Formal Riemannian Structure of the Wasserstein metric

The Wasserstein metric is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }