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arXiv · 2201.03359

Riemannian Surfaces with Simple Singularities

Marc Troyanov
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Abstract

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.

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Marc Troyanov. 2022-01-10. Riemannian Surfaces with Simple Singularities. https://arxiv.org/abs/2201.03359

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