Support by the Project of Stable Support for Youth Team in Basic Research Field,CAS(Grant No.YSBR-001);NSFC(Grant Nos.12271495,11971450 and 12071449).
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...