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Do I have the following correct?

If you define manifolds as subspaces of $\mathbb{R}^N$ as is done in Guillemin and Pollack, then you don't need charts and atlases. All you need are smooth parametrizations to describe a topological manifold as smooth. However, if you are dealing with abstract manifolds where the topological space is not a subset of $\mathbb{R}^N$, then you need charts with smooth "overlaps" to show a that manifold is smooth.

Here's a link that discusses why the two different characterizations of smooth are ultimately the same, Guillmin & Pollack's Definition of a Manifold. My question is more of a practical nature.

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    $\begingroup$ "... then you don't need charts and atlases". I'm not sure what you could mean by this statement, because there are many applications of charts and atlases even for submanifolds of $\mathbb R^N$. $\endgroup$ – Lee Mosher Feb 1 at 17:44
  • $\begingroup$ I meant "... then you don't need charts and atlases to describe a topological manifold as smooth." $\endgroup$ – TJCrow Feb 2 at 13:37
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Definition: A manifold $M$ of dimension $k$ in the sense of Guillemin and Pollack is a subset $X$ of $\mathbb{R}^N$ such that for all $x\in X$ there is an open neighborhood $V\subset\mathbb{R}^N$ of $x$ so that $V\cap X$ is diffeomorphic to an open subset $U$ of $\mathbb{R}^k$. Such a diffeomorphism $\phi:U\to V$ is called a parametrization.

The trick here is that a diffeomorphism $f$ from $X\subset\mathbb{R}^N$ to $Y\subset\mathbb{R}^M$ means a smooth bijection with smooth inverse. But smooth means that, locally, $f:X\to Y$ always extends to a map $F:W\to Y\subset\mathbb{R}^M$, $W\subset\mathbb{R}^N$ which is smooth in the calculus 3 sense (all partials of all orders exist).

If you have a collection of parametrizations $\phi_\lambda$ that cover $X$, you have an atlas $\lbrace \phi_\lambda^{-1}\rbrace$. It is worth trying to show this so that you understand. The non-trivial condition that smooth maps have these local extensions is what allows us to not have to "check the overlaps". It trades one thing to check for another.

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  • $\begingroup$ "The non-trivial condition that smooth maps have these local extensions is what allows us to not have to "check the overlaps". It trades one thing to check for another." Right. You are pointing out how the two systems are related. But concerning my question, I seems you agree with what I $\endgroup$ – TJCrow Feb 2 at 13:41
  • $\begingroup$ "The non-trivial condition that smooth maps have these local extensions is what allows us to not have to "check the overlaps". It trades one thing to check for another." Right. You are in part pointing out how the two systems (Parameterizations vs. Charts) are equivalent when the manifold is a subset of \mathbb{R}^N. But concerning my question, as a practical matter, do you think what I said in the op is true? $\endgroup$ – TJCrow Feb 2 at 13:49
  • $\begingroup$ Yes, I agree with what you've written. $\endgroup$ – Prototank Feb 3 at 0:57
  • $\begingroup$ Thanks @Prototank. That helps. I've been teaching myself out of G&P and ran into trouble trying to show an abstract quotient space was smooth. I concluded that I'd have to use charts and atlases. Thanks for the confirmation. $\endgroup$ – TJCrow Feb 3 at 16:53
  • $\begingroup$ If this answer is sufficient, would you mind marking it as such? $\endgroup$ – Prototank Feb 4 at 15:34

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