  This thesis contains an introduction to the method of average in Finsler
geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity
conditions. We prove that the Levi-Civita connection of any Riemannian metric
affine equivalent to the Berwald metric leaves invariant the indicatrix of th
Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure
such that there is a Riemannian connection whose Levi-Civita leaves invariant
by parallel transport the indicatrix of the Finsler structure, then the
structure (M,F) is Berwald. As an application we obtain a necessary condition
for a structure to be Landsberg but not Berwald.

Differential Geometry