Rigidity of Riemannian embeddings of discrete metric spaces

• 成果类型: Article
• 署名作者: Eilat, Matan; Klartag, Bo'az
• 署名单位: Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01048-y
• 发表日期: 2021
• 页码: 349-391
• 关键词: boundary distance; MANIFOLDS; SURFACES; graph

摘要

Let M be a complete, connected Riemannian surface and suppose that S subset of M is a discrete subset. What can we learn about M from the knowledge of all Riemannian distances 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 Z(2) x{0} cannot be isometrically embedded in any complete Riemannian surface.