A smooth structure on a topological can be characterized as a sheaf of local rings, see for example the discussion here.

Q: Is there a way to characterize a Riemannian structure on a smooth manifold by a sheaf of functions?

A most likely horrible guess to clarify the type of answers I'm thinking about: define a Riemannian manifold to be a locally ringed space that locally looks like the sheaf $(\mathbb R^n, \mathcal H_g)$ where $g$ is some non degenerate symmetric positive definite matrix and $\mathcal H_g$ is the sheaf (is it even a sheaf?) that assigns to open subsets harmonic functions solving the Laplace equation given by $g$.

Please forgive my ignorance in the above, this is not my field. Just had to do a little Riemannian geometry today and was thinking whether there's a sheaf-theoretic/functor of points way to think about things.