Journal of the
Korean Mathematical Society JKMS

In this paper, we investigate a coordinate-free study of the first approximation Matsumoto metric in a more general manner. Namely, for a Finsler metric $(M,L)$ and a one form $\mathfrak{B}$, we study some geometric objects associated with the Matsumoto metric $\widetilde{L}(x,y)=L(x,y)+\mathfrak{B}(x,y)+\frac {\mathfrak{B}^2(x,y)} {L(x,y)}$ in terms of the objects of $L$. Here we consider $L$ is Finslerian and so we call $\widetilde{L}$ the generalized Matsumoto metric. We find the metric and Cartan tensors and other geometric objects associated with $\widetilde{L}$. We characterize the non-degeneracy of the metric tensor of $\widetilde{L}$. We find the geodesic spray, Barthel connection and Berwald connection of $\widetilde{L}(x,y)$ when the one form $\mathfrak{B}$ is associated to a concurrent $\pi$-vector field. Then, we calculate the curvature of the Barthel connection of $\widetilde{L}$. To illustrate our primary results, one example is given.

Keywords: First approximation Matsumoto metric, geodesic spray, Barthel connection, Berwald connection

MSC numbers: Primary 53C60, 53B40, 58B20