The fundamental theorem of affine geometry on tori

TL;DR

This paper extends the classical Fundamental Theorem of Affine Geometry to n-dimensional tori, showing that bijections mapping lines to lines are affine automorphisms, thus characterizing affine maps on compact quotients.

Contribution

It establishes an analogous fundamental theorem for affine automorphisms specifically on n-dimensional tori, a compact quotient space.

Findings

Bijections of tori that map lines to lines are affine automorphisms.

The characterization holds for all n-dimensional tori with n ≥ 2.

Extends classical affine geometry results to compact quotient spaces.

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.