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Let $M,N$ be smooth manifolds and $f:N\to M$ a smooth map. Denote by $\mathcal{O}_M,\mathcal{O}_N$ the corresponding sheaves of smooth functions. If we regard $\mathcal{O}_M$ as the sheaf of smooth sections of the trivial bundle $M\times \mathbb{R}$ over $M,$ does the sheaf of sections of the pullback bundle coincide with the pullback sheaf $f^*\mathcal{O}_M.$ More generally does this happen for any vector bundle? I think it does, although wikipedia states that in general the sheaf of sections of the pullback bundle only coincides with the inverse image sheaf. Nonetheless, since pullback of a trivial vector bundle is a trivial vector bundle. Then the sheaf of sections of the pullback of $M\times \mathbb{R}$ must coincide with $\mathcal{O}_N.$ But this seems weird. Edit: I made a mistake in the statement the map is supposed to go like $f:N \to M.$

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