{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/coarse-ricci-curvature-as-a-function-on","title":"Coarse Ricci curvature as a function on $M\\times M$","arxiv_id":"1505.04166","date":"2015-05-15","proceeding":null,"authors":["Antonio Ache","Micah Warren"],"abstract":"We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula $$ \\mathrm{Ric}(\\gamma^{\\prime}\\left( 0\\right) ,\\gamma^{\\prime}\\left( 0\\right) )=\\frac{1}{2}\\frac{d^{2}}{ds^{2}}\\mathrm{Ric}_{\\Delta_g}(x,\\gamma\\left( s\\right) )$$ for any curve $\\gamma(s).$","url_abs":"https://arxiv.org/abs/1505.04166v1","url_pdf":"https://arxiv.org/pdf/1505.04166v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"coarse-ricci-curvature-as-a-function-on","repo_url":"https://github.com/micah541/ricci","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}