• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.921-934
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• 2019

The object of the present paper is to study generalized pseudo-projectively symmetric manifolds and Einstein generalized pseudo-projectively symmetric manifolds. Finally, the existence of generalized pseudo-projectively symmetric manifolds have been proved by two non-trivial examples.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1053-1068
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• 2021

We investigate the spacelike hypersurface Mⁿ with constant mean curvature (CMC) in a Lorentz Einstein manifold Lⁿ⁺¹₁, which is supposed to obey some appropriate curvature constraints. Applying a suitable Simons type formula jointly with the well known generalized maximum principle of Omori-Yau, we obtain some rigidity classification theorems and pinching theorems of hypersurfaces.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.53-64
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• 2008

Lower dimensional cases of Einstein's connection were already investigated by many authors for n = 2, 3, 4, 5, 6. In the following series of two papers, we present a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor: I. The recurrence relations in 8-g-UFT II. The Einstein 's connection in 8-g-UFT In our previous paper [1], we investigated some algebraic structure in Einstein's 8-dimensional unified field theory (i.e., 8-g-UFT), with emphasis on the derivation of the recurrence relations of the third kind which hold in 8-g-UFT. This paper is a direct continuation of [1]. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 8-g-UFT and to display a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in the first paper [1]. All considerations in this paper are restricted to the first class only of the generalized 8-dimensional Riemannian manifold $X_8$, since the cases of the second class are done in [2], [3] and the case of the third class, the simplest case, was already studied by many authors.

• Honam Mathematical Journal
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• v.27 no.1
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• pp.131-140
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• 2005

Lower dimensional cases of Einstein's connection were already investigated by many authors for n = 2, 3, 4, 5, 6, 7. In the following series of two papers, we present a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor: I. The recurrence relations in 8-g-UFT II. The Einstein's connection in 8-g-UFT In our previous paper [1], we investigated some algebraic structure in Einstein's 8-dimensional unified field theory (i.e., 8-g-UFT), with emphasis on the derivation of the recurrence relations of the third kind which hold in 8-g-UFT. This paper is a direct continuation of [1]. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 8-g-UFT and to display a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in the first paper [1]. All considerations in this paper are restricted to the second class only of the generalized 8-dimensional Riemannian manifold $X_8$, since the case of the first class are done in [2], [3] and the case of the third class, the simplest case, was already studied by many authors.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.795-808
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• 2012

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds with conservative curvature tensor and also 3-dimensional conformally flat trans-Sasakian manifolds. Next we consider compact connected $\eta$-Einstein 3-dimensional trans-Sasakian manifolds. Finally, an example of a 3-dimensional trans-Sasakian manifold is given, which verifies our results.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1255-1265
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• 2009

We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M, g) of constant radius $\gamma$ reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the $\eta$-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.679-692
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• 2006

On a compact oriented n-dimensional manifold $(M^n,\;g)$, it has been conjectured that a metric g satisfying the critical point equation (2) should be Einstein. In this paper, we prove that if a manifold $(M^4,\;g)$ is a 4-dimensional oriented compact warped product, then g can not be a solution of CPE with a non-zero solution function f.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.149-156
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• 2015

We analyze the $(LCS)_n$-manifolds endowed with some symmetric properties, focusing on Ricci tensor and the 1-form ${\gamma}$. We study some properties of special Weakly Ricci-Symmetric $(LCS)_n$-manifolds and also shown that Weakly ${\phi}$-Ricci Symmetric $(LCS)_n$-manifold is an ${\eta}$-Einstein manifold.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.597-607
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• 2021

The present paper deals with the study of Fischer-Marsden conjecture on a Kenmotsu manifold. It is proved that if a Kenmotsu metric satisfies 𝔏^*_g(λ) = 0 on a (2n + 1)-dimensional Kenmotsu manifold M²ⁿ⁺¹, then either ξλ = -λ or M²ⁿ⁺¹ is Einstein. If n = 1, M³ is locally isometric to the hyperbolic space H³ (-1).

• Honam Mathematical Journal
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• v.37 no.4
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• pp.457-468
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• 2015

The aim of present paper is to investigate 3-dimensional ${\xi}$-projectively flt and $\tilde{\varphi}$-projectively flt normal almost paracontact metric manifolds. As a first step, we proved that if the 3-dimensional normal almost paracontact metric manifold is ${\xi}$-projectively flt then ${\Delta}{\beta}=0$. If additionally ${\beta}$ is constant then the manifold is ${\beta}$-para-Sasakian. Later, we proved that a 3-dimensional normal almost paracontact metric manifold is $\tilde{\varphi}$-projectively flt if and only if it is an Einstein manifold for ${\alpha},{\beta}=const$. Finally, we constructed an example to illustrate the results obtained in previous sections.