Universität zu Köln

Struve, Rolf and Struve, Horst (2019). The Thomsen-Bachmann correspondence in metric geometry II. J. Geom., 110 (1). BASEL: SPRINGER BASEL AG. ISSN 1420-8997

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Abstract

We continue the investigations of 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'. In the first part (H. Struve and R. Struve in J Geom, 2019. https://doi.org/10.1007/s00022-018-0465-8) of this paper it was shown that the Thomsen-Bachmann correspondence can be precisely stated in a framework of first-order logic. We now prove that 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 that these geometries and the theory of their group of motions are mutually faithfully interpretable (and bi-interpretable, but not definitionally equivalent). 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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Struve, Rolf UNSPECIFIED UNSPECIFIED UNSPECIFIED Struve, Horst UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-152813
DOI:	10.1007/s00022-019-0467-1
Journal or Publication Title:	J. Geom.
Volume:	110
Number:	1
Date:	2019
Publisher:	SPRINGER BASEL AG
Place of Publication:	BASEL
ISSN:	1420-8997
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/15281