Optimal Transport and Ricci curvature

Introduction and Motivation

This article provides a brief introduction into a connection of optimal transport and the curvature tensor of a Riemannian manifold. In fact, we are going to study the transport map $T_{t}(x):={\text{exp}}_{x}(t\xi (x)),$ where $\xi$ denotes a $C^{1}$ vector field on the manifold $(M,g).$

These kind of maps appear very naturally in the context of optimal transport. Recall that in optimal transport one is particularly interested in the Monge Problem, being the following optimization problem: Let $(M,g)$ be a compact and connected Riemannian manifold. Let furthermore, $\mu ,\nu$ denote two probability measures on $M$, the Monge Problem is then given by

               $\inf _{T\#\mu =\nu }\int d(x,T(x))^{2}\,dV,$

where the infimum is taken among all measurable maps $T:M\rightarrow M$ and $d$ denotes the Metric on $(M,g)$ induced by $g.$

Theorem

Let $(M,g)$ be a compact and connected Riemannian manifold. Let furthermore, $\mu =fdV,\nu =gdV$ denote two probability measures on $M$ which are absolutely continuous with respect to the measure on the manifold, induced by the metric. Then the Monge Problem admits a unique solution $T.$ Moreover, in that case

         $T(x)=\exp _{x}(\nabla \psi (x))$
         

for some $\psi .$