Inverse problem for geodesic

Question

If I know the expressions for geodesic distance between any points $x$ and $y$:

$$L=L(x^\mu,y^\nu) \ .$$

How do I find the metric of the corresponding space?

Answer 1

$$g_{\mu\nu}(p) = -\frac{1}{2}\partial^{(p)}_\mu \partial^{(q)}_\nu L(p,q)^2|_{p=q}$$ In every sufficiently small coordinate patch around $p$. The proof is not so easy, even if the identity is easily expected from the analogous result in the flat manifold case. It is a property of the so called Synge world function $L(p,q)^2$ defined in convex normal neighbourhoods of a Riemannian/Lorentzian manifold $(M,g)$ or in sufficiently small neighbourhoods of the diagonal of $M\times M$.

There is a section completely devoted to it in my lecture notes on geometric methods in mathematical physics here.