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 Riemannian distances 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.

令M为一个完整的连通黎曼曲面，并假定\（{\ mathcal {S}} \ subset M \）是一个离散子集。我们可以从\（{\ mathcal {S}} \）两对点之间的所有黎曼距离的知识中学到M吗？我们证明，如果\（{\ mathcal {S}} \）中的距离对应于二维晶格中的距离，或更普遍地，它对应于\（{\ mathbb {R}} ^ 2 \，则M与欧几里得平面等距。因此，我们发现某些离散度量空间的黎曼嵌入非常僵化。结果是\（{\ mathbb {Z}} ^ 3 \）的子集严格包含\（{{mathbb {Z}} ^ 2 \ times \ {0 \} \）不能等轴嵌入到任何完整的黎曼曲面中。