  The collection of distances between pairs of points in a discrete set can
provide information about a Riemannian manifold. For example, in
[arXiv:2004.08621] it was shown that if $M$ is a complete and connected
Riemannian surface, and the distances between points in some discrete subset of
$M$ correspond to those of a net in $\mathbb{R}^{2}$, then $M$ is isometric to
the Euclidean plane. Whether the $n$-dimensional analog of this theorem holds
true is still unknown. In this paper we prove that if $M$ is a complete and
connected $n$-dimensional Riemannian manifold, and $X \subset M$ is a discrete
subset such that the distances between points of $X$ correspond to those of a
net in $\mathbb{R}^{n}$, then $X$ is a net in $M$. Loosely speaking, this means
that there are no mesoscopic portions of the manifold that are oblivious of the
embedding.

Metric Geometry;Differential Geometry