On the c-concavity with respect to the quadratic cost on a manifold

Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition ${\mathrm{\nabla }}^{2}f<g$ 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.