A logical characterization of the Euclidean plane, sphere and cylinder

Abstract

Within a mathemtaical theory, a structured set is called homogeneous,if its ‘parts’ cannot be distinguished from each other by statements of that theory. In Euclidean space geometry, among all convex surfaces and among all complete connected orientable 2‐manifolds with at least one regular point, there are only three types of homogeneous objects: the Euclidean plane, sphere and cylinder.