Finsler space subjected to a Kropina change with an h-vector

TL;DR

This paper investigates the effects of a Kropina change with an h-vector on Finsler spaces, establishing conditions for when their Cartan connections coincide and when the change is projective.

Contribution

It provides necessary and sufficient conditions for the invariance of Cartan connection coefficients and projectiveness under a Kropina change with an h-vector in Finsler geometry.

Findings

Conditions for identical Cartan connection coefficients.

Criteria for the Kropina change to be projective.

Extension of Finsler space transformations with h-vectors.

Abstract

In this paper, we discuss the Finsler spaces $(M^n,L)$ and $(M^n,\,^{*}L)$, where $^{*}L(x,y)$ is obtained from $L(x,y)$ by Kropina change $^{*}L(x,y)=\frac{L^2(x,y)}{b_i(x,y)\,y^i}$ and $b^{}_{i}(x,y)$ is an $h$-vector in $(M^n,L)$. We find the necessary and sufficient condition when the Cartan connection coefficients for both spaces $(M^n,L)$ and $(M^n,\,^{*}L)$ are the same. We also find the necessary and sufficient condition for Kropina change with an $h$-vector to be projective.