifold ${\bar{M}}$ admitting a half lightlike submanifold M such that the structure vector field of ${\bar{M}}$ belongs to the transversal vector bundle of M. We prove several classification theorems of such an indefinite trans-Sasakian manifold.

• Journal of Korea Multimedia Society
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• v.16 no.2
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• pp.160-168
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• 2013

In this paper an algorithm that represents shape sequences with moving frames parallel along the sequences are developed. According to Levi-Civita connection, it is not easy to measure the variation of the vector fields on non-Euclidean spaces without tools to parallel transport them. Thus, parallel transport of the vector fields along the shape sequences is implemented using the theories of principal frame bundle and analyzed via extensive simulation.