Tangents and Rectifiability of Ahlfors Regular Lipschitz Differentiability Spaces

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [Ch99]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchheim.