• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.869-881
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• 2011

The main goal in the present paper is to obtain a necessary and sufficient condition for a new connection with zero torsion vector to be an Einstein's connection and derive some useful representation of the vector defining the Einstein's connection in even-dimensional UFT $X_n$.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.167-174
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• 2013

The main goal in the present paper is to obtain a solution of Einstein's unified field equations for the third class in $X_4$.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.133-145
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• 2006

We study Riemannian or pseudo-Riemannian manifolds which admit a closed conformal vector field. Subject to the condition that at each point $p{\in}M^n$ the set of conformal gradient vector fields spans a non-degenerate subspace of TpM, using a warped product structure theorem we give a complete description of the space of conformal vector fields on arbitrary non-Ricci flat Einstein spaces.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.445-453
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• 2021

The purpose of the paper is to study of Para-Kenmotsu metric as a 𝜂-Ricci soliton. The paper is organized as follows: • If an 𝜂-Einstein para-Kenmotsu metric represents an 𝜂-Ricci soliton with flow vector field V, then it is Einstein with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with the flow vector field V being an infinitesimal paracontact transformation, then V is strict and the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with non-zero flow vector field V being collinear with 𝜉, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Finally, we cited few examples to illustrate the results obtained.

• Korean Journal of Mathematics
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• v.31 no.2
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• pp.139-144
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• 2023

In this paper, we show that a recurrent manifold with harmonic curvature tensor is locally symmetric and that an Einstein and conformally recurrent manifold is locally symmetric. As a consequence, Einstein and recurrent manifolds must be locally symmetric. On the other hand, we have obtained some results for a (conformally) recurrent manifold with parallel vector field and also investigated some results for a (conformally) recurrent manifold with concircular vector field.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1315-1325
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• 2019

The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.207-215
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• 2015

The main goal in the present paper is to obtain a particular solution $g_{{\lambda}{\mu}}$, ${\Gamma}^{\nu}_{{\lambda}{\mu}}$ and an algebraic solution $\bar{g}_{{\lambda}{\mu}}$, $\bar{\Gamma}^{\nu}_{{\lambda}{\mu}}$ by means of $g_{{\lambda}{\mu}}$, ${\Gamma}^{\nu}_{{\lambda}{\mu}}$ in UFT $X_4$.

• 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.11 no.4
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• pp.1047-1053
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• 1996

In this paper, we obtain a solution of Einstein's unified field equations on a generalized n-dimensional Riemannian manifold $X_n$.

• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.413-420
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• 2008

The object of the present paper is to study some properties of a quasi Einstein manifold. A non-trivial concrete example of a quasi Einstein manifold is also given.