Riemannian Surfaces with Simple Singularities

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 simple sin-gularities.Wefirstdescribethemlocallyandthengloballyusingthenotionof(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.