Porous double spaces (Q1119907)

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 set of lines \(A\in {\mathcal L}\) such that (i) \(A\|_{\ell}B\wedge A\|_ rB\) for some \(B\neq A\), and (ii) for some \(c\in P\) the incidence closure of \((c\|_{\ell}A)\cup \{c\|_ rA)\) in (P,\({\mathcal L})\) does not contain all possible left and right parallels. Main results are: Assuming \(\Phi\) nonvoid, implies that D is a prism space of dimension 3, and \(\|_{\ell}\neq \|_ r\); then either (P,\({\mathcal L})\) must be a projective space in which case \({\mathcal A}\) is a quaternion division algebra, or the 2-set S is a ruled quadric, and \({\mathcal A}\) the algebra of \(2\times 2\)-matrices over a commutative field, cf. the author [Symp. Math. 11, Algebra commut., Geometria, Convegni 1971/1972, 413-439 (1973; Zbl 0284.50017), and loc. cit.], the author and G. Kist [loc. cit.]. For \(\Phi\) void, the following disjunction is proved to be complete: (I) \({\mathcal L}={\mathcal L}_ 1\) (i.e. (P,\({\mathcal L})\) is an affine space); (IIa) \({\mathcal L}\neq {\mathcal L}_ 1\neq \emptyset \wedge {\mathcal L}_ 2=\emptyset\) (i.e. (P,\({\mathcal L})\) is a proper slit space); (IIb) \({\mathcal L}\neq L_ 1 \wedge {\mathcal L}_ 2\neq \emptyset\); (IIc) \({\mathcal L}={\mathcal L}_ 0\) (i.e. (P,\({\mathcal L})\) is a projective space). It is refined by assuming (E) \(\|_{\ell}=\|_ r\) or its negation (\(\neg E)\). The results are \((\Delta:=\dim (P,{\mathcal L})):\) (I) \(\wedge (\neg E) \Rightarrow \Delta \geq 3\), (P,\({\mathcal L})\) is pappian; (IIa) \(\Rightarrow \Delta \geq 3\), dim(S)\(\geq 1\), \(co\dim (S)>1\); (IIa) \(\wedge (E) \Rightarrow characteristic\) of \((\tilde P,\tilde {\mathcal L})=2\); (IIb) \(\Rightarrow {\mathcal L}_ 0=\emptyset\), \({\mathcal L}_ 1\neq \emptyset\), (\(\neg E)\), the 2-set S is the union of two distinct hyperplanes in (P,\({\mathcal L})\), i.e. \((P,{\mathcal L})\) is a stripe space; (IIc) \(\Rightarrow\) (E), characteristic of \((P,{\mathcal L})=2\). Furthermore, D is a prism space if one of the following conditions holds: (I) \(\wedge (\neg E)\), (I) \(\wedge\) (E) \(\wedge(\Delta \geq 3\) or \((P,{\mathcal L})\) is a translation plane), (IIa), (IIb) \(\wedge \Delta \geq 3\); (IIb) \(\wedge (P,{\mathcal L})\) is the dual of a planar nearfield plane, (IIc). The corresponding kinematic algebras are identified among the types occuring in the classification mentioned above.