2023/05/30 by Eilat, Matan
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2305.19415
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 ℝn. 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 ℝ2. 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.