  Given a Finsler space (M,F), one can define natural average Riemannian
metrics on M by averaging on the indicatrix I_x the fundamental tensor g of the
Finsler function $F$. In this paper we determine explicitly the Levi-Civita
connection for these average Riemannian metrics. We apply the result to the
case when (M,F) is a Landsberg space. Using a particular averaging procedure,
the invariance of the average metric along a homotopy in the space of Finsler
structures over M is shown. As a consequence of such invariance, we prove that
any C^5 regular Landsberg space is a Berwald space.
