On the existence of Bertrand pairs

  • A. M. d’Azevedo BredaEmail author
  • F. J. Craveiro de Carvalho
  • Bernd Wegner
Original Paper


In this note we introduce the Bertrand Group for a curve. While there is no mention of this group in it, Saban (Simon Stevin 57:37–45, 1983) contains a statement which would mean that for a simple, closed, twisted curve that group would be trivial. However there appears to be a flaw in the proof given there. Here we are able to show that for a simple, twisted curve that group is either trivial or Z 2. If this latter group can occur we do not know. We also show that for some classes of curves the non-triviality of that group forces the curve to be plane.


Curves in Euclidean space Frenet-Serret frame Bertrand mates 

Mathematics Subject Classification (2000)

53A04 51H30 53A55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Craveiro de Carvalho F.J., Robertson S.A.: Self-Parallel curves. Math. Scand. 65, 67–74 (1989)MathSciNetzbMATHGoogle Scholar
  2. Craveiro de Carvalho F.J., Wegner B.: Diametrical submanifolds. Periodica Mathematica Hungarica 40, 1–11 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. Hsiung C.-C.: A first couse in differential geometry. Wiley, Hoboken (1981)Google Scholar
  4. Irwin M.C.: Transnormal circles. J. London Math. Soc. 42, 545–552 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Kosniowski C.: A first course in algebraic topology. Cambridge University Press, Cambridge (1980)zbMATHCrossRefGoogle Scholar
  6. Saban G.: Barbier’s theorem through self-mated closed Bertrand curves. Simon Stevin 57, 37–45 (1983)MathSciNetzbMATHGoogle Scholar
  7. Wegner B.: Self-parallel and transnormal curves. Geom Dedicata 38, 175–191 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  • A. M. d’Azevedo Breda
    • 1
    Email author
  • F. J. Craveiro de Carvalho
    • 2
  • Bernd Wegner
    • 3
  1. 1.Departamento de MatemáticaUniversidade de AveiroAveiroPortugal
  2. 2.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  3. 3.Fachbereich MathematikBerlinGermany

Personalised recommendations