TITLE: Manifold of probability measures: connections between two types of metrics
QUESTION [37 upvotes]: The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with optimal transport, and metric (2) is connected with information geometry. 
Question: What are the connections between these two metrics? We know that the Fisher-Rao metric is characterized by the Fisher information matrix, but what is the corresponding characterization for the Wasserstein metric? Any references are much appreciated.

REPLY [7 votes]: Just a quick follow-up: Very recently (well, actually in 2015) three teams came up independently and almost simultaneously with the same construction of a new "optimal-transport-like" distance on the space of Radon measures $\mathcal M^+$, which somehow interpolates continuously between Wasserstein and Fisher-Rao. This distance now goes by the name of Wasserstein-Fisher-Rao (WFR) metrics, sometimes also Hellinger-Kantorovich (HK) distance, and gave rise to a whole new topics generally referred to as unbalanced optimal transport. As pointed out by @GabeK in his excellent answer the Wasserstein and Fisher-Rao structures interact in an interesting way and lead to unsuspected behaviour (well, at least for me). The underlying structure possesses several rich and geometric underlying formulations. In a nutshell, the WFR distance can be seen as an infimal-convolution of Wasserstein and Fisher-Rao distances.
For recent developments see the citations of the original trhee papers below (I belong to the first team)

[1] Kondratyev, S., Monsaingeon, L., & Vorotnikov, D. (2016). A new optimal transport distance on the space of finite Radon measures. Advances in Differential Equations, 21(11/12), 1117-1164.
[2] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2018). An interpolating distance between optimal transport and Fisher–Rao metrics. Foundations of Computational Mathematics, 18(1), 1-44.
[3] Liero, M., Mielke, A., & Savaré, G. (2018). Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones mathematicae, 211(3), 969-1117.