Seiedeh Sedigheh Alavi - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran. Email: sedighehalavi۵۴@gmail.com

It is proved that every locally flat Finsler manifold is a locally flat Riemannian manifold. Some low dimensional locally Finsler manifolds are classified. It is also proved that in a categorical sense, there is a correspondence between locally flat Finsler manifolds and locally hessian Riemannian manifolds

Finsler, flat manifold, low dimensional