  The curvature-dimension condition is a generalization of the Bochner
inequality to weighted Riemannian manifolds and general metric measure spaces.
It is now known to be equivalent to evolution variational inequalities for the
heat semigroup, and quadratic Wasserstein distance contraction properties at
different times. On the other hand, in a compact Riemannian manifold, it
implies a same-time Wasserstein contraction property for this semigroup. In
this work we generalize the latter result to metric measure spaces and more
importantly prove the converse: contraction inequalities are equivalent to
curvature-dimension conditions. Links with functional inequalities are also
investigated.
