• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.979-991
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• 2016

The object of the present paper is to study N(k)-quasi-Einstein manifolds. We study an N(k)-quasi-Einstein manifold satisfying the curvature conditions $R({\xi},X){\cdot}Z=0$, $Z(X,{\xi}){\cdot}R=0$, and $P({\xi},X){\cdot}Z=0$, where R, P and Z denote the Riemannian curvature tensor, the projective curvature tensor and Z tensor respectively. Next we prove that the curvature condition $C{\cdot}Z=0$ holds in an N(k)-quasi-Einstein manifold, where C is the conformal curvature tensor. We also study Z-recurrent N(k)-quasi-Einstein manifolds. Finally, we construct an example of an N(k)-quasi-Einstein manifold and mention some physical examples.

• 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.12 no.3
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• pp.685-690
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• 1997

Our study focuses on the condition under which a subspace of complex projective space can become an Einstein space. We prove that a subspace becomes an Einstein space if it's codimension is less than n-1 and its curvature tensor and Ricci tensor satisfies Ryan's condition.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.70-81
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• 2024

We consider the *-general critical equation on LP Sasakian manifolds, and show that such a manifold is generalized η-Einstein. After then, we consider LP Sasakian manifolds with *-conformally semisymmetric condition, and show that such manifolds are *-Einstein. Moreover, we show that the *-conformally semisymmetric LP Sasakian manifold is locally isometric to Eⁿ⁺¹(0) × Sⁿ(4).

• 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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1683-1691
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• 2013

In this paper, we obtain the criteria that the Riemannian manifold B is Einstein or a gradient Ricci soliton from the information of the second derivative of $f$ in the warped product space $R{\times}_fB$ with gradient Ricci solitons. Moreover, we construct new examples of non-Einstein gradient Ricci soliton spaces with an Einstein or non-Einstein gradient Ricci soliton leaf using our main theorems. Finally we also get analogous criteria for the Lorentzian version.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.43-51
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• 2018

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

• Journal of the Korean Mathematical Society
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• v.41 no.5
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• pp.875-882
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• 2004

Let (M,g) be a compact m dimensional Einstein manifold with smooth boundary. Let $\Delta$$_{p}$,B be the realization of the p form valued Laplacian with a suitable boundary condition B. Let Spec($\Delta$$_{p}$,B) be the spectrum where each eigenvalue is repeated according to multiplicity. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.ant.

• 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.