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.

我们在二维平面上对所有平滑平坦的黎曼度量进行分类。在完整情况下，众所周知这些度量与欧几里得度量是等距的。在不完整的情况下，存在大量自然而相关且有用的非等距度量。值得注意的是，平面上所有平坦黎曼度量的研究和分类（作为一个主题）对文献来说都是新的。我们的许多研究都集中于形式为\（e ^ {2 \ varphi} g_0 \）的保形度量，其中\（\ varphi：{\ mathbb {R}} ^ 2 \ rightarrow {\ mathbb {R}} \）是谐波函数，\（g_0 \）是\（{\ mathbb {R}} ^ 2 \）的标准欧几里得度量。我们发现所有这些度量（我们称为“谐波”）都源自黎曼曲面。