Saglam, Ismail2025-01-062025-01-0620191300-00981303-614910.3906/mat-1806-232-s2.0-85064173419https://doi.org/10.3906/mat-1806-23https://search.trdizin.gov.tr/tr/yayin/detay/335636https://hdl.handle.net/20.500.14669/2413We 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.eninfo:eu-repo/semantics/openAccessFlat metricthe Gauss-Bonnet formulasurfaces with puncturesthe Hopf-Rinow theoremArticle8322Q281333563643WOS:000462461500018Q3