The Intrinsic Geometry of Surfaces

By intrinsic geometry of a surface we mean those properties that depend exclusively on measurements made on the surface but not on how the surface is embedded in the ambient space. Hence, the length of a curve on a sheet of paper is an intrinsic property, because it doesn´t change when we fold the sheet; but the distance in space between two points on that same sheet is not intrinsic. In this chapter we study the intrinsic properties of surfaces and the mappings that preserve such properties (isometries).