Generalized first approximation Matsumoto metric

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 B , we study some geometric objects associated with the Matsumoto metric L˜(x,y)=L(x,y)+B(x,y)+B2(x,y)/L(x,y) in terms of the objects of L . Here we consider L is Finslerian and so we call L˜ the generalized Matsumoto metric. We find the metric and Cartan tensors and other geometric objects associated with L˜ . We characterize the non-degeneracy of the metric tensor of L˜ . We find the geodesic spray, Barthel connection and Berwald connection of L˜(x,y) when the one form B is associated to a concurrent π -vector field. Then, we calculate the curvature of the Barthel connection of L˜ . To illustrate our primary results, one example is given.