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 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 $\nu$. These plans are used to define the Kantorovich Problem. 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, in the context of ${\mathcal {P}}_{2}(X)$. Since the Wasserstein metric is studied and applied in optimal transport, this relationship can also be used to apply optimal transport tools to Riemannian geometry.