  In this paper we introduce a natural definition for the affine maps between
two Finsler manifolds $(M, F)$ and $(N,\tilde F)$ and we give some geometrical
properties of these affine maps. Starting from the equations of the affine
maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet
space $J^1(TM;N)$, in the sense of a Berwald nonlinear connection
$\Gamma^b_jet$, a Berwald $\Gamma^b_jet$-linear d-connection $B\Gamma^b_jet$,
together with its d-torsions and d-curvatures, which geometrically
characterizes the initial affine maps between Finsler manifolds.
