Rigidity of Riemannian embeddings of discrete metric spaces

Abstract: 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.

analysis of PDEsdifferential geometrymetric geometry

Audience: researchers in the topic

( paper )

---

Online asymptotic geometric analysis seminar

Series comments: The link: technion.zoom.us/j/99202255210

If you are interested in giving a talk, please let one of the organizers know. Also, please suggest speakers which you would like to hear talk. Most talks are 50 minutes, but some 20-minute talks will be paired up as well. The talks will be video recorded conditioned upon the speakers' agreement.

---