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I want to show that

The $C^\infty$-differential structures of $\mathbb{R}$ up to $C^\infty$-diffeomorphism is unique.

For differential structures given by a single chart $(\varphi,\mathbb{R})$, I think the diffeomorphism to the differential structure given by $(\mathrm{id},\mathbb{R})$ can be found by taking its inverse.

But I don't know how to find the diffeomorphism for a differential structure given by multiple charts, say $(\varphi_1,U_1)$, $(\varphi_2,U_2)$. I can take their inverse repectively, but what should I do on the intersections?

Thanks in advance!

William von Schwarz
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  • what do you mean by upto diffeomorphism smooth atlas is unique ? – Balaji sb Mar 24 '23 at 03:17
  • I mean for any two $C^\infty$ differential structures on the real line, there exists a $C^\infty$-diffeomorphism between them. @Balajisb – William von Schwarz Mar 24 '23 at 04:26
  • You mean to say $\phi_i(f(m)) = \psi_j(m)$ for a global single diffeomorphism $f$ on manifold and any 2 charts $\phi_i,\psi_j$ with support at $m$ from the 2 different atlas ? – Balaji sb Mar 24 '23 at 04:33
  • I think what I mean is for any smooth atlas on $\mathbb{R}$, there is a global diffeomorphism $f$ such that $\varphi_i\circ f\circ\mathrm{id}^{-1}$ and $\mathrm{id}\circ f\circ\varphi_i^{-1}$ is smooth function for each chart $(\varphi_i,U_i)$. – William von Schwarz Mar 24 '23 at 04:59
  • At the end of the day by the definition of smoothness, $\phi_i(f(m))$ is smooth iff $\phi_i(f(\phi_i^{-1}(u)))$ is smooth and $f(\phi_i^{-1}(u))$ is smooth iff $\phi_i(f(\phi_i^{-1}(u)))$ is smooth by definition of smoothness on a manifold as smoothness is always defined w.r.t charts and $f$ is a diffeomorphism of manifolds iff $\phi_i(f(\phi_i^{-1}(u)))$ is smooth. So what you want will be satisfied ? Can you edit the question and make it clear on what you want ? Good Luck ! – Balaji sb Mar 24 '23 at 05:20
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    It takes some effort; you can find references here, and here. – Moishe Kohan Mar 24 '23 at 23:07

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