Suppose we have a compact manifold $M$, like for example $\mathbb{C}P^n.$ We can cover this manifold by finitely many open geodesic balls $B_r(p)$ of radius $r$.
If for example $r$ is equal to the injectivity radius of $M$, can we say how many balls are needed to cover our manifold? What about the case $M=\mathbb{C}P^n$ with $\omega_{FS}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log(1+|z_1|^2+\ldots+|z_n|^2)$ ?