- International Electronic Journal of Geometry
- Vol: 16 Issue: 1
- Some Aspects on a Special Type of $(\\alpha,\\beta )$-metric
Some Aspects on a Special Type of $(\\alpha,\\beta )$-metric
Authors : Laurian-loan Piscoran, Cătălin Barbu
Pages : 295-303
Doi:10.36890/iejg.1265041
View : 6 | Download : 3
Publication Date : 2023-04-30
Article Type : Research
Abstract :The aim of this paper is twofold. Firstly, we will investigate the link between the condition for the functions $\\phi(s)$ from $(\\alpha, \\beta)$-metrics of Douglas type to be self-concordant and k-self concordant, and the other objective of the paper will be to continue to investigate the recently new introduced $(\\alpha, \\beta)$-metric ([17]): $$ F(\\alpha,\\beta)=\\frac{\\beta^{2}}{\\alpha}+\\beta+a \\alpha $$ where $\\alpha=\\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $\\beta=b_{i}y^{i}$ is a 1-form, and $a\\in \\left(\\frac{1}{4},+\\infty\\right)$ is a real positive scalar. This kind of metric can be expressed as follows: $F(\\alpha,\\beta)=\\alpha\\cdot \\phi(s)$, where $\\phi(s)=s^{2}+s+a$. In this paper we will study some important results in respect with the above mentioned $(\\alpha, \\beta)$-metric such as: the Kropina change for this metric, the Main Scalar for this metric and also we will analyze how the condition to be self-concordant and k-self-concordant for the function $\\phi(s)$, can be linked with the condition for the metric $F$ to be of Douglas type. self-concordant functions, Kropina change, main scalar.Keywords : Finsler $(\\alpha¸ \\beta)$-metric, self-concordant functions, main scalar, Kropina change