 

Jacob Shulkin and Wouter van Limbeek
The fundamental theorem of affine geometry on tori view print


Published: 
June 1, 2017 
Keywords: 
Fundamental theorem of geometry, affine torus, collineation, affine automorphism 
Subject: 
Primary 51M04; Secondary 51M05, 51N10, 53A15 


Abstract
The classical Fundamental Theorem of Affine Geometry states that for n≧ 2, any bijection of ndimensional 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 ndimensional torus (n≧ 2) is affine if and only if it maps lines to lines.


Acknowledgements
This work was completed as part of the REU program at the University of Michigan, for the duration of which JS was supported by NSF grant DMS1045119.


Author information
Jacob Shulkin:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
jshulkin@umich.edu
Wouter van Limbeek:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
wouterv@umich.edu

