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Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I don't known whether there is any paper that using this result, and maybe another condition on $M$, that obtain a topological result about $M$.

On the other hand, is there any similar result for the scalar curvature?, sometime I've heard that a lower bound on the scalar curvature is easier to treat than a lower bound on the Ricci curvature but I don't know why, if anybody could help me to understand this it would be very helpful.

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  • $\begingroup$ For example, can we obtain Bonnet-Myers theorem just using the convexity of the entropy on $\mathcal{P}^2(M)$?. By analogous I mean similar. $\endgroup$
    – Mario
    Commented May 21, 2014 at 11:57
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    $\begingroup$ Have you read the papers by Sturm dx.doi.org/10.1007/s11511-006-0002-8 dx.doi.org/10.1007/s11511-006-0003-7 and by Lott and Villani dx.doi.org/10.4007/annals.2009.169.903 ? $\endgroup$
    – Tapio Rajala
    Commented May 21, 2014 at 12:32
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    $\begingroup$ It is not so easy to answer your question. We do have Bonnet-Myers in CD(K,N) spaces, but here the convexity-type condition is using the Rényi entropy and not the Shannon entropy. $\endgroup$
    – Tapio Rajala
    Commented May 21, 2014 at 12:38
  • $\begingroup$ I have not yet read the papers you mentioned above but I'm studying a course of Nicola Gigli on the subject, I will read those papers very soon. Do you have the reference of Bonnet-Myers in $CD(K,N)$? $\endgroup$
    – Mario
    Commented May 21, 2014 at 12:53
  • $\begingroup$ On the other hand, is there a link between $c$-cyclically monotone sets (from optimal transport theory) and rectifiability? $\endgroup$
    – Mario
    Commented May 21, 2014 at 12:54

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