• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.337-346
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• 2000

The geodesic equation in a two-dimensional Finsler space is given by the differential equation of the Weierstrass form. In the present paper, we express the differential equations of geodesics in a two-dimensional Finsler space with a generalized Kropina metric.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.209-221
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• 2008

In the present paper, first we define a Douglas space of the second kind of a Finsler space with an (${\alpha},{\beta}$)-metric. Next we find the conditions that the Finsler space with an (${\alpha},{\beta}$)-metric be a Douglas space of the second kind and the Finsler space with a Matsumoto metric be a Douglas space of the second kind.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2003.09a
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• pp.5.2-5
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• 2003

In this paper, we are to find the condition that a two-dimensional Finsler space with Matsumoto metric satisfying L(${\alpha}$,${\beta}$)=${\alpha}$$^2$/(${\alpha}$-${\beta}$) be a Landsberg space and the differential equations of geodesics.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.425-441
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• 2006

We have two concepts of Douglas spaces and Lands-berg spaces as generalizations of Berwald spaces. S. Bacso gave the definition of a weakly-Berwald space [2] as another generalization of Berwald spaces. In the present paper, we find the conditions that the Finsler space with an (${\alpha},{\beta}$)-metric be a weakly-Berwald space and the Finsler spaces with some special (${\alpha},{\beta}$)-metrics be weakly-Berwald spaces, respectively.

• Journal of the Korean Mathematical Society
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• v.41 no.3
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• pp.567-589
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• 2004

In the present paper, we treat an infinite series ($\alpha$, $\beta$)-metric L =$\beta$$^2$/($\beta$-$\alpha$). First, we find the conditions that a Finsler metric F$^{n}$ with the metric above be a Berwald space, a Douglas space, and a projectively flat Finsler space, respectively. Next, we investigate the condition that a two-dimensional Finsler space with the metric above be a Landsbeg space. Then the differential equations of the geodesics are also discussed.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.407-418
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• 2019

In this paper, we study conformal transformations between special class of Finsler metrics named C-reducible metrics. This class includes Randers metrics in the form $F={\alpha}+{\beta}$ and Kropina metric in the form $F={\frac{{\alpha}^2}{\beta}}$. We prove that every conformal transformation between locally dually flat Randers metrics must be homothetic and also every conformal transformation between locally dually flat Kropina metrics must be homothetic.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1293-1307
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• 2017

General (${\alpha},{\beta}$)-metrics form a rich class of Finsler metrics. They include many important Finsler metrics, such as Randers metrics, square metrics and spherically symmetric metrics. In this paper, we find equations which are necessary and sufficient conditions for such Finsler metric to be locally projectively flat. By solving these equations, we obtain all of locally projectively flat general (${\alpha},{\beta}$)-metrics under certain condition. Finally, we manufacture explicitly new locally projectively flat Finsler metrics.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1485-1502
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• 2014

The explicit formula for the hyperbolic metric ${\lambda}_{{\alpha},{\beta},{\gamma}}(z){\mid}dz{\mid}$ on the thrice-punctured sphere $\mathbb{P}{\backslash}\{0,1,{\infty}\}$ with singularities of order 0 < ${\alpha}$, ${\beta}$ < 1, ${\gamma}{\leq}1$, ${\alpha}+{\beta}+{\gamma}$ > 2 at 0, 1, ${\infty}$ was given by Kraus, Roth and Sugawa in [9]. In this article we investigate the asymptotic properties of the higher order derivatives of ${\lambda}_{{\alpha},{\beta},{\gamma}}(z)$ near the origin and give more precise descriptions for the asymptotic behavior of ${\lambda}_{{\alpha},{\beta},{\gamma}}(z)$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1457-1469
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• 2016

In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat (${\alpha}$, ${\beta}$)-metric where the underline Riemannian metric ${\alpha}$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge-$Amp{\acute{e}}re$ type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.