In this paper, we derive equations analyzing the projective change between the generalized Matsumoto metric \(\varvec{F = \frac{\alpha ^{m+1}}{(\alpha - \beta )^{m}},\;m\ne 0,-1,1}\) and the Kropina metric \(\varvec{\tilde{F} =\frac{\tilde{\alpha }^{2}}{\tilde{\beta }}}\) , where \(\varvec{\alpha }\) and \(\varvec{\tilde{\alpha }}\) are two distinct Riemannian metrics, and \(\varvec{\beta }\) and \(\varvec{\tilde{\beta }}\) are nonzero one-forms, on a manifold of dimension \(\varvec{n>2}\) . We also investigate how some curvature properties, such as weakly and Berwald curvatures, remain such projective changes. The geometric behavior of the generalized Matsumoto and Kropina metrics under these transformations is investigated in depth. Our discoveries add to a deeper understanding of projective equivalence in Finsler geometry, with potential applications in geometric analysis and theoretical physics.
