Motivation Consider $S^1=\{x \in \mathbb{R}^2: \left|x\right|^2 = 1\}$ and $\mathbb{R}/\mathbb{Z}$ as smooth manifolds, when $S^1$ is considered as a smooth submanifold of $\mathbb{R}^2$ (here, with the natural smooth sructure of $\mathbb{R}^2$) and $\mathbb{R}/\mathbb{Z}$ has its smooth structure defined by the atlas obtained from restricting the projection map $p: \mathbb{R} \to \mathbb{R}/\mathbb{Z}$ to interval of length half.

Now, suppose we want to show that $S^1$ and $\mathbb{R}/\mathbb{Z}$ are diffeomorphic. We can take $$ \begin{align} f: \ \mathbb{R}/\mathbb{Z} &\longrightarrow S^1\\ [x] & \longmapsto (\cos(2\pi x), \sin(2\pi x)) \end{align}$$ and show that it is smooth in both directions.

The natural way to show this will be to consider another atlas on $S^1$, For example, the one obtained by pairs $(U_\alpha, \varphi_\alpha)$ where $U_\alpha$ are image of invervals of the form $(\alpha, \alpha + 2 \pi) \subset \mathbb{R}$ under the map $\gamma: t \mapsto (\cos t, \sin t)$, and $\varphi_\alpha = (\gamma |_{(\alpha, \alpha + 2 \pi)})^{-1}$ are inverses of the restriction of $\gamma$.

Problem \ Question: The atlas that makes it easy to show the diffeomorphism between the two manifolds is not necessarily equivalent to the structure of $S^1$ taken as a submanifold of $\mathbb{R}^2$. How can we (easily) know that the smooth structure obtained on $S^1$ as a submanifold of $\mathbb{R}^2$ is equivalent to the one obtained by the atlas $\{(U_\alpha,\varphi_\alpha)\}_{\alpha \in \mathbb{R}}$?

As a submanifold of $\mathbb{R^2}$, the smooth structure on $S^1$ is defined by diffeomorphisms $\psi: W \to Q$ where $W,Q \subset \mathbb{R}^2$ and $\psi(W \cap S^1) \subset \mathbb{R}\times\{0\}$. Then, the atlas considered on $S^1$ is obtained by charts of the form $(\psi |_{W \cap S^1}, W \cap S^1)$. In our case, the existence of such diffeomorphisms (of $\mathbb{R}^2$) can be established by the theorem about regular zero sets of smooth functions (i.e. $ F(x_1,x_2) = x_1^2+x_2^2 -1$ and $S^1 = F^1(\{0\})$).

In this specific case, it's fairly easy to show by computation that transition maps between some $(\psi |_{W \cup S^1}, W \cup S^1)$ and $(\varphi_\alpha, U_\alpha)$ is smooth, but it can become a mess in more complicated cases.

Stated more generally: Is there a simple way to show that an atlas obtained from a 'natural' homeomorphisms from open subsets $\mathbb{R}^k$ to subsets of some $k$-dimensional manifold $M \subset \mathbb{R}^d$, establishes on $M$ the a smooth structure that is the same as the one it inherits from $\mathbb{R}^d$?

  • $\begingroup$ This sounds like a possible application of invariance of domain, but I am not entirely sure. $\endgroup$ – Prototank Oct 25 '18 at 13:31

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