On tangent cones in Wasserstein space
  If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $\mu$ is an absolutely continuous measure on S, then we compute the tangent cone of P(M) at $\mu$. 
John Lott
math.DG