Communications in Analysis and Geometry

Volume 30 (2022)

Number 1

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

Pages: 1 – 51



Nicola Gigli (SISSA, Trieste, Italy)

Enrico Pasqualetto (SISSA, Trieste, Italy; and Department of Mathematics and Statistics, University of Jyväskylä, Finland)


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.

Received 24 May 2018

Accepted 19 December 2018

Published 22 July 2022