Connections between metric differentiability and rectifiability

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection C of Banach (or metric) spaces: if a metric measure space X bi-Lipschitz embeds in some element in C, and if every Lipschitz map X→ Y∈ C is differentiable, then X is rectifiable. This gives a simple proof of the rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to arbitrary targets are metrically differentiable, the domain is rectifiable. We moreover establish the compatibility of metric and w^*-differentials of maps from metric spaces in the spirit of Ambrosio--Kirchheim.