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I am confused on the difference between a parameterization of a manifold $M$ and local charts on $M$. If $M$ has dimension $n$, we may find a subset $U \subset \mathbb{R}^n$ such that there exists a homeomorphism $X: U \rightarrow M$. The map $X$ is said to be a parametrization of $M$. Thus it appears to me that the points in $U$ give us coordinates on $M$. If so, how is this any different than a coordinate chart $V$ on $M$? Secondly, wouldn't the existence of such a homeomorphism mean that $M$ is trivial (can be covered by a single chart)?

Ted Shifrin
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CBBAM
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    chart/parametrization are almost identical. I (as learnt from many books) use chart to mean a map (with the suitable conditions) going from the manifold to an open set in $\Bbb{R}^n$, and a parametrization for its inverse. Why is the language of charts good? Because we are dealing with one fixed manifold $M$ and just restricting various objects to various subsets of one fixed $M$. Why is the language of parametrizations good? Because many times in practice that’s what we can ‘easily write down’ (e.g $t\mapsto (\cos t,\sin t)$ is a parametrization of $S^1$, suitably restricted) – peek-a-boo Dec 05 '23 at 22:49
  • @peek-a-boo Thank you, so when we say we are doing things in coordinates we really mean we are using a parameterization? I never thought of it this way and always thought of coordinates as being the inverse of the chart diffeomorphism (which I guess is the same thing based on your comment). – CBBAM Dec 05 '23 at 22:51
  • ‘doing things in coordinates’ is again an author-by-author thing. Many books I read to things “manifold to R^n”, but there are of course authors who write the other way. But with some practice, you should be able to translate for yourself which is the intended meaning, and which is ‘easier to phrase’ in the given context (of course neither is truly ‘easier’ since one is obtained from the other by taking the inverse, so you’re just one bijective step away in both cases…). – peek-a-boo Dec 05 '23 at 22:53
  • @peek-a-boo: your example of a parametrization of $S^1$ is not a homeomorphism: and that's quite right. I think a parametrization is only required to be a local homeomorphism. – Rob Arthan Dec 05 '23 at 22:58
  • @peek-a-boo What is still confusing to me is that some authors use both in the same text. For example, in Lee's book he says "Suppose $S \subset M$ is an immersed $k$-dimensional manifold. A local parameterization of $S$ is a continuous map $X: U \rightarrow M$ whose domain is an open subset $U \subset \mathbb{R}^k$, whose image is an open subset of $S$, and which, considered as a map into $S$, is a homeomorphism onto its image." He defines a coordinate map $\phi: U \rightarrow \hat{U}$ to be a homeomorphism from $U \subset M$ to $\hat{U} \subset \mathbb{R}^n$. Why bother defining both? – CBBAM Dec 05 '23 at 22:59
  • @RobArthan I did say ‘suitably restricted’, so ok explicitly, $f:(0,2\pi)\to S^1$, $f(t)=(\cos t,\sin t)$ is a homeomorphism from $(0,2\pi)$ onto $S^1\setminus{(1,0)}$. To cover that remaining point, we consider a different domain for $f$, say $\tilde{f}:(-\pi,\pi)\to S^1$. – peek-a-boo Dec 05 '23 at 23:06
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    What Lee said is exactly what I said as well (indeed, Lee is one of the books I learnt from): parametrization goes from open set in $\Bbb{R}^k$ into the manifold, while charts go from manifold to open set in some $\Bbb{R}^n$. Oh and this discussion reminds me of this answer of mine, where I talk about things in great detail. – peek-a-boo Dec 05 '23 at 23:08
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    @peek-a-boo: I think we are in agreement. Perhaps you should write up an answer explaining to the OP that parametrizations are not homeomorphisms with the whole manifold (which is what the wording of the question suggests). – Rob Arthan Dec 05 '23 at 23:12
  • @peek-a-boo Thank you, I was confused on why two different terms exist to describe the same thing, even by the same author. But I guess it is a matter of convenience. – CBBAM Dec 05 '23 at 23:44
  • @RobArthan Thank you. Yes you are right, I was confused by the wording but I now understand that parameterizations are local. Thus they seem even more identical to coordinate maps. – CBBAM Dec 05 '23 at 23:45
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    The very simple answer (which is basically what @peek-a-boo has been saying) is that parametrizations are inverses of coordinate maps. And vice versa. They're useful for different reasons in different settings, and it's a good idea to keep the distinction in mind. That's why they have different names. – Jack Lee Dec 06 '23 at 01:02
  • @JackLee Thank you very much for the comment and for writing such wonderful books! – CBBAM Dec 06 '23 at 02:18
  • Parametrizations are particularly useful when you want to pull back differential forms and integrate. Charts are more convenient for writing vector fields. Notice a covariant/contravariant theme? :D – Ted Shifrin Dec 06 '23 at 18:24

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