The fundamental theorems of affine and projective geometry revisited

Shiri Artstein-Avidan, Boaz A. Slomka

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n + 2 fixed projective points in real n-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

Original languageEnglish
Article number1650059
JournalCommunications in Contemporary Mathematics
Volume19
Issue number5
DOIs
StatePublished - 1 Oct 2017

Keywords

  • Fundamental theorem
  • affine-additive maps
  • collineations

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