{"paper":{"title":"Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Enrico Pasqualetto, Nicola Gigli","submitted_at":"2016-11-29T14:27:09Z","abstract_excerpt":"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'.\n  The class of spaces $({\\rm X},{\\sf d},{\\mathfrak m})$ we consider are PI spaces that for every $\\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\\mathfrak m}$-a.e.\\ ${\\rm X}$ and corresponding $(1+\\varepsilon)$-biLipschitz maps $\\varphi_i:U_i\\to{\\mathbb R}^{k_i}$ such that $(\\varphi_i)_*{\\mathfrak m}\\lower3pt\\hbox{$|_{U_i}$}\\ll\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09645","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}