Porous double spaces
type: Item
instance of: scholarly article
title: Porous double spaces
zbMATH Open document ID: 0671.51001
DOI: 10.1007/BF01224235
published in: Journal of Geometry
publication date: 1989-01-01T00:00:00Z
review text: The concept of double space was defined by the author, \textit{H.-J. Kroll} and \textit{K. Sörensen} [J. Reine Angew. Math. 262-263, 153-157 (1973; Zbl 0265.50003)] as a generalization of Clifford parallelism in real projective 3-space, cf. the author and \textit{H.-J. Kroll} [Geschichte der Geometrie seit Hilbert (Wissenschaftliche Buchgesellschaft, Darmstadt) (1988)]. A subset \(S\neq \tilde P\) of a projective space \((\tilde P,\tilde {\mathcal L})\) such that \(| X\cap S| \leq 2\) or \(X\subseteq S\) for all \(X\in \tilde {\mathcal L}\) is called a 2-set; for \(P:=\tilde P\setminus S\), \(| X| \geq 4\) \(\forall X\in \tilde {\mathcal L}\), \(i\in \{0,1,2\}\), \({\mathcal L}_ i:=\{X\cap P:\) \(X\in \tilde {\mathcal L}\), \(X\varsubsetneq S\), \(| X\setminus P| =i\}\), \({\mathcal L}:={\mathcal L}_ 0\cup {\mathcal L}_ 1\cup {\mathcal L}_ 2\) the trace geometry (P,\({\mathcal L})\) is a so-called porous space, cf. \textit{G. Kist} [Result. Math. 3, 192-211 (1980; Zbl 0477.51004)] for an internal description of such spaces. The author determines and classifies those porous spaces \(D:=(P,{\mathcal L},\|_{\ell},\|_ r)\) which are double spaces at the same time and satisfy \(| X| \geq 5\) \(\forall X\in {\mathcal L}\). \((\|_{\ell}\), \(\|_ r\) denote the left resp. right parallelism given on L, \((a\|_{\ell}X)\) the left parallel to X containing the point a.) To any double space which satisfies the prism axioms (:=prism space) there corresponds a kinematic algebra \({\mathcal A}\), cf. \textit{L. Bröcker} [Geom. Dedicata 1, 241-268 (1973; Zbl 0249.50014)], and the author, H.-J. Kroll, and K. Sörensen [loc. cit.]. The author uses the classification of kinematic algebras, - cf. the author [Abh. Math. Sem. Univ. Hamburg 41, 158-171 (1974; Zbl 0296.50019)], the author and \textit{G. Kist} [Rings and Geometry, Proc. NATO Adv. Study Inst., Istanbul/Turkey 1984, NATO ASI Ser., Ser. C 160, 437-509 (1985; Zbl 0598.51012)] - to carry out the project: Let \(\Phi\) denote the se…
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51J15
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 16Y99
zbMATH DE Number: 4098171
zbMATH Keywords: porous double space
zbMATH Keywords: kinematic space
zbMATH Keywords: prism space
zbMATH Keywords: kinematic algebra
author: Helmut Karzel
reviewed by: Horst Szambien
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4098171