  We prove a refined contraction inequality for diffusion semigroups with
respect to the Wasserstein distance on a compact Riemannian manifold taking
account of the dimension. The result generalizes in a Riemannian context, the
dimensional contraction established in [BGG13] for the Euclidean heat equation.
It is proved by using a dimensional coercive estimate for the Hodge-de Rham
semigroup on 1-forms.
