论文标题
简单奇点的黎曼表面
Riemannian Surfaces with Simple Singularities
论文作者
论文摘要
在本说明中，我们讨论了带有一组单数点的Riemannian表面的几何形状。我们假设保形结构通过奇异性扩展，并且曲率是可以集成的。这样的点称为\ emph {简单的奇点}。我们首先使用（真实）除数的概念在本地描述它们，然后在全球范围内描述它们。我们制定了高斯河网配方，并将其与某些渐近等级比率相关联。我们证明了平面度量的分类定理，在紧凑的表面上具有简单的奇异性，并讨论了伯格 - 尼伦贝格的问题，在具有除数的表面上。我们最终讨论了与球形多面体的关系。
In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple singularities}. We first describe them locally and then globally using the notion of (real) divisor. We formulate a Gauss-Bonnet formula and relate it to some asymptotic isoperimetric ratio. We prove a classifications theorem for flat metrics with simple singularities on a compact surface and discuss the Berger--Nirenberg Problem on surfaces with a divisor. We finally discuss the relation with spherical polyhedra.
