# Maximal smooth atlas

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!

• 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 – user555729 May 6 at 9:32

## 1 Answer

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 $$M=R^n$$.