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If $M$ is a smooth manifold and $E \to M$ a vector bundle is it possible to recover $M$ from the sheaf $\Gamma$ of smooth sections of $E$?

This might need some additional conditions on $M$ such as compactness or orientability and I'm welcome to hear if any additional requirements make this possible.

I've tried to find whether or not this is true online, but could not find any resources having this as a theorem.

Nathaniel Johonson
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    If you only consider it as a sheaf of abelian groups, no. If you consider it as a sheaf of $C^{\infty}(M)$-modules, then yes. This perspective is probably more prevalent among complex or algebraic geometers than among differential geometers. This has been discussed many times before, e.g. here, but perhaps this is not phrased in the right language. – Thorgott Dec 26 '23 at 12:00
  • Thanks for the comment. I'm not very well versed in algebraic geometry and therefore looking for a more differential geometric notion. I'm also happy with considering $\Gamma$ as a sheaf of $C^\infty(M)$-modules. @Thorgott – Nathaniel Johonson Dec 26 '23 at 15:46
  • Do you know about cocycles? Essentially, a vector bundle over $M$ is the same thing as a locally free sheaf of $\mathcal{C}^{\infty}(M)$, because both satisfy a "local triviality" condition and the global non-triviality can be expressed in terms of a cocycle that explains how the trivial parts are glued together. – Thorgott Dec 26 '23 at 16:51
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    I don't think you read the question correctly @Thorgott, OP is asking whether $M$ can be recovered from the sheaf $U\mapsto \Gamma(U,E)$, not whether $E$ can be recovered. – Quaere Verum Dec 28 '23 at 11:32
  • Ah, I did in fact misread. A sheaf only makes sense if you specify on which space the sheaf is defined, though. So is the question whether the sheaf of sections of $E$ (which is defined on the topological space underlying $M$) determines the smooth structure of $M$? – Thorgott Dec 29 '23 at 03:13

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