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I have the impression that the answer to this should be elementary and maybe I am missing something obvious; however, a quick search in the literature gave no results.

Question. Does it exist an example of (possibly closed) non-orientable smooth manifold covered by exactly two charts? If so, what is a reference?

Of course, the intersection of these charts should be non-connected, because the transition function cannot change sign on a connected set.

Edit. Gae. S.'nice answer shows that the Klein bottle can be written as the union of two open cylinders (with disconnected intersection). So, let me ask the following variation of the above question:

Question 2. Does it exist a closed, non-orientable smooth manifold that can be written as the union of exactly two simply-connected charts?

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Well, the open Mobius strip and, for the closed case, the Klein bottle should do it, should they not?

For Mobius, you cut it in two stripes. For Klein, you cut it in two cylinders, which are diffeomorphic to $\Bbb R^2\setminus\{0\}$.

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  • $\begingroup$ Uhm...for the Mobius strip, maybe I see it now. The charts should be two rectangular strips with disconnected intersection. For the Klein bottle, I have to think a bit, but maybe it works, too. Thank you for the answer. $\endgroup$ Apr 20, 2021 at 6:35
  • $\begingroup$ @FrancescoPolizzi It should be noted that for Klein they aren't charts onto a ball, because my definition of chart only requires to be onto an open subset of $\Bbb R^n$. $\endgroup$
    – user239203
    Apr 20, 2021 at 6:41
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    $\begingroup$ Yes, this is also my definition. $\endgroup$ Apr 20, 2021 at 6:41
  • $\begingroup$ I added an edit to the question, asking if (in the closed case) there are examples with simply-connected charts. $\endgroup$ Apr 20, 2021 at 8:29

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