• Title/Summary/Keyword: special $(alpha,beta)$-metric

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ON THE BERWALD CONNECTION OF A FINSLER SPACE WITH A SPECIAL $({\alpha},{\beta})$-METRIC

  • Park, Hong-Suh;Park, Ha-Yong;Kim, Byung-Doo
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.355-364
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    • 1997
  • In a Finsler space, we introduce a special $(\alpha,\beta)$-metric L satisfying $L^2(\alpha,\beta) = c_1\alpha^2 + 2c_2\alpha\beta + c_3\beta^2$, which $c_i$ are constants. We investigate the Berwald connection in a Finsler space with this special $\alpha,\beta)$-metric.

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ON TWO-DIMENSIONAL LANDSBERG SPACE WITH A SPECIAL (${\alpha},\;{\beta}$)-METRIC

  • Lee, Il-Yong
    • The Pure and Applied Mathematics
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    • v.10 no.4
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    • pp.279-288
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    • 2003
  • In the present paper, we treat a Finsler space with a special (${\alpha},\;{\beta}$)-metric $L({\alpha},\;{\beta})\;\;C_1{\alpha}+C_2{\beta}+{\alpha}^2/{\beta}$ satisfying some conditions. We find a condition that a Finsler space with a special (${\alpha},\;{\beta}$)-metric be a Berwald space. Then it is shown that if a two-dimensional Finsler space with a special (${\alpha},\;{\beta}$)-metric is a Landsberg space, then it is a Berwald space.

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FINSLER SPACES WITH CERTAIN ($\alpha$,$\beta$)-METRIC OF DOUGLAS TYPE

  • Park, Hong-Suh;Lee, Yong-Duk
    • Communications of the Korean Mathematical Society
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    • v.16 no.4
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    • pp.649-658
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    • 2001
  • We shall find the condition for a Finsler space with a special ($\alpha$.$\beta$)-metric L($\alpha$.$\beta$) satisfying L$^2$ =2$\alpha$$\beta$ to be a Douglas space. The special Randers change of the above Finsler metric by $\beta$ is also studied.

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A Finsler space with a special metric function

  • Park, Hong-Suh;Lee, Il-Young
    • 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.

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On the projectively flat finsler space with a special $(alpha,beta)$-metric

  • Kim, Byung-Doo
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.407-413
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    • 1996
  • The $(\alpha, \beta)$-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\Beta$; it has been sometimes treat in theoretical physics. In particular, the projective flatness of Finsler space with a metric $L^2 = 2\alpha\beta$ is considered in detail.

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WEAKLY BERWALD SPACE WITH A SPECIAL (α, β)-METRIC

  • PRADEEP KUMAR;AJAYKUMAR AR
    • 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 Bmm is a 1-form. We have shown that under certain geometric and algebraic circumstances, it transforms into a weakly Berwald space.

ON THE LANDSBERG SPACES OF DIMENSION TWO WITH A SPECIAL ($\alpha$, $\beta$)-METRIC

  • Park, Hong-Suh;Lee, Il-Yong
    • 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.

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ON PROJECTIVELY FLAT FINSLER SPACES WITH $({\alpha},{\beta})$-METRIC

  • Park, Hong-Suh;Lee, Il-Yong
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.373-383
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    • 1999
  • The ($\alpha$,$\beta$)-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-from $\beta$;it has been sometimes treated in theoretical physics. The condition for a Finsler space with an ($\alpha$,$\beta$)-metric L($\alpha$,$\beta$) to be projectively flat was given by Matsumoto [11]. The present paper is devoted to studying the condition for a Finsler space with L=$\alpha$\ulcorner$\beta$\ulcorner or L=$\alpha$+$\beta$\ulcorner/$\alpha$ to be projectively flat on the basis of Matsumoto`s results.

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GEOMETRY OF LOCALLY PROJECTIVELY FLAT FINSLER SPACE WITH CERTAIN (𝛼, 𝛽)-METRIC

  • AJAYKUMAR ABBANIRAMAKRISHNAPPA;PRADEEP KUMAR
    • 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.

ON WEAKLY-BERWALD SPACES OF SPECIAL (α, β)-METRICS

  • Lee, Il-Yong;Lee, Myung-Han
    • 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.