Coarse Ricci curvature as a function on $M\times M$

Antonio Ache; Micah Warren

Coarse Ricci curvature as a function on $M\times M$

Differential Geometry 2015-05-18 v1

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 $\gamma(s).$

Keywords

coarse Ricci curvature smooth metric measure space logarithmic Sobolev inequality Ricci tensor Riemannian manifold Bakry-Emery framework

Cite

@article{arxiv.1505.04166,
  title  = {Coarse Ricci curvature as a function on $M\times M$},
  author = {Antonio Ache and Micah Warren},
  journal= {arXiv preprint arXiv:1505.04166},
  year   = {2015}
}

Comments

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

Coarse Ricci curvature as a function on $M\times M$

Abstract

Keywords

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