# Choosing Coordinate Charts — Smooth Functions on Manifolds

I'm trying to prove the following claim from Lee's Intro to Smooth Manifolds.

$$\textbf{Claim:}$$ For smooth manifolds M and N, show that $$F:M \rightarrow N$$ is smooth if and only if $$F^*(C^{\infty}(N)) \subseteq C^{\infty}(M).$$

The textbook defines a map $$F:M \rightarrow N$$ between manifolds to be smooth if for every $$p \in M,$$ there exist smooth charts $$(U, \phi)$$ containing p and $$(V, \psi)$$ containing $$F(p)$$ such that $$F(U) \subseteq V$$ and the composite map $$\psi \circ F \circ \phi^{-1}$$ is smooth in the sense of ordinary calculus (i.e. $$\psi \circ F \circ \phi^{-1} \in C^{\infty}(\mathbb{R}^n)).$$ The problem defines $$C(M)$$ as the algebra of continuous functions $$f:M \rightarrow \mathbb{R}$$, and for any continuous map $$F:M \rightarrow N$$, it defines $$F^*:C(N) \rightarrow C(M)$$ by $$F^*(f)=f \circ F$$.

One direction is easy: Suppose $$F:M \rightarrow N$$ is smooth, and take any $$f \in C^{\infty}(N)$$. Then $$F^*(f)=f \circ F \in C(M)$$ is smooth because it's the composition of smooth maps $$f$$ and $$F$$. Since $$f \in C^{\infty}(N)$$ was arbitrary, we get $$F^*(C^{\infty}(N)) \subseteq C^{\infty}(M).$$

But I have a question concerning the other direction, as I'm still trying to understand charts: Suppose $$F^*(C^{\infty}(N)) \subseteq C^{\infty}(M).$$ For any $$p \in M,$$ we can find a smooth chart $$(U, \phi)$$ for M with $$p \in U$$ simply because $$M$$ is a smooth manifold. Now since $$F(p) \in N$$, we can find a chart $$(V, \psi)$$ containing $$F(p),$$ but since I'm trying to show $$F$$ is smooth, I feel that my only option is to $$choose$$ a chart $$(V, \psi)$$ for $$N$$ such that $$F(U) \subseteq V,$$ and then go on to (try to) show that $$\psi \circ F \circ \phi^{-1}$$ is smooth.

$$\textbf{Question}:$$ What allows me to just select a chart $$(V, \psi)$$ on $$N$$ satisfying $$F(U) \subseteq V$$? Phrased differently, what guarantees that such a chart exists? Could I have also selected a chart $$(V, \psi)$$ such that $$F(U)=V$$? A related question: can I select any diffeomorphism defined on the set $$V$$ to be the diffeomorphism in my chart? The essence of my question is: does $$any$$ open set $$V \subseteq N$$ and $$any$$ diffeomorphism $$\psi :V \rightarrow \mathbb{R^n}$$ constitute a chart? If not, what allows me to choose such charts? For the n-sphere, for example, it seems like the standard charts are defined for specific open sets and specific diffeomorphisms, and that any open set (with diffeomorphism) does not constitute a smooth chart, so I'm slightly confused as to how we seem to be able to choose charts that satisfy conditions we need. Sorry for writing like 6 questions but they're all in essence the same; I just wanted to be clear as to exactly where my confusion lies. Any relevant insight is very much appreciated.

• If $(V,\psi)$ is a chart, then for any open subset $W\subseteq V$, you have that $(W,\psi|_W)$ is also a chart. Smoothness is a local property, so restricting to small open sets doesn't affect the compatibility of charts. So you're free to consider $U\cap F^{-1}(V)$ on $M$ as your neighborhood. Note also that since $\psi$ is only locally defined, it doesn't meet your hypothesis for $\psi\circ F$ to be smooth. However a bump function should fix that. – Matt Sep 27 '18 at 8:57
• Thanks for the reply. You've definitely provided some clarity. Although helpful, I don't see why we are able to choose the chart $(V,\psi)$ for $N$ such that $F(U) \subseteq V$ based on the fact that, given a chart, we can restrict the chart's map to any open subset and get another chart. I don't see how we know that the image of $U$ will land in some domain $V$ of a chart $(V,\psi)$ for $N$. – innerproduct Sep 27 '18 at 22:34

Suppose $$F^*(C^\infty(N))\subseteq C^\infty(M)$$, that is, for any $$g\in C^\infty(N)$$, we have that $$g\circ F\in C^\infty(M)$$.
Fix $$p\in M$$, and let $$(\psi:W\subseteq N\to\tilde{W}\subseteq\mathbb{R}^n)$$ be a coordinate chart about $$F(p)$$ in $$N$$. Let $$(\phi:U\subseteq M\to\tilde{U}\subseteq\mathbb{R}^m)$$ be a coordinate chart about $$p$$ in $$M$$. Let $$V\subset W$$ be some compactly contained neighborhood of $$p$$ in $$W$$, and let $$\tilde{V}=\psi(V)$$. Let $$\theta$$ be a bump function on $$\overline{V}$$ supported in $$W$$, and let $$\tilde{\psi}=\theta\psi\in C^\infty(N)$$.
Then $$F^{-1}(V)$$ is an open neighborhood of $$p$$ by continuity, and so $$U\cap F^{-1}(V)$$ is an open neighborhood of $$p$$ and is a subset of $$U$$. Hence $$\phi|_{F^{-1}(V)\cap U}:F^{-1}(V)\cap U\to\phi(F^{-1}(V)\cap U)$$ is still a chart map. So we need only show the following map is smooth $$\psi\circ F\circ\left(\left.\phi\right|_{F^{-1}(V)\cap U}\right)^{-1}:\phi(F^{-1}(V)\cap U)\to\tilde{V}.$$
However, since $$\psi\circ F=\tilde{\psi}\circ F$$ on $$V$$, and $$\tilde{\psi}\circ F$$ is smooth by assumption, the result follows.