A Concurrent Generalized Kropina Change

Abstract

This paper investigates a generalized Kropina metric featuring a specific $π$-form. Start with a Finsler manifold $(M,F)$ admits a concurrent $π$-vector field $\overlineφ$, then, examine the $ϕ$-concurrent generalized Kropina change defined by $\widehat{F}=\frac{F^{m+1}}{Φ^{m}},\,\, Φ^{m}>0$, where $Φ$ represents the corresponding $1$-form. We investigate the fundamental geometric objects associated with $\widehat{F}$ in an intrinsic manner after adopting this modification and present an example of a Finsler metric that admits a concurrent vector field along with $\widehat{F}$. Also, we prove that the geodesic sprays of $F$ and $\widehat{F}$ can never be projectively related. Moreover, we show $\overlineφ$ is not concurrent with respect to $\widehat{F}$. Eventhough, we give a sufficient condition for $\overlineφ$ to be concurrent with respect to $\widehat{F}$. Finally, we prove that the $ϕ$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$.