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If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are smooth.

How much can be said about the differentiable structure of $M$ if it is known which mappings $\mathbf{R}\to M$ are smooth? I suspect that this is not enough to determine the differentiable structure of $M$. If so, can there be a number $k <\operatorname{dim}M$ such that the smooth mappings $\mathbf{R}^k\to M$ completely determine the differentiable structure of $M$?

I think my first question can be rephrased as follows: if $f\colon\mathbf{R}^n\to\mathbf{R}$ has the property that for every smooth $\gamma\colon\mathbf{R}\to\mathbf{R}^n$, the composition $f\circ\gamma\colon\mathbf{R}\to\mathbf{R}$ is smooth, does this imply that $f$ is smooth?

I conjecture that if $f\colon\mathbf{R}^n\to\mathbf{R}$ has the property that for every smooth $\sigma\colon\mathbf{R}^2\to\mathbf{R}^n$, the composition $f\circ\sigma\colon\mathbf{R}^2\to\mathbf{R}$ is smooth, then $f$ is smooth.

Alexey
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  • https://math.stackexchange.com/questions/2170970/differentiability-of-composition-with-every-path-implies-differentiability – Moishe Kohan Oct 26 '23 at 18:53
  • @MoisheKohan, this seems completely unrelated. – Alexey Oct 27 '23 at 04:22
  • After thinking about it, it seems intuitively that smooth mappings $\mathbf{R}^2\to M$ must determine the differentiable structure on $M$... – Alexey Oct 27 '23 at 14:06
  • Sorry, it was a wrong reference. Your question is essentially one of real analysis: Call a map $f$ of open subsets SOP if it's percomposition with every smooth path is smooth. Suppose that $h$ is a homeomorphism of open subsets of $R^n$ such that $h$ and it's inverse are SOP. Does it follow that $h$ is smooth? – Moishe Kohan Oct 27 '23 at 15:24
  • Maybe a continuous map which is smooth on every affine line segment is smooth. I do not know any counter examples. – Moishe Kohan Oct 27 '23 at 15:29
  • @MoisheKohan, I've updated the question. – Alexey Oct 27 '23 at 15:54
  • I think you should add the continuity assumption in the last paragraph. – Moishe Kohan Oct 27 '23 at 15:56
  • @MoisheKohan, there might be two different questions: with and without the continuity assumption. – Alexey Oct 27 '23 at 15:59
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    A similar question was asked here without a satisfactory answer. Maybe you should try MO. – Moishe Kohan Oct 27 '23 at 16:10
  • @MoisheKohan, I posted the question on MO, and it received a satisfactory answer: https://mathoverflow.net/a/457443 – Alexey Oct 31 '23 at 06:50
  • Great........... – Moishe Kohan Oct 31 '23 at 09:34
  • I suggest that you write an answer containg your update. Then you can accept it and the question will disappear from the unanswered queue. – Paul Frost Nov 02 '23 at 10:24

1 Answers1

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I posted the question on MO and it received a satisfactory answer:

This is true for infinitely differentiable curves: if a map sends smooth curves to smooth curves, then it is smooth, by a theorem of Boman from 1967:

Jan Boman. Differentiability of a Function and of its Compositions with Functions of One Variable. Mathematica Scandinavica 20 (1967), 249–268.

Boman also proves (Theorem 3) that in the case of curves $C^∞$-functions in his theorem cannot be replaced by $C^k$-functions for a finite $k$.

However, if we switch from curves to surfaces, Theorem 8 in the cited paper shows that maps that send $C^k$-surfaces to $C^k$-surfaces are $C^k$-differentiable, for both finite and infinite $k$, answering the original question in the affirmative.

Alexey
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