A Concurrent Generalized Kropina Change

Abstract

This paper investigates a generalized Kropina metric featuring a specific $\pi$-form. Start with a Finsler manifold $(M,F)$ admits a concurrent $\pi$-vector field $\overline{\varphi}$, then, examine the $\phi$-concurrent generalized Kropina change defined by $\widehat{F}=\frac{F^{m+1}}{\Phi^{m}},\,\, \Phi^{m}>0$, where $\Phi$ 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 $\overline{\varphi}$ is not concurrent with respect to $\widehat{F}$. Eventhough, we give a sufficient condition for $\overline{\varphi}$ to be concurrent with respect to $\widehat{F}$. Finally, we prove that the $\phi$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$.