A Concurrent Generalized Kropina Change
A. Soleiman, Ebtsam H. Taha
TL;DR
This work analyzes the $\phi$-concurrent generalized Kropina change $\widehat{F}=F^{m+1}\Phi^{-m}$ on a Finsler manifold admitting a concurrent $\pi$-vector field $\overline{\varphi}$. It derives intrinsic relations between the transformed and original geometric objects, including $\widehat{\ell}$, $\widehat{\hbar}$, $\widehat{g}$, and $\widehat{T}$, and expresses the geodesic spray $\widehat{G}$ in terms of $G$, revealing a nondegeneracy condition $mF^2||\overline{\varphi}||_g-(m-1)\Phi^2\neq0$ and that $G$ and $\widehat{G}$ are not projectively related in general. The paper shows that $\overline{\varphi}$ is not concurrent with respect to $\widehat{F}$ except under a sufficient condition, and provides a framework to determine quando $\overline{\varphi}$ becomes concurrent via a vanishing $\mathbb{F}$. It then demonstrates that the change preserves the almost rational nature of the initial metric: $\widehat{F}$ is rational if $m\in\mathbb{Z}$ and almost rational otherwise, with explicit expressions for the transformed metric tensor components. Overall, the results offer a rigorous intrinsic treatment of a generalized Kropina-type change in Finsler geometry and its impact on fundamental geometric objects and rationality properties.
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 $\overline{\varphi}$, 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 $\overline{\varphi}$ 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 $φ$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$.