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

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.Comment: 5 pages, 0 figure