• 대한수학회논문집
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• 제15권4호
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• pp.641-648
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• 2000

The purpose of this paper is to give a necessary and sufficient condition for a smooth manifold to admit a Lorentzian metric. As an application of this result, on Lorentzian manifolds we have shown that the existence of a 1-dimensional distribution is equivalent to the existence of a non-vanishing vector field.

• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.191-203
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• 2021

Let M be an almost Kenmotsu manifold of dimension 2n + 1 having non-vanishing ��-sectional curvature such that trℓ > -2n - 2. We prove that any second order parallel tensor on M is a constant multiple of the associated metric tensor and obtained some consequences of this. Vector fields keeping curvature tensor invariant are characterized on M.

• 대한수학회보
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• 제29권1호
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• pp.73-80
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• 1992

We discuss transverse harmonic fields on compact foliated Riemannian manifolds, and give a necessary and sufficient condition for a transverse field to be a transverse harmonic one and the non-existence of transverse harmonic fields. 1. On a foliated Riemannian manifold, geometric transverse fields, that is, transverse Killing, affine, projective, conformal fields were discussed by Kamber and Tondeur([3]), Molino ([5], [6]), Pak and Yorozu ([7]) and others. If the foliation is one by points, then transverse fields are usual fields on Riemannian manifolds. Thus it is natural to extend well known results concerning those fields on Riemannian manifolds to foliated cases. On the other hand, the following theorem is well known ([1], [10]): If the Ricci operator in a compact Riemannian manifold M is non-negative everywhere, then a harmonic vector field in M has a vanishing covariant derivative. If the Ricci operator in M is positive-definite, then a harmonic vector field other than zero does not exist in M.

• 호남수학학술지
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• 제45권2호
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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.

• 대한수학회지
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• 제56권6호
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• pp.1475-1488
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• 2019

In this paper, we firstly define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton $(M^n,g)(n{\geq}3)$ is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

• 대한수학회보
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• 제40권3호
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• pp.457-464
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• 2003

Given a Riemannian manifold (M, $\alpha$) with an almost Hermitian structure f and a non-vanishing covariant vector field b, consider the generalized Randers metric $L\;=\;{\alpha}+{\beta}$, where $\beta$ is a special singular Riemannian metric defined by b and f. This metric L is called an (a, b, f)-metric. We compute the inverse and the determinant of the fundamental tensor ($g_{ij}$) of an (a, b, f)-metric. Then we determine the maximal domain D of $TM{\backslash}O$ for an (a, b, f)-manifold where a y-local Finsler structure L is defined. And then we show that any (a, b, f)-manifold is quasi-C-reducible and find a condition under which an (a, b, f)-manifold is C-reducible.

• Kyungpook Mathematical Journal
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• 제58권2호
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• pp.347-359
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• 2018

A Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is called a generalized ${\phi}-recurrent$ if its curvature tensor R satisfies $${\phi}^2(({\nabla}_wR)(X,Y)Z)=A(W)R(X,Y)Z+B(W)G(X,Y)Z$$ for all $X,\;Y,\;Z,\;W{\in}{\chi}(M)$, where ${\nabla}$ denotes the operator of covariant differentiation with respect to the metric g, i.e. ${\nabla}$ is the Riemannian connection, A, B are non-vanishing 1-forms and G is given by G(X, Y)Z = g(Y, Z)X - g(X, Z)Y. In particular, if A = 0 = B then the manifold is called a ${\phi}-symmetric$. Now, a Kenmotsu manifold $M^n({\phi},\;{\xi},\;{\eta},\;g)$, (n = 2m + 1 > 3) is said to be generalized ${\phi}-Ricci$ recurrent if it satisfies $${\phi}^2(({\nabla}_wQ)(Y))=A(X)QY+B(X)Y$$ for any vector field $X,\;Y{\in}{\chi}(M)$, where Q is the Ricci operator, i.e., g(QX, Y) = S(X, Y) for all X, Y. In this paper, we study generalized ${\phi}-recurrent$ and generalized ${\phi}-Ricci$ recurrent Kenmotsu manifolds with respect to quarter-symmetric metric connection and obtain a necessary and sufficient condition of a generalized ${\phi}-recurrent$ Kenmotsu manifold with respect to quarter symmetric metric connection to be generalized Ricci recurrent Kenmotsu manifold with respect to quarter symmetric metric connection.