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.