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

Antonio Ache; Micah Warren

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arXiv · 1505.04166

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

Antonio Ache·Micah Warren

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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 $\gamma(s).$

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Antonio Ache, Micah Warren. 2015-05-15. Coarse Ricci curvature as a function on $M\times M$. https://arxiv.org/abs/1505.04166

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