Given two smooth manifolds $M$、 $N$, and a $C^1$ diffeomorphism $f:M \rightarrow N$. Is there a smooth diffeomorphism between $M$ and $N$?
2025-01-13 00:12:44.1736727164
Smooth diffeomorphism and $C^1$ diffeomorphism
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Yes.
Take two smooth manifolds $M$ and $N$ and a $C^1$-diffeomorphism between them. Then, you can consider the manifold $M'$ which is nothing but $M$ but with a different smooth structure, namely the pullback of the one on $N$. You know have (tautologically) a smooth diffeomorphism $f : M' \to N$. Since $f$ is $C^1$, $M'$ is actually $C^1$-diffeomorphic to $M$.
Here comes the zinger: Whitney proved that a $C^1$-manifold has a unique smooth structure, so $M$ and $M'$ are actually smoothly diffeomorphic, and consequently so are $M$ and $N$.