Rigidity of Riemannian embeddings of discrete metric spaces

Let M be a complete, connected Riemannian surface and suppose that S⊂ M is a discrete subset. What can we learn about M from the knowledge of all Riemannian distances between pairs of points of S? We prove that if the distances in S correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in R², then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of Z³ that strictly contains Z²× { 0 } cannot be isometrically embedded in any complete Riemannian surface.