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Let $F:M\to N$ be a map of smooth manifolds. Show that the following are equivalent:

  1. $F$ is smooth,
  2. For each $p\in M$ there exist smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ \varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi(V)$,
  3. $F$ is continuous and there exist smooth atlases $\{(U_\alpha,\varphi_\alpha)\}$ and $\{(V_\beta,\psi_\beta)\}$ for $M$ and $N$, respectively, such that for each $\alpha$ and $\beta$, $\psi_\beta\circ F\circ \varphi_\alpha^{-1}$ is smooth from $\varphi_\alpha(U_\alpha\cap F^{-1}(V_\beta))$ to $\psi_\beta(V_\beta)$.

For those interested, this is exercise 2.7 from Lee's text on Smooth Manifolds (pg35).

I can show 1 $\iff$ 2, and 3 $\implies 2$. For 2 $\implies$ 3, I can show that $F$ is continuous. By hypothesis we get a smooth atlas $\{(U_p,\varphi_p)\}_{p\in M}$ for $M$ and a collection of smooth charts $\mathcal N:=\{(V_p,\psi_p)\}_{p\in M}$ for $N$ satisfying the desired conclusion. But $\mathcal N$ isn't necessarily a smooth atlas for $N$ since the domains don't necessarily cover $N$.

I feel like I should be able to take the given smooth structure for $N$ since $\mathcal N$ will (uniquely) determine this structure anyways. But, how do I know that the coordinate representations will satisfy the desired conclusion (assuming this approach works, of course)?

Or is there some way to "extend" $\mathcal N$ to an atlas (not necessarily maximal) for $N$ with the desired property?

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  • $\begingroup$ Extend the collection of charts $\mathcal{N}$ to an atlas by throwing in some extra charts. Also include the intersections of the domains of the extra charts with those of the charts in $\mathcal{N}$. On these intersections, the $\psi_p$'s remain the same. $\endgroup$ – InTransit Aug 1 '14 at 8:30
  • $\begingroup$ @InTransit: Sorry, but I don't see how the domains in the extended collection of charts will cover $N$ since each domain will be contained in some domain $V_p$ of $\mathcal N$, which possibly doesn't cover $N$. $\endgroup$ – user59083 Aug 1 '14 at 8:33
  • $\begingroup$ $N$ has an atlas, so take whatever you need from it. I didn't say to use only the intersections with the $\mathcal{N}$ charts because that would cover only $F(M)$. That's why I used the word "also" in my first comment. $\endgroup$ – InTransit Aug 1 '14 at 8:50
  • $\begingroup$ @InTransit: I can see how this might work after your edit, but I still have concerns. Say we extend $\mathcal N$ by throwing in sufficiently many charts, call them $(V_i,\psi_i)$, to cover $N$. How do we know that the coordinate representation will be smooth on $\varphi(U\cap F^{-1}(V_i))$? For instance, what if $V_i$ does not intersect any $V_p$? $\endgroup$ – user59083 Aug 1 '14 at 8:51
  • $\begingroup$ That's even less of a problem! You will effectively be concentrating only on $F(M)$. Non-intersecting charts would be present only because you want an atlas. $\endgroup$ – InTransit Aug 1 '14 at 8:52

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