• 대한수학회논문집
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• 제35권1호
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• pp.269-278
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• 2020

This paper aims to study the geometrical properties of the conharmonic curvature tensor of a locally conformal almost cosymplectic manifold. The necessary and sufficient conditions for the conharmonic curvature tensor to be flat, the locally conformal almost cosymplectic manifold to be normal and an η-Einstein manifold were determined.

• 대한수학회논문집
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• 제35권2호
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• pp.603-611
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• 2020

The purpose of this note is to introduce a type of Riemannian manifold called an almost quasi Ricci symmetric manifold and investigate the several properties of such a manifold on which some geometric conditions are imposed. And the existence of such a manifold is ensured by a proper example.

• 대한수학회논문집
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• 제24권3호
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• pp.415-424
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• 2009

The object of the present paper is to study the generalized quasi-Einstein manifolds satisfying some conditions. Finally the existence of such manifolds is ensured by several interesting examples.

• Korean Journal of Mathematics
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• 제27권4호
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• pp.1133-1147
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• 2019

In this paper, let M = B×_f² F be an Einstein Lorentzian warped product manifold with 2-dimensional base. We study the geodesic completeness of some metric with constant curvature. First of all, we discuss the existence of nonconstant warping functions on M. As the results, we have some metric g admits nonconstant warping functions f. Finally, we consider the geodesic completeness on M.

• 호남수학학술지
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• 제32권1호
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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.

• Korean Journal of Mathematics
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• 제27권1호
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• pp.9-15
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• 2019

In this paper, we study a type of Riemannian manifold, namely quasi Ricci symmetric manifold. Among others, we show that the scalar curvature of a quasi Ricci symmetric manifold is constant. In addition if the manifold is Einstein, then its Ricci tensor is zero. Also we prove that if the associated vector field of a quasi Ricci symmetric manifold is either recurrent or concurrent, then its Ricci tensor is zero.

• 호남수학학술지
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• 제40권3호
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• pp.447-456
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• 2018

In this paper, we consider nonconstant warping functions on Einstein Lorentzian warped product manifolds $M=B{\times}_{f^2}F$ with an 1-dimensional base B which has a negative definite metric. As the results, we discuss that on M the resulting Einstein Lorentzian warped product metric is a future (or past) geodesically complete one outside a compact set.

• 호남수학학술지
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• 제46권1호
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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).

• 대한수학회지
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• 제48권4호
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• pp.669-689
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• 2011

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.281-289
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• 1998

On a generalized Riemannian manifold $X_n$, we may impose a particular geometric structure by the basic tensor field $g_{\lambda\mu}$ by means of a particular connection ${\Gamma}{_\lambda}{^\nu}_{\mu}$. For example, Einstein's manifold $X_n$ is based on the Einstein's connection defined by the Einstein's equations. Many recurrent connections have been studied by many geometers, such as Datta and Singel, M. Matsumoto, and E.M. Patterson. The purpose of the present paper is to study some relations between a generalized semisymmetric $g$-recurrent manifold $GSX_n$ and its submanifold. All considerations in this present paper deal with the general case $n{\geq}2$ and all possible classes.