The fundamental theorem of affine geometry on tori

Abstract

The classical Fundamental Theorem of Affine Geometry states that for $n\geq 2$, any bijection of $n$-dimensional Euclidean space that maps lines to lines (as sets) is given by an affine map. We consider an analogous characterization of affine automorphisms for compact quotients, and establish it for tori: A bijection of an n-dimensional torus ($n\geq 2$) is affine if and only if it maps lines to lines.