On Generalized Matsumoto Metrics with a Special $\pi$-form

Abstract

We explore a generalization of Matsumoto metric intrinsically. Given a Finsler manifold $(M,F)$ which admits a concurrent $\pi$-vector field $\overline{\phi }$, we consider the change $\hat{F}(x,y)=\frac{F^2(x,y)}{F(x,y)-\Phi (x,y)}$, where $\Phi$ is the associated concurrent $\pi$-form with $F(x,y)>\Phi (x,y)$ for all $(x,y) \in \T M$. We find the condition under which the generalized $\varphi$-Matsumoto metric $\hat{F}$ is a Finsler metric. Moreover, the relations between the associated Finslerian geometric objects of $\hat{F}$ and $F$ are obtained, namely, the relations between angular metric tensors, metric tensors, Cartan torsions, geodesic sprays, Barthel connections (along with its curvature) and Berwald connections. Further, we prove that the Finsler metrics $F$ and $\hat{F}$ can never be projectively related. Also, a condition for the $\pi$-vector field $\overline{\phi }$ to be concurrent with respect to $\hat{F}$ is acquired. Moreover, an example of a rational Finsler metric admitting a concurrent $\pi$-vector field together with the associated change $\hat{F}$ is provided. Finally, we find the conditions that preserve the almost rationality property of a Finsler metric $F$ under the $\varphi$-Matsumoto change.

20 pages, Dedicated to the memory of Professor Nabil L. Youssef