Formal Riemannian Structure of the Wasserstein metric

Given a closed and convex space $X\subseteq R^{d}$, two probability measures on the same space, $\mu ,\nu$, 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.