Projective changes between two Finsler spaces with $(\alpha, \beta )$-metrics

TL;DR

This paper characterizes the conditions under which two specific types of Finsler metrics, $( ext{alpha}, eta)$-metrics and Matsumoto metrics, are projectively related, especially when certain curvature conditions are met.

Contribution

It provides new conditions for projective changes between these two classes of Finsler metrics, extending understanding of their geometric relations.

Findings

Derived conditions for projective equivalence between $( ext{alpha}, eta)$-metrics and Matsumoto metrics.

Analyzed projective changes under special curvature assumptions.

Extended the theory of projective relations in Finsler geometry.

Abstract

In the present paper, we find the conditions to characterize projective change between two $(\alpha, \beta)$-metrics, F = $\alpha + \epsilon \beta + k\frac{\beta^2}{\alpha}$ ($\epsilon$ and k $\neq$ 0 are constants) and a Matsumoto metric $\bar{F}=\frac{\bar{\alpha}^2}{\bar{\alpha} -\bar{\beta}}$ on a manifold with dimension $n \geq 3$ where $\alpha$ and $\bar{\alpha}$ are two Riemannian metrics, $\beta$ and $\bar{\beta}$ are two non-zero 1-forms. Moreover, we study such projective changes when F has some special curvature properties.