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For each nonnegative integer $n$, the Euclidean space $\mathbb{R}^n$ is a smooth $n$-manifold with the smooth structure determined by the atlas $\mathcal{A}=(\mathbb{R}^n,\mathbb{1}_{\mathbb{R}^n})$.

Any chart $(U,\varphi)$ contained in the given maximal smooth atlas is called smooth chart or smooth coordinate chart.

Since $\mathcal{A}$ is a smooth atlas, is contained in a unique maixmal smooth atlas, called smooth structure determined by $\mathcal{A}.$

Who are the smooth coordinate chart $(U,\varphi)$ for $\mathbb{R}^n$ respect to this smooth smooth structure?

It has to happen that $\varphi\circ\mathbb{1}_{\mathbb{R}^n}^{-1}\colon\mathbb{1}_{\mathbb{R}^n}(\mathbb{R}^n\cap U)\to \varphi(\mathbb{R}^n\cap U)$ is $C^{\infty}$, that is $\varphi\colon U\to \hat{U}:=\varphi(U)$ is $C^{\infty}$, same reasoning for $\mathbb{1}_{\mathbb{R}^n}\circ\varphi^{-1}$. Therefore, with respect to this smooth structure, the smooth coordinate charts for $\mathbb{R}^n$ are exactly those charts $(U,\varphi)$ such that $\varphi$ is a diffeomorphism.

So, in the case of $\mathbb{R}^n$, then we can describe the maximal atlas, right?

Question. Are there other cases of manifolds in which we can fully describe some maximal atlas?

Thanks!

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  • $\begingroup$ Your answer to the first question is correct, diffeomorphisms describe the maximal smooth structure. As to your second question, consider the same ambient space $\mathbb R$ with the smooth structure determined by the homeomorphism $x^3$; this does not have a type of classification of the maximal smooth structure as in the previous case $\endgroup$ – user555729 May 6 at 9:32
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Maximal smooth atlases is just a bookkeeping device; for all practical purposes, it suffices to work with non-maximal ones and add more smooth charts on "as needed'' basis.

But suppose that you have a topological $n$-manifold $M$ equipped with a smooth atlas ${\mathcal A}$. Then the maximal smooth atlas ${\mathcal A}'$ on $M$ containing ${\mathcal A}$ consists of pairs $(U,\phi)$ where $U$ is an open subset of $M$ and $\phi: U\to \hat{U}\subset R^n$ is a homeomorphism which is a diffeomorphism with respect to the atlas ${\mathcal A}$. In other words, for every $x\in M$, $\phi$ is required to satisfy the condition that $$ \phi\circ \psi^{-1} $$ is a diffeomorphism between suitable open subsets in $R^n$. Here $(V,\psi)$ is a chart in ${\mathcal A}$ such that $x\in U\cap V$.

This is pretty much the same "description" as you gave in the case when $M=R^n$.

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