# Diameter comparison for manifolds with $\text{Ric}\ge 0$

Let $$M$$ be a Riemannian manifold with $$\text{Sec}\ge 0$$. From Topogonov Theorem follows that for every $$p \in M$$ the quantity $$\frac{\text{Diam}(B_r(p))}{r}$$ is non-increasing in $$(0,\infty)$$. Does this hold also if we only assume $$\text{Ric}\ge 0$$? I suspect that this is no longer true, however I am not able to build a counterexaple.

Clarification: $$\text{Diam}(B_r(p)):=\sup_{x,y\in B_r(p)}d(x,y).$$

• It is a theorem due to Abresch-Gromoll that diameter growth is linear. Any complete Riemannian manifold with $Ric \geq 0$ has diameter growth of order $O(r)$ with respect to any point. – C.F.G Sep 11 at 13:06
• The notion of diameter growth in that paper is not the one I am interested in, here by $\text{Diam}(B_r)$ I mean the supremum of $d(x,y)$ among all $x,y \in B_r$, which clearly implies $\text{Diam}(B_r)\le 2r.$ – mrprottolo Sep 11 at 13:19