We prove that each complete flat cone metric on a surface with regular or irregular punctures can be triangulated with finitely many types of triangles. We derive the Gauss-Bonnet formula for this kind of cone metrics. In addition, we prove that each free homotopy class of paths has a geodesic representative.
Flat metric, the Gauss-Bonnet formula, surfaces with punctures, the Hopf-Rinow theorem
"Complete flat cone metrics on punctured surfaces,"
Turkish Journal of Mathematics: Vol. 43:
2, Article 18.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss2/18