Coarse Ricci curvature as a function on M× M

2015/05/15 by Antonio G. Ache, Antonio Ache, Ache, Antonio +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #math.DG

paper · pdf · doi:10.48550/arxiv.1505.04166

12 pages. A previous post arXiv:1410.3351 by the same authors has been expanded and split into 3 parts

arxiv created 2015/05/15 · openalex publication_date 2015/05/15 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Ric(γ^′( 0) ,γ^′( 0) )=(1)/(2)\fracd²ds²Ric_Δg(x,γ( s) ) for any curve γ(s).

Citations

• Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces
• Ricci curvature for metric-measure spaces via optimal transport
• Towards a theoretical foundation for Laplacian-based manifold methods
• Ricci curvature of Markov chains on metric spaces
• Geodesic convexity of the relative entropy in reversible Markov chains
• Logarithmic Sobolev Inequalities
• Ricci Curvature and the Manifold Learning Problem

Related

• Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian
• Coarse Ricci curvature of weighted Riemannian manifolds
• Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces
• Approximating coarse Ricci curvature on submanifolds of Euclidean space
• Coarse Ricci curvature on the space of probability measures