Publication
During the years 1940–1970, Alexandrov and the “Leningrad School” have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of curvature, which is a Radon measure. This class of surfaces has good convergence properties and is remarkably stable with respect to various geometrical constructions (gluing, etc.). It includes polyhedral surfaces as well as Riemannian surfaces of class upper C squaredC2, and both of these classes are dense families of Alexandrov’s surfaces. Any singular surface that can be reasonably thought of is an Alexandrovsurfaceandanumberofgeometricpropertiesofsmoothsurfacesextend and generalize to this class. The goal of this paper is to give an introduction to Alexandrov’s theory, to provide some examples and state some of the fundamental facts of the theory. We discuss theconformal viewpointintroducedbyYuriG.Reshetnyak andexplainhowitleads to a classification of compact Alexandrov’s surfaces.