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Journal of Geometry

, 110:9 | Cite as

The Thomsen–Bachmann correspondence in metric geometry I

  • Rolf Struve
  • Horst Struve
Article
  • 10 Downloads

Abstract

We study in this two part paper the Thomsen–Bachmann correspondence between metric geometries and groups which is often summarized by the phrase ‘Geometry can be formulated in the group of motions’. We show that (1) the correspondence can be precisely stated in a framework of first-order logic, (2) the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley–Klein geometries, isotropic and equiform geometries, and (3) these geometries and the theory of their group of motions are not only mutually interpretable but also bi-interpretable. Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method. In the first part of the paper we introduce the fundamental logical and geometric notions and show that the Thomsen–Bachmann correspondence can be rephrased in first-order logic by ‘The theory of the group of motions is a conservative extension of the underlying geometric theory’.

Keywords

Thomsen–Bachmann correspondence calculus of reflections reflection group Bachmann group absolute geometry Cayley–Klein geometries sentential equivalence bi-interpretability symmetric space 

Mathematics Subject Classification

03B30 51F05 51F15 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BochumGermany
  2. 2.Seminar für Mathematik und ihre DidaktikUniversität zu KölnCologneGermany

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