The classification of flat Riemannian metrics on the plane

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-arising, 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.