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

Abstract

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 class of spaces $(\mathrm{X},\mathsf{d},\mathfrak{m})$ we consider are PI spaces that for every $\epsilon >0$ admit a countable collection of Borel sets $(U_i)$ covering $\mathfrak{m}$-a.e.\ $\mathrm{X}$ and corresponding $(1+\epsilon )$-biLipschitz maps ${\phi }_i:U_i\to {\mathbb{R}}^{k_i}$ such that $(φ_i)_*{\mathfrak m}\lower3pt\hbox{$|_{U_i}$}\ll\mathcal L^{k_i}$. This class is known to contain ${\mathsf{RCD}}^{\ast }(K,N)$ spaces. Part of the work we carry out is that to give a meaning to notion of $L^2$-sections of the Gromov-Hausdorff tangent bundle, in particular explaining what it means to have a measurable map assigning to $\mathfrak{m}$-a.e.\ $x\in \mathrm{X}$ an element of the pointed-Gromov-Hausdorff limit of the blow-up of $\mathrm{X}$ at $x$.