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I came across this statement in one of the introductions to differential geometry that some of the manifolds cannot be expressed with a single global coordinate system and one of the examples is the surface of a sphere.

A global coordinate system is one where we have one-to-one mapping from all points on manifold (S) to $\mathbb R^n$. I can map each point on a sphere to $\mathbb R^3$, hence I should have a global coordinate system?

Am I missing something very obvious?

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  • $\begingroup$ The dimension of the manifold must be the same as the dimension $n$ of $\Bbb R^n$ containing the image of the coordinate chart(s). $\endgroup$
    – Travis Willse
    Dec 16, 2019 at 4:32
  • $\begingroup$ But then I can just say the dimension of surface of the sphere is 3? $\endgroup$
    – Umang Gupta
    Dec 16, 2019 at 4:39
  • $\begingroup$ No, the chart must be a local homeomorphism, which is why the above condition holds. $\endgroup$
    – Travis Willse
    Dec 16, 2019 at 4:44

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A chart for a manifold $M$ ($\dim M = n$) is a pair $(U,\varphi)$ where $U\subseteq M$ is open and $\varphi\colon U \to \varphi[U]\subseteq \Bbb R^n$ is a homeomorphism onto the open set $\varphi[U] \subseteq \Bbb R^n$. If $M$ is compact, the existence of a global chart $(M,\varphi)$ means that the image $\varphi[M] \subseteq \Bbb R^n$ is open, non-empty, and compact (by continuity of $\varphi$). This is impossible.

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  • $\begingroup$ Thanks! I think I understand your argument and on reading further I figured it out too but it makes me wonder if there is a particular intuition or reason why we define the range space to be open set and not closed? $\endgroup$
    – Umang Gupta
    Dec 16, 2019 at 4:42
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    $\begingroup$ The same reason why when dealing with Calculus in general, one considers open sets: when computing the limit of a function as $x$ tends to some point $p$ in the open set, one can approach $p$ from all possible directions (as the open set contains some ball centered at $p$). And derivatives are, in the end of the day, limits. Our goal is to do Calculus on the manifold, by transferring everything from $\Bbb R^n$ via charts. $\endgroup$
    – Ivo Terek
    Dec 16, 2019 at 4:44

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