• Communications of the Korean Mathematical Society
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• v.27 no.2
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• pp.317-326
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• 2012

We consider $N(k)$-quasi Einstein manifolds satisfying the conditions $C({\xi},\;X).S=0$, $\tilde{C}({\xi},\;X).S=0$, $\bar{P}({\xi},\;X).C=0$, $P({\xi},\;X).\tilde{C}=0$ and $\bar{P}({\xi},\;X).\tilde{C}=0$ where $C$, $\tilde{C}$, $P$ and $\bar{P}$ denote the conformal curvature tensor, the quasi-conformal curvature tensor, the projective curvature tensor and the pseudo projective curvature tensor, respectively.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.149-161
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• 2019

In the present paper, we study curvature properties of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds. We obtain some results of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds satisfying $R({\xi},X).C=0$, $R({\xi},X).{\tilde{M}}=0$, $R({\xi},X).P=0$, $R({\xi},X).{\tilde{C}}=0$ and $R({\xi},X).H=0$, where $C,\;{\tilde{M}},\;P,\;{\tilde{C}}$ and H are a quasi-conformal curvature tensor, a M-projective curvature tensor, a pseudo-projective curvature tensor, and a concircular curvature tensor and conharmonic curvature tensor, respectively.

• Honam Mathematical Journal
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• v.45 no.2
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• pp.316-324
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• 2023

The P-curvature tensor has been studied in the space-time of general relativity and it is found that the contracted part of this tensor vanishes in the Einstein space. It is shown that Rainich conditions for the existence of non-null electro variance can be obtained by P_𝛼𝛽. It is established that the divergence of tensor G_𝛼𝛽 defined with the help of P_𝛼𝛽 and scalar P is zero, so that it can be used to represent Einstein field equations. It is shown that for V₄ satisfying Einstein like field equations, the tensor P_𝛼𝛽 is conserved, if the energy momentum tensor is Codazzi type. The space-time satisfying Einstein's field equations with vanishing of P-curvature tensor have been considered and existence of Killing, conformal Killing vector fields and perfect fluid space-time has been established.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.81-99
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• 2006

We investigate a curvature-like tensor defined by (3.1) in Sasakian manifold of $dimension{\geq}$ 5, and show that this tensor satisfies some properties. Especially, we determine compact Sasakian manifolds with vanishing this tensor and improve some theorems concerning contact conformal curvature tensor and spectrum of Laplacian acting on $p(0{\leq}P{\leq}2)-forms$ on the manifold by using this tensor component.

• 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.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.433-448
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• 2022

The Cartan's second curvature tensor Pⁱ_jkh is a positively homogeneous of degree-1 in yⁱ, where yⁱ represent a directional coordinate for the line element in Finsler space. In this paper, we discuss the decomposition of Cartan's second curvature tensor Pⁱ_jkh in two spaces, a generalized 𝔅P-recurrent space and generalized 𝔅P-birecurrent space. We obtain different tensors which satisfy the recurrence and birecurrence property under the decomposition. Also, we prove the decomposition for different tensors are non-vanishing. As an illustration of the applicability of the obtained results, we finish this work with some illustrative examples.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.795-806
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• 2007

We investigate F-traceless component of the conformal curvature tensor defined by (3.6) in $K\ddot{a}hler$ manifolds of dimension ${\geq}4$, and show that the F-traceless component is invariant under concircular change. In particular, we determine $K\ddot{a}hler$ manifolds with parallel F-traceless component and improve some theorems, provided in the previous paper([2]), which are concerned with the traceless component of the conformal curvature tensor and the spectrum of the Laplacian acting on $p(0{\leq}p{\leq}2)$-forms on the manifold by using the F-traceless component.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.537-562
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• 2019

The aim of the present paper is to study the ${\delta}$-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting ${\eta}$-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ${\phi}$-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying ${\bar{R}}.{\bar{S}}=0$, ${\bar{P}}.{\bar{S}}=0$ is an ${\eta}$-Einstein manifold. Moreover, we obtain the conditions for the ${\delta}$-Lorentzian trans-Sasakian manifolds with a semisymmetric-metric connection to be conformally flat and ${\xi}$-conformally flat.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.341-350
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• 1995

Let (M,g) be a compact manifold of dimension n with metric tensor g. Let $\Delta^p = d\delta + \deltad$ be the Laplace-Beltrami operator acting on the space of smooth p-forms.

• Kyungpook Mathematical Journal
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• v.15 no.1
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• pp.67-71
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• 1975