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arXiv 2001.03888math.DG
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平面上平坦黎曼度量的分类

The classification of flat Riemannian metrics on the plane

Vincent E. Coll,, Lee B. Whitt

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摘要：

我们对二维平面上的光滑平坦黎曼度量进行了分类。在完全情况下，众所周知，这些度量与欧几里得度量等距。在不完整的情况下，有大量的自然产生的，非等距指标是相关的和有用的。值得注意的是，研究和分类的所有平面黎曼度量的平面-作为一个主题-是新的文献。我们的研究主要集中在$e^{2φ}g_0$形式的共形度量上，其中$φ: \mathbb{R}^2 \rightarrow \mathbb{R)$是调和函数，$g_0$是$\mathbb{R}^2$上的标准欧几里得度量。我们发现，所有这样的度量，我们称之为“调和”，产生于黎曼曲面。

英文摘要：

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 $e^{2φ}g_0$, where $φ: \mathbb{R}^2 \rightarrow \mathbb{R)$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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