• non-branching
• Ricci curvature
• metric measure spaces
• optimal transport
• optimal maps

In 1987, Brenier proved the existence and uniqueness of optimal transport maps in the Euclidean space, with cost the squared distance. After this result, many others were proven, for example the one by McCann on a Riemannian setting, also in this case with cost the squared distance. In this work, we extend this kind of result in a more general setting.
We introduce the notion of 'Ricci curvature bounded from below' (briefly a CD space) on a general metric measure space and we prove some important properties of these spaces, as the compatibility with the Riemannian case and the stability with respect to Sturm's convergence.
Then we prove the existence and uniqueness of optimal maps on a metric measure space with Ricci curvature bounded from below, which satisfies at least one of this two property: it is non-branching or it is a strong CD space, which is a stronger notion of Ricci curvature bounded from below for a metric measure space. In particular, to prove the theorem on the second assumption, we will see that the space is essentially non-branching.