Rigidity of Riemannian embeddings of discrete metric spaces

Abstract

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S}\subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We prove that if the distances in $\mathcal{S}$ correspond to the distances in a $2$-dimensional lattice, or more generally in an arbitrary net in ${\mathbb{R}}^2$, 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 ${\mathbb{Z}}^3$ that strictly contains ${\mathbb{Z}}^2\times \{0\}$ cannot be isometrically embedded in any complete Riemannian surface.

36 pages

References in corpus (1)

• Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics