• Bulletin of the Korean Mathematical Society
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• v.37 no.1
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• pp.85-90
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• 2000

On Riemannian manifolds of dimension 4 the Einstein condition is equivalent to the Thorpe condition. In this paper, we construct a few metrics which we Einstein but not Thorpe, and vice versa in dimensions larger than 4.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.167-176
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• 2010

Let (B, $\check{g}$) and (N, $\hat{g}$) be Einstein manifolds. Then, we get a complete (necessary and sufficient) condition for the warped product manifold $B\;{\times}_f\;N\;:=\;(B\;{\times}\;N,\;\check{g}\;+\;f{\hat{g}}$) to be Einstein, and obtain a complete condition for the Einstein warped product manifold $B\;{\times}_f\;N$ to be weakly stable. Moreover, we get a complete condition for the map i : ($B,\;\check{g})\;{\times}\;(N,\;\hat{g})\;{\rightarrow}\;B\;{\times}_f\;N$, which is the identity map as a map, to be harmonic. Under the assumption that i is harmonic, we obtain a complete condition for $B\;{\times}_f\;N$ to be Einstein.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.315-336
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• 1999

In the paper [12] we have introduced the new kind of pseudo-einstein ruled real hypersurfaces in complex space forms $M_n(c), c\neq0$, which are foliated by pseudo-Einstein leaves. The purpose of this paper is to give a geometric condition for non-proper pseudo-Einstein ruled real hypersurfaces to be totally geodesic in the sense of Kimura [8] for c> and Ahn, Lee and the present author [1] for c<0.

• 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 $X_n$.

• Korean Journal of Mathematics
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• v.18 no.1
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• pp.21-28
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• 2010

In the unified field theory(UFT), many works on the solutions of Einstein's equation have been published. The main goal in the present paper is to obtain some geometric results on a particular solution of Einstein's equation under some condition in even-dimensional UFT $X_n$.

• Korean Journal of Mathematics
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• v.25 no.1
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• pp.127-135
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• 2017

The manifold $^*g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{{\lambda}{\nu}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 5-dimensional $^*g-ESX_5$ and to display a surveyable tnesorial representation of 5-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.313-321
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• 2015

The manifold $^*g-ESX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^*g^{{\lambda}{\nu}}$ through the ES-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 3-dimensional $^*g-ESX_3$ and to display a surveyable tnesorial representation of 3-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.1-11
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• 2005

The purpose of the present paper is to introduce a new concept of the E($^*{\kappa}$)-connection ${\Gamma}^{\nu}_{{\lambda}{\mu}}$, which is both Einstein and ($^*{\kappa}$)-connection, and to obtain a necessary and sufficient condition for the existence of the unique E($^*{\kappa}$)-connection in $n-^*g$-UFT. Next, under this condition, we shall obtain a surveyable tensorial representation of the unique E($^*{\kappa}$)-connection in $n-^*g$-UFT.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.641-648
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• 2000

It has been conjectured that, on a compact 3-dimensional manifold, a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics of volume 1 is Einstein. In this paper we find a sufficient condition that a critical point is Einstein. This condition is equivalent for a critical point ot be conformally flat. Its relationship with the Fisher-Marsden conjecture is also discussed.