# Covering number estimates on closed Riemannian manifolds

Let $$(M^n,g)$$ be an $$n$$-dimensional compact and connected Riemannian manifold with sectional curvature bounded above and below by $$c,C$$. Is it possible/known how to express the external covering number of $$M^n$$ as a function of $$c,C$$, $$n$$, $$\operatorname{diam}(M^n)$$, and possibly some other numerical data associated to $$(M^n,g)$$?

This must be known, but I have no idea where to look.

Definition: For $$\varepsilon>0$$, the $$\varepsilon$$-external covering number of $$M^n$$, denoted by $$N_{\varepsilon}(M^n)$$, is the smallest number of points $$p_1,\dots,p_N\in M^n$$ which are $$\varepsilon$$-dense wrt. $$(M^n,g)$$'s Riemannian distance $$d_g$$. That is, if $$p\in M^n$$ then there is some $$p_i$$, as above, such that $$d_g(p,p_i)\le \varepsilon .$$

• For lower bound you can use the Bishop-Gromov volume comparison, not sure about an upper bound. Feb 4, 2022 at 5:41
• @DmytroYeroshkin How can this be done? I feel that I'm missing something. Also, for the upper bound I found the following unpublished article but it only holds for small balls. arxiv.org/pdf/1305.1529.pdf May 16, 2022 at 7:36
• The simplest lower bound can be achieved as follows: By Bishop-Gromov, we know that the volume of an $\epsilon$-ball in $M$ is at most the volume of an $\epsilon$-ball in the model space with constant curvature $c$. Therefore in order for $N$ $\epsilon$-balls to cover $M$, $N$ must be at least $Vol(M)$ divided by the volume of an $\epsilon$-ball in the model space. I suspect there are ways to improve the bound by understanding how big the overlaps have to be, but I've not actually looked at the details. May 16, 2022 at 11:25
• @DmytroYeroshkin Thanks, the lower-bound is obvious now; thanks for the helping details. May 16, 2022 at 11:28