Riemannian Surfaces with Simple Singularities

TL;DR

This paper explores the geometry of Riemannian surfaces with simple singularities, providing classifications, formulas, and relations to polyhedral structures, under the assumption of extended conformal structures and integrable curvature.

Contribution

It introduces a classification of flat metrics with simple singularities and formulates a Gauss-Bonnet type relation for such surfaces.

Findings

Classification theorem for flat metrics with simple singularities

Gauss-Bonnet formula adapted to singular surfaces

Relation between simple singularities and spherical polyhedra

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.