论文标题
飞机上的扁平riemannian指标的分类
The classification of flat Riemannian metrics on the plane
论文作者
论文摘要
我们对二维平面上的所有平滑扁平riemannian指标进行了分类。在完整的情况下，众所周知，这些指标是欧几里得度量标准的等距。在不完整的情况下，相关且有用的自然非等法指标有很多自然的非等法指标。值得注意的是，作为主题的所有平面riemannian指标的研究和分类是文献的新事物。我们的许多研究都集中在$ e^{2φ} g_0 $的形式的共形度量上，其中$φ：\ mathbb {r}^2 \ rightarrow \ rightArrow \ mathbb {r）$是谐波函数，$ g_0 $是$ g_0 $是$ \ mathbb的标准euclidean衡量标准。我们发现，所有这些称为“谐波”的指标都来自Riemann表面。
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.
