Abstract and concrete tangent modules on Lipschitz differentiability spaces

TL;DR

This paper establishes an isometric embedding between abstract and concrete tangent modules in Lipschitz differentiability spaces, characterizes when this embedding is an isomorphism, and links Lipschitz conditions to differentiable structures.

Contribution

It constructs an isometric embedding of tangent modules, characterizes when it is an isomorphism, and connects Lipschitz conditions with differentiable structures in metric spaces.

Findings

Embedding of tangent modules is isometric

Lipschitz condition implies existence of differentiable structure

Embedding into Gromov--Hausdorff tangent module without reflexivity assumptions

Abstract

We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli's tangent module admits an isometric embedding into the so-called Gromov--Hausdorff tangent module, without any a priori reflexivity assumptions.