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I am trying to get a better understanding of the reasons why (local) coordinate systems of a manifold are doomed to remain local. I apologize for the lack of rigor in the reasonnig, I am really trying to get an intuitive feel for what is happening.

Let me explain what I mean with an example: the one dimensional circle cannot be described with only one coordinate system. My feeling is that the "reason" is topological in nature : the circle "loops arround", which coordinate systems cannot do.

However, (in the case of a Riemannian manifold at least), it also feels like another reason for this failure could be related to the fact that the exponential map is not global (finite injecivity radius). Both notion seem (to me at least) to make rigorous the intuitive idea that "you cannot straigten / flatten a manifold at will". And indeed, the exponential map being a diffeomorphism, it locally defines a coordinate system.

Questions: Is this a good intuition ? Are these (topological & metric) the only reasons for the impossibility to have global coordinates of manifolds ? Are there other reasons, and if so, what are they intuitively?

Thank you.

G. Fougeron
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  • If a manifold has a global coordinate system, it is then globally diffeomorphic to an open subset of $\mathbb{R}^n$: this is a topological statement – Didier Jun 01 '21 at 09:44
  • Yes. Good point. This is clear to me. Can this be extended in any way? What about the minimal number of maps in an atlas for instance? – G. Fougeron Jun 01 '21 at 09:51
  • There isn't much of a connection with the Riemannian case, since the exponential map corresponds to normal coordinates, which are quite constrained compared to an arbitrary coordinate patch. One can have manifolds which admit a global coordinate patch, and yet $\exp$ always fails to be a global diffeomorphism. – Kajelad Jun 01 '21 at 18:16
  • Related, but offering little insight (as I don't understand the answers fully) : https://math.stackexchange.com/questions/2097690/minimal-number-of-charts-covering-a-manifold and https://math.stackexchange.com/questions/75594/surface-where-number-of-coordinate-charts-in-atlas-has-to-be-infinite?noredirect=1&lq=1 . In particular, the minmum number of charts is related to CW complexes built on the manifold, and is hence topological in nature. – G. Fougeron Jun 02 '21 at 07:58

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