TY - JOUR

T1 - The fundamental theorems of affine and projective geometry revisited

AU - Artstein-Avidan, Shiri

AU - Slomka, Boaz A.

N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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.

AB - 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.

KW - Fundamental theorem

KW - affine-additive maps

KW - collineations

UR - http://www.scopus.com/inward/record.url?scp=84986601861&partnerID=8YFLogxK

U2 - 10.1142/S0219199716500590

DO - 10.1142/S0219199716500590

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AN - SCOPUS:84986601861

SN - 0219-1997

VL - 19

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 5

M1 - 1650059

ER -