Formal Riemannian Structure of the Wasserstein metric

Given a closed and convex space $X\subseteq R^{d}$, two probability measures on the same space, $\mu ,\nu \in {\mathcal {P}}_{2}(X)$, the 2-Wasserstein metric is defined as

$W_{2}(\mu ,\nu ):=\min _{\gamma \in \Gamma (\mu ,\nu )}\left(\int |x_{1}-x_{2}|^{2}\,d\gamma (x_{1},x_{2})\right)^{1/2}$

where $\Gamma (\mu ,\nu )$ is a transport plan from $\mu$ to $\nu$. The Wasserstein metric is indeed a metric in the sense that it satisfies the desired properties of a distance function between probability measures on ${\mathcal {P}}_{2}(X)$. Moreover, the Wasserstein metric can be used to define a Riemannian metric on ${\mathcal {P}}_{2}(X)$. Such a metric allows one to define angles and lengths of vectors at each point in our ambient space. This structure can then be used to apply tools and phenomena found in Riemannian geometry, such as geodesics, to the field of optimal transport.