• Title/Summary/Keyword: Landsberg manifold

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GENERALIZED LANDSBERG MANIFOLDS OF SCALAR CURVATURE

  • Aurel Bejancu;Farran, Hani-Reda
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.543-550
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    • 2000
  • We prove that every generalized Landsberg manifold of scalar curvature R is a Riemannian manifold of constant curvature, provided that $R\neq\ 0$.

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AN INTRINSIC PROOF OF NUMATA'S THEOREM ON LANDSBERG SPACES

  • Salah Gomaa Elgendi;Amr Soleiman
    • Journal of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.149-160
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    • 2024
  • In this paper, we study the unicorn's Landsberg problem from an intrinsic point of view. Precisely, we investigate a coordinate-free proof of Numata's theorem on Landsberg spaces of scalar curvature. In other words, following the pullback approach to Finsler geometry, we prove that all Landsberg spaces of dimension n ≥ 3 of non-zero scalar curvature are Riemannian spaces of constant curvature.

ON THE GENERALIZED RANDERS CHANGE OF BERWALD METRICS

  • Lee, Nany
    • Korean Journal of Mathematics
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    • v.18 no.4
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    • pp.387-394
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    • 2010
  • In this paper, we study the generalized Randers change $^*L(x,y)=L(x,y)+b_i(x,y)y^i$ from the Brewald metric L and the h-vector $b_i$. And in search for a non-Berwald Landsberg metric, we obtain the conditions on $b_i(x,y)$ under which $^*L$ is a Landsberg metric.

ON CONFORMAL TRANSFORMATIONS BETWEEN TWO ALMOST REGULAR (α, β)-METRICS

  • Chen, Guangzu;Liu, Lihong
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.1231-1240
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    • 2018
  • In this paper, we characterize the conformal transformations between two almost regular (${\alpha},{\beta}$)-metrics. Suppose that F is a non-Riemannian (${\alpha},{\beta}$)-metric and is conformally related to ${\widetilde{F}}$, that is, ${\widetilde{F}}=e^{{\kappa}(x)}F$, where ${\kappa}:={\kappa}(x)$ is a scalar function on the manifold. We obtain the necessary and sufficient conditions of the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature. Further, when both F and ${\widetilde{F}}$ are regular, the conformal transformation between F and ${\widetilde{F}}$ preserving the mean Landsberg curvature must be a homothety.

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.

INTRINSIC THEORY OF Cv-REDUCIBILITY IN FINSLER GEOMETRY

  • Salah Gomaa Elgendi;Amr Soleiman
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.187-199
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    • 2024
  • In the present paper, following the pullback approach to Finsler geometry, we study intrinsically the Cv-reducible and generalized Cv-reducible Finsler spaces. Precisely, we introduce a coordinate-free formulation of these manifolds. Then, we prove that a Finsler manifold is Cv-reducible if and only if it is C-reducible and satisfies the 𝕋-condition. We study the generalized Cv-reducible Finsler manifold with a scalar π-form 𝔸. We show that a Finsler manifold (M, L) is generalized Cv-reducible with 𝔸 if and only if it is C-reducible and 𝕋 = 𝔸. Moreover, we prove that a Landsberg generalized Cv-reducible Finsler manifold with a scalar π-form 𝔸 is Berwaldian. Finally, we consider a special Cv-reducible Finsler manifold and conclude that a Finsler manifold is a special Cv-reducible if and only if it is special semi-C-reducible with vanishing 𝕋-tensor.