Zoom link: https://princeton.zoom.us/j/594605776

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.