{"paper":{"title":"Bakry-\\'Emery curvature-dimension condition and Riemannian Ricci curvature bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG","math.PR"],"primary_cat":"math.FA","authors_text":"Giuseppe Savar\\'e, Luigi Ambrosio, Nicola Gigli","submitted_at":"2012-09-25T22:26:41Z","abstract_excerpt":"The aim of the present paper is to bridge the gap between the Bakry-\\'{E}mery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form ${{\\mathcal{E}}}$ admitting a Carr\\'{e} du champ $\\Gamma$ in a Polish measure space $(X,\\mathfrak{m})$ and a canonical distance ${\\mathsf{d}}_{{{\\mathcal{E}}}}$ that induces the original topology of $X$. We first characterize the distinguished class of Riemannian Energy measure spaces, where ${\\mathcal{E}}$ coincides with the Cheeger energy induced by ${\\mathsf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5786","kind":"arxiv","version":4},"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"}