A Concurrent Generalized Kropina Change

Abstract

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

2o pages, To the memory of Professor Nabil L. Youssef, some typos are fixed