Let M be a complete, connected Riemannian surface and
suppose that S is a discrete subset of M. What can we learn about M
from the knowledge of all distances in the surface 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 R^2, 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 Z^3 that strictly
contains a two-dimensional lattice cannot be isometrically embedded in
any complete Riemannian surface. This is a joint work with M. Eilat.