Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces

We prove that for a suitable class of metric measure spaces the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of $L^2$-sections of the ‘Gromov–Hausdorff tangent bundle’. The key assumption that we make is a form of rectifiability for which the space is ‘almost isometrically’ rectifiable (up to m-null sets) via maps that keep under control the reference measure. We point out that $\mathsf{RCD}^\ast (K,N)$ spaces fit in our framework.

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Received 24 May 2018

Accepted 19 December 2018

Published 22 July 2022