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 $\omega_{FS}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log(1+|z_1|^2+\ldots+|z_n|^2)$ ?

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    $\begingroup$ Do you know about injectivity radius? And can you answer this immediately for $n=1$? $\endgroup$ – Ted Shifrin Dec 12 '16 at 1:07
  • $\begingroup$ Naturally you first have to fix a metric (e.g., a particular scaling of Fubini-Study, to refer back to an earlier question of yours). Do you mean you have a specific $r$ in mind, or are you looking for the number of balls as a function of $r$ (which I suspect is Difficult)? $\endgroup$ – Andrew D. Hwang Dec 12 '16 at 1:07
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    $\begingroup$ Oops ... I was thinking about the largest possible $r$. :) Hi, Andy :) $\endgroup$ – Ted Shifrin Dec 12 '16 at 1:19
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    $\begingroup$ @JoeAaron: That's the scaling for which a projective line has unit area, so the diameter is $\sqrt{\pi}/2$; as Ted is hinting, there's an easy way to count the minimum number of open geodesic balls of radius $\sqrt{\pi}/2$ that cover. These balls have considerable overlap: The complement of each is a complex hyperplane. Offhand, it looks as if each can be shrunk (keeping the same centers) to radius $\sqrt{\pi}/4 + \varepsilon$ while still covering. $\endgroup$ – Andrew D. Hwang Dec 12 '16 at 1:28
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    $\begingroup$ @JoeAaron: Does Ted's suggestion for $n = 1$ help? (Think of open geodesic balls on a round sphere of intrinsic diameter $d$; how many balls of radius [sic] $d$ does it take to cover? What is the complement of each ball?) Then (for $n > 1$) show that if your Fubini-Study metric has intrinsic diameter $d$, the complement of the geodesic ball of center $[p]$ and radius $d$ is the projective hyperplane orthogonal to $p$. $\endgroup$ – Andrew D. Hwang Dec 13 '16 at 14:01

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