  A notion of differentiability is being proposed for maps between Wasserstein
spaces of order 2 of smooth, connected and complete Riemannian manifolds. Due
to the nature of the tangent space construction on Wasserstein spaces, we only
give a global definition of differentiability, i.e. without a prior notion of
pointwise differentiability. With our definition, however, we recover the
expected properties of a differential. Special focus is being put on
differentiability properties of pushforward maps induced by smooth maps between
the underlying manifolds, and on convex mixing of differentiable maps, with an
explicit construction of the differential.
