The fundamental theorem of affine geometry on tori

2016/12/17 by Jacob Shulkin, Shulkin, Jacob, Wouter van Limbeek +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1612.05819

openalex publication_date 2016/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical Fundamental Theorem of Affine Geometry states that for n≥ 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≥ 2) is affine if and only if it maps lines to lines.

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