A smooth structure on a topological can be characterized as a sheaf of local rings, see for example [the discussion here](https://mathoverflow.net/questions/88056/is-there-a-sheaf-theoretical-characterization-of-a-differentiable-manifold).

**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.