• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.217-224
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• 1999

In this paper the properties of the Finsler metrics compatible with an f-structure are investigated.

• 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 C^v-reducible and generalized C^v-reducible Finsler spaces. Precisely, we introduce a coordinate-free formulation of these manifolds. Then, we prove that a Finsler manifold is C^v-reducible if and only if it is C-reducible and satisfies the 𝕋-condition. We study the generalized C^v-reducible Finsler manifold with a scalar π-form 𝔸. We show that a Finsler manifold (M, L) is generalized C^v-reducible with 𝔸 if and only if it is C-reducible and 𝕋 = 𝔸. Moreover, we prove that a Landsberg generalized C^v-reducible Finsler manifold with a scalar π-form 𝔸 is Berwaldian. Finally, we consider a special C^v-reducible Finsler manifold and conclude that a Finsler manifold is a special C^v-reducible if and only if it is special semi-C-reducible with vanishing 𝕋-tensor.

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.551-560
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• 2009

A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated $Poincar{\acute{e}}$ Recurrence Theorem is pointed out.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1359-1380
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• 2018

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.