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 $\overline{\varphi}$, we consider the change $\widehat{F}(x,y)=\frac {F^2 (x,y)} {F(x,y)-\Phi(x,y)}$, where $\Phi$ is  the associated concurrent $\pi$-form with $F(x,y) > \Phi(x,y)$ for all $(x,y) \in \mathcal{T} M$.  We find the condition under which the generalized $\phi$-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 $\pi$-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 $\pi$-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 $\phi$-Matsumoto change.

Keywords

• Finsler metric
• generalized Matsumoto metric
• geodesic spray
• concurrent vector field
• Berwald connection
• almost rational Finsler metric

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