On the rigidity of Wasserstein contraction along heat flows

We establish an equivalence between the rigidity of Wasserstein contraction along heat flows and the rigidity of Bakry--\'Emery gradient estimates for Lipschitz functions. Applying results of Ambrosio--Bru\'e--Semola and Han, we show that if an $\rcd$ space with Ricci lower bound $K\in[0,\infty)$ admits two distinct points $x,y$ such that the $2$-Wasserstein distance between the associated heat kernels satisfies $$W_2(p_t(x,\cdot), p_t(y,\cdot)) = e^{-Kt} d(x,y),$$ then the space splits off a line. Moreover, for weighted smooth manifolds, we provide a direct proof of the rigidity theorem for all curvature bounds $K \in \mathbb{R}$. In particular, we characterize a class of weighted Euclidean spaces as the only spaces where the Wasserstein contraction is sharp for all pairs of points.