Euclidean nets under isometric embeddings

Abstract

Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$. What can then be derived about the geometry of $M$? In arXiv:2004.08621 it was shown that if $n=2$ then $M$ is isometric to $\mathbb{R}^{2}$. In this paper we show two consequential geometric properties that the manifold $M$ shares with the Euclidean space in any dimension. The first property is that $X$ is a net with respect to the Riemannian distance in $M$. The second property is that all geodesics in $M$ are distance minimizing, and there are no conjugate points in $M$. This demonstrates the possibility of inferring infinitesimal qualities from discrete data, even in higher dimensions. As a corollary we obtain that the large-scale geometry of $M$ is asymptotically Euclidean.