Mathematics

In this seminar, we present the construction (made by Ambrosio and Gigli) of the parallel transport in the Wasserstein space. First of all, we give a brief presentation of the differential structure associated to the Wasserstein space. Then we present a construction of the parallel transport in the case of a manifold embedded in a euclidean space, which fits better for a generalization in the Wasserstein setting than the usual construction in Riemannian geometry. We describe the geometric similarities and the differences between the Wasserstein space and an embedded Riemannian manifold. As a consequence, using this analogy with the Riemannian case, we describe the construction of the parallel transport in the Wasserstein space. Then, if time allows, we will describe some examples: the case of geodesics and an example of non existence of parallel transport.