A Concurrent Generalized Kropina Change

TL;DR

This paper explores a new generalized Kropina change in Finsler geometry involving a $ abla$-form, analyzing its geometric properties, invariance, and the behavior of concurrent vector fields under this transformation.

Contribution

It introduces a $ abla$-form-based generalized Kropina change, investigates its geometric effects, and establishes conditions for concurrency and invariance properties.

Findings

Geodesic sprays of $F$ and $ ilde{F}$ are not projectively related.

The $ abla$-form change preserves the almost rational property of the Finsler metric.

Conditions under which the concurrent vector field remains concurrent after the 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 ${F}$.