论文标题
离散度量空间的Riemannian嵌入的刚度
Rigidity of Riemannian embeddings of discrete metric spaces
论文作者
论文摘要
令$ m $为完整的，连接的Riemannian Surface，并假设$ \ Mathcal {s} \子集M $是离散的子集。我们可以从$ \ Mathcal {s} $的点之间的所有距离的知识中学到什么$ m $？我们证明，如果$ \ MATHCAL {S} $中的距离对应于$ 2 $维晶格中的距离，或者更一般地在$ \ mathbb {r}^2 $中的任意网中，则$ m $是欧几里得平面的等值。因此，我们发现某些离散度量空间的Riemannian嵌入非常僵化。推论的是，严格包含$ \ mathbb {z}^2 \ times \ {0 \} $的$ \ mathbb {z}^3 $的子集不能被嵌入在任何完整的riemannian表面中。
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.
