Abstract
Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space \({{\rm PG}(3,\mathbb{R})}\) whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein’s definition). Our new access to the topic ‘Clifford parallelism’ is free of complexification and involves Klein’s correspondence λ of line geometry together with a bijective map γ from all regular spreads of \({{\rm PG}(3,\mathbb{R})}\) onto those lines of \({{\rm PG}(5,\mathbb{R})}\) having no common point with the Klein quadric; a regular parallelism P of \({{\rm PG}(3,\mathbb{R})}\) is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of \({{\rm PG}(3,\mathbb{R})}\) are Clifford (D3 = dimensionality definition). Submission of (D2) to λ−1 yields a complexification free definition of a Clifford parallelism which uses only elements of \({{\rm PG}(3,\mathbb{R})}\): A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group \({Aut_e({\bf P}_{\bf C})}\) of all automorphic collineations and dualities of the Clifford parallelism P C and show \({Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}\).
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References
Betten D.: Nicht-desarguessche 4-dimensionale Ebenen. Arch. Math. 21, 100–102 (1970)
Betten D., Riesinger R.: Topological parallelisms of the real projective 3-space. Result. Math. 47, 226–241 (2005)
Betten D., Riesinger R.: Constructing topological parallelisms of PG\({(3,\mathbb{R})}\) via rotation of generalized line pencils. Adv. Geom. 8, 11–32 (2008)
Betten D., Riesinger R.: Generalized line stars and topological parallelisms of the real projective 3-space. J. Geom. 91, 1–20 (2008)
Betten D., Riesinger R.: Hyperflock determining line sets and totally regular parallelisms of \({{\rm PG}(3,\mathbb{R})}\). Mh. Math. 161, 43–58 (2010)
Betten D., Riesinger R.: Parallelisms of \({{\rm PG}(3,\mathbb{R})}\) composed of non-regular spreads. Aequationes Math. 81, 227–250 (2011)
Blunck A., Pasotti St., Pianta S.: Generalized Clifford parallelisms. Innov. Incidence Geom. 11, 197–212 (2010)
Brauner H.: Geometrie projektiver Räume I. Bibliographisches Institut, Mannheim (1976)
Brauner H.: Geometrie projektiver Räume II. Bibliographisches Institut, Mannheim (1976)
Clifford W.K.: Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. (1) 4, 381–395 (1873)
Giering O.: Vorlesungen über höhere Geometrie. Vieweg, Braunschweig-Wies-baden (1982)
Grundhöfer T., Löwen R.: Linear topological geometries. In: Buekenhout, F. (eds) Handbook of incidence geometry., Elsevier, Amsterdam (1995)
Hirschfeld J.W.P.: Finite projective spaces of three dimensions. Clarendon Press, Oxford (1985)
Hughes D.R., Piper F.C.: Projective planes. Springer, New York (1973)
Johnson N.L.: Parallelisms of projective spaces. J. Geom. 76, 110–182 (2003)
Johnson N.L.: Combinatorics of spreads and parallelisms. Pure and Applied Mathematics (Boca Raton), vol. 295. CRC Press, Boca Raton (2010)
Karzel H., Kroll H.-J.: Eine inzidenzgeometrische Kennzeichnung projektiver kinematischer Räume. Arch. Math. (Basel) 26, 107–112 (1975)
Karzel H., Kroll H.-J.: Geschichte der Geometrie seit Hilbert. Wiss. Buchges., Darmstadt (1988)
Karzel H., Kroll H.-J., Sörensen K.: Invariante Gruppenpartitionen und Doppelräume. J. Reine Angew. Math. 262/263, 153–157 (1973)
Karzel H., Kroll H.-J., Sörensen K.: Projektive Doppelräume. Arch. Math. (Basel) 25, 206–209 (1974)
Kroll H.-J.: Bestimmung aller projektiven Doppelräume. Abh. Math Sem. Univ. Hamburg 44, 139–142 (1975)
Lenz H.: Vorlesungen über projektive Geometrie. Akad. Verlagsges. Geest & Portig, Leipzig (1965)
Pickert G.: Analytische Geometrie. 7. Auflage. Akad. Verlagsges. Geest & Portig, Leipzig (1976)
Pasotti St.: Regular parallelisms in kinematic spaces. Discrete Math. 310, 3120–3125 (2010)
Pottmann H., Wallner J.: Computational Line Geometry. Springer, Berlin (2001)
Riesinger R.: Beispiele starrer, topologischer Faserungen des reellen projektiven 3-Raums. Geom. Dedicata 40, 145–163 (1991)
Salzmann H.R., Betten D., Grundhöfer T., Hähl H., Löwen R., Stroppel M.: Compact projective planes. De Gruyter, Berlin (1995)
Schaal H.: Lineare Algebra und Analytische Geometrie II. Vieweg, Braunschweig (1976)
Schaal H., Glässner E.: Lineare Algebra und Analytische Geometrie III. Vieweg, Braunschweig (1977)
Tyrrell J.A., Semple J.G.: Generalized Clifford parallelism. University Press, Cambridge (1971)
Veblen O., Young J.W.: Projective geometry I. Blaisdell Publishing company, New York (1946)
Veblen O., Young J.W.: Projective geometry II. Blaisdell Publishing company, New York (1946)
Weisstein, E.W.: “Rotation Matrix”. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/RotationMatrix.html
Wong Y.-C.: Clifford parallels in elliptic (2n−1)-spaces and isoclinic n-planes in Euclidean 2n-space. Bull. Am. Math. Soc. 66, 289–293 (1960)
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Betten, D., Riesinger, R. Clifford parallelism: old and new definitions, and their use. J. Geom. 103, 31–73 (2012). https://doi.org/10.1007/s00022-012-0118-2
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DOI: https://doi.org/10.1007/s00022-012-0118-2