Some Aspects on a Special Type of $(alpha,beta )$-metric

Authors : Laurianloan PISCORAN, Cătălin BARBU

Pages : 295-303

Doi:10.36890/iejg.1265041

View : 50 | Download : 6

Publication Date : 2023-04-30

Article Type : Research Paper

Abstract :The aim of this paper is twofold. Firstly, we will investigate the link between the condition for the functions $\\phiinsert ignore into journalissuearticles values(s);$ from $insert ignore into journalissuearticles values(\\alpha, \\beta);$-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 $insert ignore into journalissuearticles values(\\alpha, \\beta);$-metric insert ignore into journalissuearticles values([17]);: $$ Finsert ignore into journalissuearticles values(\\alpha,\\beta);=\\frac{\\beta^{2}}{\\alpha}+\\beta+a \\alpha $$ where $\\alpha=\\sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $\\beta=b_{i}y^{i}$ is a 1-form, and $a\\in \\leftinsert ignore into journalissuearticles values(\\frac{1}{4},+\\infty\\right);$ is a real positive scalar. This kind of metric can be expressed as follows: $Finsert ignore into journalissuearticles values(\\alpha,\\beta);=\\alpha\\cdot \\phiinsert ignore into journalissuearticles values(s);$, where $\\phiinsert ignore into journalissuearticles values(s);=s^{2}+s+a$. In this paper we will study some important results in respect with the above mentioned $insert ignore into journalissuearticles values(\\alpha, \\beta);$-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 $\\phiinsert ignore into journalissuearticles values(s);$, 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

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