• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.649-661
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• 2003

The ($\alpha,\;\beta$)-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\beta$. In this paper, we discuss the projective flatness of Finsler spaces with certain ($\alpha,\;\beta$)-metrics ([5]) in a locally Minkowski space.

• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.415-421
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• 1996

In this paper, we shall find the conditions that the Finsler space with a special $(\alpha,\beta)$-metric be a Riemannian space and a Berwald space.

• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.353-362
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• 2019

In this paper, we study general (${\alpha}$, ${\beta}$) Finsler metrics and prove that every general (${\alpha}$, ${\beta}$)-metric is semi C-reducible but not C2-like. As a consequence of this result we prove that every general (${\alpha}$, ${\beta}$)-metric satisfying the Ricci flow equation is Einstein.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.329-352
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• 2019

In this paper, we study conformally flat (${\alpha}$, ${\beta}$)-metrics in the form $F={\alpha}(1+{\sum_{j=1}^{m}}\;a_j({\frac{\beta}{\alpha}})^j)$ with $m{\geq}2$, where ${\alpha}$ is a Riemannian metric and ${\beta}$ is a 1-form on a smooth manifold M. We prove that if such conformally flat (${\alpha}$, ${\beta}$)-metric F is of weakly isotropic scalar curvature, then it must has zero scalar curvature. Moreover, if $a_{m-1}a_m{\neq}0$, then such metric is either locally Minkowskian or Riemannian.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.491-502
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• 2023

As a generalization of Berwald spaces, we have the ideas of Douglas spaces and Landsberg spaces. S. Bacso defined a weakly-Berwald space as another generalization of Berwald spaces. In 1972, Matsumoto proposed the (α, β) metric, which is a Finsler metric derived from a Riemannian metric α and a differential 1-form β. In this paper, we investigated an important class of (α, β)-metrics of the form $F={\mu}_1\alpha+{\mu}_2\beta+{\mu}_3\frac{\beta^2}{\alpha}$, which is recognized as a special form of the first approximate Matsumoto metric on an n-dimensional manifold, and we obtain the criteria for such metrics to be weakly-Berwald metrics. A Finsler space with a special (α, β)-metric is a weakly Berwald space if and only if B^m_m is a 1-form. We have shown that under certain geometric and algebraic circumstances, it transforms into a weakly Berwald space.

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.975-984
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• 1997

In the Finsler space with an $(\alpha, \beta)$-metric, we can consider the difference tensors of the Finsler connection. The properties of the conformal transformation of these difference tensors are investigated in the present paper. Some conformal invariant tensors are formed in the Finsler space with an $(\alpha, \beta)$-metric related with the difference tensors.

• Journal of the Korean Mathematical Society
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• v.37 no.1
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• pp.73-84
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• 2000

The present paper is devoted to studying the condition that a two-dimensional Finsler space with a special (${\alpha}$, ${\beta}$)-metric be a Landsberg space. It is proved that if a Finsler space with a special (${\alpha}$, ${\beta}$)-metric is a Landsberg space, then it is a Berwald space.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.115-128
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• 2014

In this paper, we study the second approximate Matsumoto metric F = ${\alpha}+{\beta}+{\beta}^2/{\alpha}+{\beta}^3/{\alpha}^2$ on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.415-424
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• 2018

General (${\alpha},\;{\beta}$)-metrics form a rich and important class of Finsler metrics. In this paper, we obtain a differential equation which characterizes a general (${\alpha},\;{\beta}$)-metric with isotropic E-curvature, under a certain condition. We also solve the equation in a particular case.

• Journal of applied mathematics & informatics
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• v.41 no.1
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• pp.193-203
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• 2023

In view of solution to the Hilbert fourth problem, the present study engages to investigate the projectively flat special (𝛼, 𝛽)-metric and the generalised first approximate Matsumoto (𝛼, 𝛽)-metric, where 𝛼 is a Riemannian metric and 𝛽 is a differential one-form. Further, we concluded that 𝛼 is locally Projectively flat and have 𝛽 is parallel with respect to 𝛼 for both the metrics. Also, we obtained necessary and sufficient conditions for the aforementioned metrics to be locally projectively flat.