Differential Geometry and Geometric Structures
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Geometry over Skew Fields

We investigate reguli, Segre manifolds, normal rational curves, and their generalisations in projective spaces over arbitrary skew fields.


 

Twisted cubic
Twisted cubic

Normal Rational Curves

Normal rational curves in pappian projective spaces (with a commutative ground field F) allow several equivalent definitions. Some of them no longer do make sense if the ground field is allowed to be non-commutative, others will work without being equivalent. Thus there are various concepts of normal curves over skew fields: One may distinguish non-degenerate and degenerate ones, or normal curves of first and second kind. The details are rather involved.

Twisted Cubic. The picture on the left hand side shows a twisted cubic (in red) on a parabolic cylinder together with an osculating tetrahedron (yellow vertices and green edges). The tetrahedron arises by intersecting the tangent at one point of the curve with the osculating plane at an other point, and vice versa. Twisted cubics are the non-degenerate normal rational curves in 3-dimensional projective spaces.

Geometric Algebra

We aim at finding geometric counterparts for algebraic properties of the ground field F in terms of normal rational curves. For example, (non-)commutativity of F, the degree over the centre of F, the conjugacy class and the centraliser of an element of F are now well understood in terms of non-degenerate normal curves of first kind. Another problem is to investigate how the classical results on tangents, osculating subspaces, or automorphic collineations of normal curves will alter in the general case.


Publications

  • R. Riesinger: Entartete Steinerkegelschnitte in nichtpapposschen Desarguesebenen, Monatsh. Math. 89 (1980), 243-251.
  • R. Riesinger: Normkurven in endlichdimensionalen Desarguesräumen, Geom. Dedicata 10 (1981), 427-449.
  • R. Riesinger: Geometrische Überlegungen zum rechten Eigenwert-Problem über Schiefkörpern, Geom. Dedicata 12 (1982), 401-405.
  • H. Havlicek: Normisomorphismen und Normkurven endlichdimensionaler projektiver Desargues-Räume, Monatsh. Math. 95 (1983), 203-218.
  • H. Havlicek: Die automorphen Kollineationen nicht entarteter Normkurven, Geom. Dedicata 16 (1984), 85-91.
  • H. Havlicek: Erzeugnisse projektiver Bündelisomorphismen, Ber. math.- stat. Sekt. Forschungszentr. Graz Nr. 215 (1984).
  • H. Havlicek: Applications of results on generalized polynomial identities in desarguesian projective spaces. In: R. Kaya, P. Plaumann, K. Strambach (Eds.): Rings and Geometry, D. Reidel, Dordrecht, 1985.
  • H. Havlicek: Durch Kollineationsgruppen bestimmte projektive Räume, Beitr. Algebra Geom. 27 (1988), 175-184.
    Preprint (PDF)
  • H. Havlicek: Lineare Geradenkomplexe über Schiefkörpern, Arch. Math. 51 (1988), 181-187.
  • H. Havlicek: Degenerate conics revisited, J. Geometry 38 (1990), 42-51.
    Preprint (PDF)
  • H. Havlicek: Baer subspaces within Segre manifolds, Results Math. 23 (1993), 322-329.
    Preprint (PDF)
  • H. Havlicek: Affine circle geometry over quaternion skew fields, Discrete Math. 174 (1997), 153-165.
    Preprint (PDF)
  • A. Blunck: Reguli and chains over skew fields, Beiträge Algebra Geometrie 41 (2000), 7-21.
  • S. Pasotti, H. Havlicek: A Survey on the Notion of Regulus in a Skew Space, Quaderni del Seminario Matematico di Brescia 17 (2003), 32 pp. (electronic).
    Preprint (PDF) | Journal

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Andrea Blunck
Hans Havlicek
Rolf Riesinger
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