The classification of flat Riemannian metrics on the plane

Abstract

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of naturally occurring, non-isometric metrics that are relevant and useful. Remarkably, the study and classification of all flat Riemannian metrics on the plane—as a subject—is new to the literature. Much of our research focuses on conformal metrics of the form \(e^{2\varphi }g_0\), where \(\varphi : {\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a harmonic function and \(g_0\) is the standard Euclidean metric on \({\mathbb {R}}^2\). We find that all such metrics, which we call “harmonic,” arise from Riemann surfaces.