  Pushing a little forward an approach proposed by Villani, we are going to
prove that in the Riemannian setting the condition $\nabla^2 f< g$ implies that
$f$ is $c$-concave with respect to the quadratic cost as soon as it has a
sufficiently small $C^1$-norm. From this, we deduce a sufficient condition for
the optimality of transport maps.
