The Gauss-Bonnet formula of a conical metric on a compact Riemann surface

TL;DR

This paper generalizes the Gauss-Bonnet formula to conical metrics on compact Riemann surfaces with Lebesgue integrable curvature and constructs examples with non-integrable curvature.

Contribution

It extends the classical Gauss-Bonnet formula to broader conical metrics and explicitly constructs metrics with non-integrable curvature.

Findings

Generalized Gauss-Bonnet formula for conical metrics with integrable curvature

Explicit examples of conical metrics with non-integrable curvature

Conditions under which the classical formula holds or fails

Abstract

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.