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 joint work with Bo'az Klartag we showed that if $n=2$ then $M$ is isometric to $\mathbb{R}^{2}$. Moreover, in any dimension the topology of the manifold is determined, meaning that it must be diffeomorphic to the flat $\mathbb{R}^{n}$. In a more recent work, we were able to show additional 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 $M$.
In this talk I will present the setting of the problem, the results and several corollaries through special cases and (counter-)intuitive examples, and discuss the proof techniques.
