  In this work, we show how to obtain for non-compact manifolds the results
that have already been done for Monge Transport Problem for costs coming from
Tonelli Lagrangians on compact manifolds. In particular, the already known
results for a cost of the type $d^r,r>1$, where $d$ is the Riemannian distance
of a complete Riemannian manifold, hold without any curvature restriction.
