A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic (along with $\mathcal{F}|_U$) to an open subset $V$ of $\mathbb{R}^n$ (along with $\mathcal{O}|_V$, where $\mathcal{O}$ is the sheaf of smooth functions on $\mathbb{R}^n$). I think we might also need to say that this satisfies a smooth-coordinate-change axiom, although maybe that's already tied up in the definition of a sheaf. In any case, here is my question:
Given two smooth manifolds
$(M,\mathcal{F})$ and
$(N,\mathcal{G})$, is there an easy
way to write the sheaf of functions on
$M\times N$ without reference to
coordinate neighborhoods?
I'm wondering this because in one of my classes we defined smooth manifolds in this way (and we defined analytic and holomorphic manifolds similarly). It seems like some people are very fond of this alternative definition because it doesn't refer to an atlas, which at first seems like it's an inherent part of the structure of the manifold. So okay fine, everyone loves a canonical definition. However, this is only going to be useful as long as we can tell our whole story in this canonical language. In class, the only way the professor was able to give the sheaf on the product was by breaking down and using coordinates. (Admittedly, he was on the spot and presumably unprepared for the question.)
This also suggests the broader, more open-ended question:
Are there longer-run advantages to the above definition (compared to the usual definition involving an atlas and perhaps a maximal atlas)?
Best Answer
Smooth manifolds are affine, thus the sheaf of smooth functions is determined by its global sections. Now C^∞(M×N)=C^∞(M)⊗C^∞(N). The tensor product here is the projective tensor product of complete locally convex Hausdorff topological algebras.