Elham Meihami - Master of Mathematics PNU of Tabriz (Pure Mathematics geometry)

In order to study Berwald space which is a bit more general than Riemannian and locally Minkowskian spaces. All tangent spaces of Berwald space, are linearly isometric to a common Minkowski space. For Berwald spaces, the chern connection defineded as linear connection on Manifold M. theorem of Szabo;s is linear connection and also, happens to be the levi- civita (christoffel) connection of a Riemannian metric on M. the purpose of this paper is reasons why such space are important. Therefore, provided some examples which are Finslerian. The result of Ichijyo;s that we have just described put Berwald spaces somewhere in between Riemannian manifolds and generic Finsler manifolds. In essence, it says that each connected Berwald space is modeled on precisely one Minkowski space, although the exact identity of the latter does vary from one Berwald space to another. So, while Euclidean Rn gives rise to the entire category of Riemannian manifolds, every single Minkowski norm gives rise to a whole family of connected Berwald spaces. Of course, when the Minkowski norm in question happens to be the norm associated with the usual dot product, the family it generates is comprised of all Riemannian manifolds.

Riemannian metric, Berwald spaces, Minkowski space, Finslerian, Chern connection, Szabo’s Theorem