Parallel and Transnormal Curves on Surfaces

  • Bernd Wegner
Part of the Mathematics and Its Applications book series (MAIA, volume 350)


The exterior parallelism of space curves in the sense of H.R. Farran and S.A. Robertson [J. London Math. Soc. (2) 35 (1987), 527-538] is studied in the case that the curves are located on a surface M in Euclidean 3-space. At first a local existence and uniqueness result is established under some genericity assumption for pairs of parallel curves passing through so-called EP-pairs on M. A surface for which every pair of points is an EP-Pair is shown to be an open part of a sphere or a plane. Continuous families of mutually parallel curves must consist of curvature lines. Cylinders, spheres and planes are the only surfaces which carry two such families. Finally, some special results are derived for parallel pairs and self-parallel resp. transnormal curves on surfaces of rotation and cylinders.


Constant Width Normal Plane Continuous Family Normal Section Parallel Transfer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H.P. Boas: Spheres and cylinders: A local geometric characterization, Illinois J. Math. 28 (1984), 120–124.MathSciNetMATHGoogle Scholar
  2. 2.
    F.J. Craveiro de Carvalho: Transnormal graphs, Portugaliae Math. 39 (1980), 285–287.MathSciNetGoogle Scholar
  3. 3.
    F.J. Craveiro de Carvalho, S.A. Robertson: Self-parallel curves, Math. Scand. 65 (1989), 67–74.MathSciNetMATHGoogle Scholar
  4. 4.
    H.R. Farran, S.A. Robertson: Parallel immersions in Euclidean space, J. London Math. Soc. (2) 35 (1987), 527–538.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    N.J. Hicks: Notes on differential geometry, van Nostrand (Princeton, 1965).MATHGoogle Scholar
  6. 6.
    M.C. Irwin: Transnormal circles, J. London Math. Soc. 42 (1967), 545–552.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    J. Schaeft: Direction preserving maps of planar smooth curves, preprint (Univ. of Calgary, 1990).Google Scholar
  8. 8.
    B. Wegner: Some remarks on parallel immersions, Coll. Math. Soc. J. Bolyai 56 (1989/1991), 707–717.MathSciNetGoogle Scholar
  9. 9.
    B. Wegner: Self-parallel and transnormal curves, Geom. Dedicata 38 (1991), 175–191.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    B. Wegner: Projections of transnormal manifolds, J. Geom. 36 (1989), 183–187.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    B. Wegner: Globale Sätze über Raumkurven konstanter Breite I,II, Math. Nachr. 53 (1972), 337–344, ibid. 67 (1975), 213-223.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    B. Wegner: A differential geometric proof of the local geometric characterization of spheres and cylinders by Boas, Math. Balk., New Ser. 2 (1988), 294–295.MathSciNetMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Bernd Wegner
    • 1
  1. 1.BerlinGermany

Personalised recommendations