Some Rigidity Conditions on Berwald Structures

Abstract: 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.