Let $M$ be a complete, connected, $n$-dimensional Riemannian manifold, and suppose that $X \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of Riemannian distances between points of $X$? We focus on the case where the distances in $X$ correspond to the distances between points of a net in $R^n$. In the two-dimensional settings, i.e. when $n=2$, we show that the geometry of $M$ is completely determined, meaning that it must be isometric to the Euclidean plane. In higher dimensions, the topology of the underlying manifold can be recovered, so that $M$ is diffeomorphic to $R^n$. Moreover, we prove that $X$ must be a net in $M$ as well.
Based on joint work with Bo'az Klartag.