On Generalized Matsumoto Metrics with a Special $π$-form
A. Soleiman, Ebtsam H. Taha
TL;DR
This work introduces an intrinsic generalized $\phi$-Matsumoto change on Finsler manifolds admitting a concurrent $\pi$-vector field $\overline{\varphi}$, defining $\widehat{F}=\frac{F^2}{F-\Phi}$ on the domain where $F>\Phi$. It derives a suite of transformed geometric objects, including $\widehat{\ell}$, $\widehat{g}$, $\widehat{\hbar}$, and $\widehat{\mathbf{T}}$, and establishes a non-degeneracy criterion $F(1+2p^2)-3\Phi\neq0$ that guarantees $\widehat{g}$ is non-degenerate; it also relates the sprays and connections (Berwald, Barthel) of $F$ and $\widehat{F}$ and proves that the metrics cannot be projectively related unless the concurrent field vanishes. The paper further shows that the transformed geodesic spray satisfies $\widehat{G}=G-f_1\mathcal{C}+f_2\gamma\overline{\varphi}$ with explicit coefficients, and gives a condition under which $\overline{\varphi}$ remains concurrent in $\widehat{F}$. An explicit rational metric example with a concurrent $\pi$-vector field is provided, and the authors prove that rationality is preserved under the change (and in fact equivalent between $F$ and $\widehat{F}$), while almost rationality is not automatically preserved. These results illuminate how generalized Matsumoto-type changes interact with concurrency and rationality in Finsler geometry, revealing new connections between curvature, sprays, and metric degeneracy.
Abstract
We explore a generalization of Matsumoto metric intrinsically. Given a Finsler manifold $(M,F)$ which admits a concurrent $π$-vector field $\overline{\varphi}$, we consider the change $\widehat{F}(x,y)=\frac {F^2 (x,y)} {F(x,y)-Φ(x,y)}$, where $Φ$ is the associated concurrent $π$-form with $F(x,y) > Φ(x,y)$ for all $(x,y) \in \T M$. We find the condition under which the generalized $φ$-Matsumoto metric $\widehat{F}$ is a Finsler metric. Moreover, the relations between the associated Finslerian geometric objects of $\widehat{F}$ 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 $\widehat{F}$ can never be projectively related. Also, a condition for the $π$-vector field $\overline{\varphi}$ to be concurrent with respect to $\widehat{F}$ is acquired. Moreover, an example of a rational Finsler metric admitting a concurrent $π$-vector field together with the associated change $\widehat{F}$ is provided. Finally, we find the conditions that preserve the almost rationality property of a Finsler metric $F$ under the $φ$-Matsumoto change.