The fundamental theorem of affine geometry

Authors

  • J.B. Sancho de Salas Departamento de Matemáticas, Universidad de Extremadura 06006 Badajoz, Spain

DOI:

https://doi.org/10.17398/2605-5686.38.2.221

Keywords:

Fundamental Theorem, semiaffine morphisms, parallel morphisms

Abstract

We deal with a natural generalization of the classical Fundamental Theorem of Affine Geometry to the case of non bijective maps. This extension geometrically characterizes semiaffine morphisms. It was obtained by W. Zick in 1981, although it is almost unknown. Our aim is to present and discuss a simplified proof of this result.

Downloads

Download data is not yet available.

References

S. Artstein-Avidan, B.A. Slomka, The fundamental theorems of affine and projective geometry revisited, Commun. Contemp. Math. 19 (5) (2017), 1650059, 39 pp.

M.K. Bennett, “ Affine and projective geometry ”, John Wiley & Sons, Inc., New York, 1995.

M. Berger, “ Geometry I ”, Springer-Verlag, Berlin, 1987.

A. Beutelspacher, U. Rosenbaum, “ Projective geometry: from foundations to applications ”, Cambridge University Press, Cambridge, 1998.

A. Chubarev, I. Pinelis, Fundamental theorem of geometry without the 1-to-1 assumption, Proc. Amer. Math. Soc. 127 (9) (1999), 2735 – 2744.

C.A. Faure, An elementary proof of the fundamental theorem of projective geometry, Geom. Dedicata 90 (2002), 145 – 151.

C.A. Faure, A. Frölicher, Morphisms of projective geometries and semilinear maps, Geom. Dedicata 53 (1994), 237 – 262.

R. Frank, Ein lokaler Fundamentalsatz für Projektionen, Geom. Dedicata 44 (1992), 53 – 66.

J. Frenkel, “ Géométrie pour l’élève-professeur ”, Hermann, Paris, 1973.

E. Kamke, Zur Definition der affinen Abbildung, Jahresbericht D.M.V. 36 (1927), 145 – 156.

H. Karzel, H.J. Kroll, “ Geschichte der Geometrie seit Hilbert ”, Wissenschaftliche Buchgesellschaft, Darmstadt, 1988.

H. Havlicek, A generalization of Brauner’s theorem on linear mappings, Mitt. Math. Sem. Giessen 215 (1994), 27 – 41.

H. Havlicek, Weak linear maps - A survey, in “ 107 Jahre Drehfluchtprinzip ”, (Proceedings of the Geometry Conference held in Vorau, June 1 – 6, 1997. Edited by O. Roschel and H. Vogler), Technische Universität Graz, Graz, 1999, 76 – 85.

H. Lenz, Einige Anwendungen der projektiven Geometrie auf Fragen der Flächentheorie, Math. Nachr. 18 (1958), 346 – 359.

A. Reventós, “ Affine maps, euclidean motions and quadrics ”, Springer, London, 2011.

O. Tamaschke, “ Projektive Geometrie II. Mit einer Einführung in die affine Geometrie ”, Bibliographisches Institut, Mannheim-Vienna-Zürich, 1972.

W. Wenzel, An axiomatic system for affine spaces in terms of points, lines, and planes, J. Geom. 107 (2016), 207 – 216.

W. Zick, Parallentreue Homomorphismen in affinen Räumen, Inst. f. Math., Universität Hannover Preprint Nr. 129 (1981), 1 – 21.

W. Zick, Der Satz von Martin in Desargues’schen affinen Räumen, Inst. f. Math., Universität Hannover Preprint Nr. 134 (1981), 1 – 13.

Downloads

Published

2023-12-01 — Updated on 2023-12-01

Versions

Issue

Section

Geometry

How to Cite

The fundamental theorem of affine geometry. (2023). Extracta Mathematicae, 38(2), 221-235. https://doi.org/10.17398/2605-5686.38.2.221