Generalized Randers metrics with special $1$-form

In this paper, we investigate a generalized Randers metric with a special $\pi$-form. Precisely, following the pullback approach to global Finsler geometry, we start with a Finsler metric $(M,L)$ that admits a concurrent $\pi$-vector field, then we consider the Randers change $\widetilde{L}=L+\mathfrak{B}$, where $\mathfrak{B}$ is the associated $1$-form. We study some of the geometric objects and properties attached to $(M,\widetilde{L})$. We prove that the corresponding metric tensor of $\widetilde{L}$ is nondegenerate without any conditions on the $1$-form $\mathfrak{B}$. By calculating the relation between the attached geodesic sprays of $L$ and $\widetilde{L}$, we establish that the change $\widetilde{L}=L+\mathfrak{B}$ preserves the geodesics of $(M,L)$, that is, the change is projective. Moreover, under a certain condition, this change preserves the deviation tensor, the $(v)h$-torsion tensor, the $(h)h$-curvature tensor and the curvature tensor of the Barthel connection. We provide an example of a Finsler metric with some details that admits a concurrent vector field together with the associated $\pi$-form.