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The fundamental theorems of affine and projective geometry revisited

Shiri Artstein-Avidan

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

E-mail Address: [email protected]

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and

Boaz A. Slomka

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

E-mail Address: [email protected]

Corresponding author.

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https://doi.org/10.1142/S0219199716500590Cited by:3 (Source: Crossref)

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.

Keywords:

AMSC: 14R10, 51A05, 51A15

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