• 대한수학회보
• /
• 제59권6호
• /
• pp.1359-1370
• /
• 2022

In this paper, we study the Finsler warped product metric which is dually flat or projectively flat. The local structures of these metrics are completely determined. Some examples are presented.

• 대한수학회보
• /
• 제60권6호
• /
• pp.1607-1620
• /
• 2023

In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of ℓ^p-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.

• 대한수학회지
• /
• 제51권6호
• /
• pp.1105-1122
• /
• 2014

Biharmonic curves are a generalization of geodesics, with applications in elasticity theory and computer science. The paper proposes a first study of biharmonic curves in spaces with Finslerian geometry, covering the following topics: a deduction of their equations, specific properties and existence of non-geodesic biharmonic curves for some classes of Finsler spaces. Integration of the biharmonic equation is presented for two concrete Finsler metrics.

• 대한수학회지
• /
• 제46권5호
• /
• pp.949-966
• /
• 2009

In this paper we introduce in study a new class of complex Finsler spaces, namely the complex Randers spaces, for which the fundamental metric tensor and the Chern-Finsler connection are determined. A special approach is devoted to $K{\ddot{a}}ahler$-Randers metrics. Using the length arc parametrization for the extremal curves of the Euler-Lagrange equations we obtain a complex nonlinear connections of Lorentz type in a complex Randers space.

• Journal of applied mathematics & informatics
• /
• 제41권1호
• /
• pp.193-203
• /
• 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.

• 대한수학회보
• /
• 제43권2호
• /
• pp.425-441
• /
• 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.

• 대한수학회지
• /
• 제46권3호
• /
• pp.551-560
• /
• 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.

• 대한수학회보
• /
• 제55권1호
• /
• pp.251-266
• /
• 2018

In this paper, considering the class of complex Kropina metrics we obtain the necessary and sufficient conditions that these are generalized Berwald and complex Douglas metrics, respectively. Special attention is devoted to a class of complex Douglas-Kropina spaces, in complex dimension 2. Also, some examples of complex Douglas-Kropina metrics are pointed out. Finally, the complex Douglas-Kropina metrics are characterized through the theory of projectively related complex Finsler metrics.

• 대한수학회보
• /
• 제53권5호
• /
• pp.1457-1469
• /
• 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.

• Kyungpook Mathematical Journal
• /
• 제52권1호
• /
• pp.81-89
• /
• 2012

In the present paper, we nd the conditions to characterize projective change between two (${\alpha}$, ${\beta}$)-metrics, such as Matsumoto metric $L=\frac{{\alpha}^2}{{\alpha}-{\beta}}$ and Randers metric $\bar{L}=\bar{\alpha}+\bar{\beta}$ on a manifold with dim $n$ > 2, where ${\alpha}$ and $\bar{\alpha}$ are two Riemannian metrics, ${\beta}$ and $\bar{\beta}$ are two non-zero 1-formas.