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Given a compact Riemannian manifold $(M,g)$, is there a subring of $C^{\infty}(M)$ that determines the isomorphism class of $(M,g)$? (In the same way that $C^{\infty}(M)$ determines the diffeomorphism class via the max spectrum.)

Same question for compact metric spaces, more generally.

I am not sure what condition to ask for on these functions... maybe something like $d(f(x), f(y)) \leq d(x,y)$? (Probably this has to be fiddled with to actually get a ring.)

What about $(M, \nabla)$, for $\nabla$ some connection?

If not ... why not?

This is just following the functions on a space determine its geometry philosophy that people in algebraic geometry like.

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    $\begingroup$ Someone told me that the right notion in this case is something called a spectral triple: IIRC it's the sections of $L^2$ sections of the bundle $\Omega^n(M)$ of forms, along with the operator $d+d^*$ on this, considered as a $C^\infty(M)$ module. I think it is a hard theorem that this recovers the isometry class of the Riemannian manifold. (I don't want to write an answer because I really know nothing about this, just stuff I've heard in passing.) In any case, the keyword 'spectral triple' should help your searches. $\endgroup$ – user98602 Jan 2 '16 at 22:56
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This appears to answer the question: https://mathoverflow.net/questions/301869/sheaf-theoretically-characterize-a-riemannian-structure

In particular, the right idea is to consider the sheaf of harmonic functions of the Laplace-Beltrami operator. (Which is only a sheaf of vector spaces, not of rings.)

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