Innovations in Incidence Geometry

Generalized Clifford parallelisms

Andrea Blunck, Stefano Pasotti, and Silvia Pianta

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We define generalized Clifford parallelisms in PG ( 3 , F ) with the help of a quaternion skew field H over a field F of arbitrary characteristic. Moreover we give a geometric description of such parallelisms involving hyperbolic quadrics in projective spaces over suitable quadratic extensions of F .

Article information

Innov. Incidence Geom., Volume 11, Number 1 (2010), 197-212.

Received: 11 December 2008
Accepted: 15 May 2010
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E04: Quadratic forms over general fields 51A15: Structures with parallelism 51J15: Kinematic spaces

Clifford parallelism quadratic forms quadrics quaternions


Blunck, Andrea; Pasotti, Stefano; Pianta, Silvia. Generalized Clifford parallelisms. Innov. Incidence Geom. 11 (2010), no. 1, 197--212. doi:10.2140/iig.2010.11.197.

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  • A. Beutelspacher and J. Ueberberg, Bruck's vision of regular spreads or What is the use of a Baer superspace?, Abh. Math. Sem. Univ. Hamburg 63 (1993), 37–54.
  • A. Blunck, S. Pasotti and S. Pianta, Generalized Clifford parallelisms, Quad. Sem. Mat. Brescia 20/07 (2007), 1–13.
  • A. Blunck and S. Pianta, Lines in $3$-space, Mitt. Math. Ges. Hamburg 27 (2008), 189–202.
  • W. Burau, Mehrdimensionale projektive und h öhere Geometrie, Math. Monogr. 5, VEB Deutscher Verlag Wiss., Berlin, 1961.
  • P. J. Cameron, Projective and Polar Spaces, QMW Maths Notes 13, QMW, London, 1991.
  • T. De Medts, A characterization of quadratic forms of type $E\sb 6,\ E\sb 7$ and $E\sb 8$, J. Algebra 252 (2002), 394–410.
  • P. K. Draxl, Skew Fields, London Math. Soc. Lecture Note Ser. 81, Cambridge Univ. Press, Cambridge, 1983.
  • O. Giering, Vorlesungen über h öhere Geometrie, Vieweg, Braunschweig, 1982.
  • T. Grundh öfer, Reguli in Faserungen projektiver Räume, Geom. Dedicata 11 (1981), 227–237.
  • A. Hahn, Quadratic Algebras, Clifford Algebras and Arithmetic Witt Groups, Springer-Verlag, Berlin, 1994.
  • H. Havlicek, Spreads of right quadratic skew field extensions, Geom. Dedicata 49 (1994), 239–251.
  • ––––, On Plücker transformations of generalized elliptic spaces, Rend. Mat. Appl. 15 (1995), 39–56.
  • ––––, A characteristic property of elliptic Plücker transformations, J. Geom. 58 (1997), 106–116.
  • H. Havlicek and S. Pasotti, A survey on the notion of regulus in a skew space, Quad. Sem. Mat. Brescia (2003), 1–32.
  • H. Karzel, Kinematic spaces, in Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), Academic Press, London, 1973, pp. 413–439.
  • H. Karzel and H.-J. Kroll, Geschichte der Geometrie seit Hilbert, Wiss. Buchges., Darmstadt, 1988.
  • N. Knarr, Translation Planes. Foundations and construction principles, Lecture Notes in Math. 1611, Springer-Verlag, Berlin, 1995.
  • T. Y. Lam, The Algebraic Theory of Quadratic Forms, Math. Lecture Note Ser., W. A. Benjamin, Inc., Reading, Mass., 1973.
  • W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer-Verlag, Berlin, 1985.
  • B. Segre, Lectures on Modern Geometry. With an appendix by Lucio Lombardo-Radice, Edizioni Cremonese, Rome, 1961.
  • J. Tits and R. Weiss, Moufang Polygons, Springer Monogr. Math., Springer-Verlag, Berlin, 2002.
  • O. Veblen and J. Young, Projective Geometry, vol. II, Ginn and Company, Boston, 1918.
  • M.-F. Vignéras, Arithmétique des Algèbres de Quaternions, Lecture Notes in Math. 800, Springer, Berlin, 1980.