  Given a Finsler space (M,F) on a manifold M, the averaging method associates
to Finslerian geometric objects affine geometric objects} living on $M$. In
particular, a Riemannian metric is associated to the fundamental tensor $g$ and
an affine, torsion free connection is associated to the Chern-Rund connection.
As an illustration of the technique, a generalization of the Gauss-Bonnet
theorem to Berwald surfaces using the average metric is presented. The parallel
transport and curvature endomorphisms of the average connection are obtained.
The holonomy group for a Berwald space is discussed. New affine, local
isometric invariants of the original Finsler metric. The heredity of the
property of symmetric space from the Finsler space to the average Riemannian
metric is proved.
