We consider the Monge optimal transport problem posed on compact manifolds (possibly with boundary) when all data are smooth and the given measures, positive. If a diffeomorphism is stationary for the total cost, it was established that it must admit a potential function. If it realizes a local minimum of the total cost, we show that the $c$-Hessian of its potential function must be non-negative, positive if we further assume that the cost function $c$ is non degenerate.