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
$$
d_g(p,p_i)\le \varepsilon
.
$$