Journal of Geometry

, Volume 108, Issue 1, pp 301–318 | Cite as

Singularities of tangent surfaces to generic space curves

  • G. IshikawaEmail author
  • T. Yamashita


We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akivis, M.A., Goldberg, V.V.: Differential Geometry of Varieties with Degenerate Gauss Maps. CMS Books in Mathematics, 18, Springer, New York (2004)Google Scholar
  2. 2.
    Arnol’d, V.I.: Catastrophe Theory, 3rd edn. Springer, New York (1992)Google Scholar
  3. 3.
    Bruce J.W., Giblin P.J.: Curves and Singularities, A Geometrical Introduction to Singularity Theory, 2nd edn. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cartan, E.: Sur les courbes de torsion nulle et les surfaces developpables dans les espaces de Riemann. C. R. Acad. Sc. 184 138 (1927) (in E. Cartan, \({\OE}\)uvres Complètes III-2, Éditions du CNRS, Paris (1984), pp. 1081–1083)Google Scholar
  5. 5.
    Cayley, A.: Mémoire sur les coubes à double courbure et les surfaces développables. Journal de Mathematique Pure et Appliquees (Liouville), 10 (1845) (245–250 = The Collected Mathematical Papers vol. I, pp. 207–211)Google Scholar
  6. 6.
    Chino S., Izumiya S.: Lightlike developables in Minkowski 3-space. Demonst. Math. 43−2, 387–399 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cleave J.P.: The form of the tangent-developable at points of zero torsion on space curves. Math. Proc. Camb. Philos. Soc. 88–3, 403–407 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eisenhart, L.P.: Non-Riemannian Geometry. In: American Mathematical Society Colloquium Publications, vol 8. American Mathematical Society, New York (1927)Google Scholar
  9. 9.
    Fischer, G., Piontkowski, J.: Ruled Varieties, An Introduction to Algebraic Differential Geometry. Advanced Lectures in Math., Friedr. Vieweg & Sohn, Hamburg (2001)Google Scholar
  10. 10.
    Fujimori, S., Rossman, W., Umehara, M., Yamada, K., Yang, S.-D.: Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-space. arXiv:1302.4315 [math.DG]
  11. 11.
    Fujimori S., Saji K., Umehara M., Yamada K.: Singularities of maximal surfaces. Math. Z. 259, 827–848 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities, Graduate Texts in Mathematics 14, Springer, New York (1973)Google Scholar
  13. 13.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Math., 80, Academic Press, New York, London (1978)Google Scholar
  14. 14.
    Ishikawa G.: Symplectic and Lagrange stabilities of open Whitney umbrellas. Invent. math. 126−2, 215–234 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ishikawa, G.: Singularities of Developable Surfaces, London Math. Soc. Lecture Note Series, 263 pp. 403–418, Cambridge Univ. Press. Cambridge (1999)Google Scholar
  16. 16.
    Ishikawa G.: Singularities of tangent varieties to curves and surfaces. J. Singular. 6, 54–83 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ishikawa G., Machida Y., Takahashi M.: Asymmetry in singularities of tangent surfaces in contact-cone Legendre-null duality. J. Singular. 3, 126–143 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ishikawa, G., Yamashita, T.: Affine connections and singularities of tangent surfaces to space curves. arXiv:1501.07341 [math.DG]
  19. 19.
    Izumiya S., Katsumi H., Yamasaki T.: The rectifying developable and the spherical Darboux image of a space curve.. Caustics ’ 98 Banach Center Publ. 50, 137–149 (1999)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Izumiya S., Nagai T., Saji K.: Great circular surfaces in the three-sphere. Differ. Geom. Appl. 29, 409–425 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol I. Intersci. Publ. Wiley, New York, London (1963)Google Scholar
  22. 22.
    Kokubu M., Rossman W., Saji K., Umehara M., Yamada K.: Singularities of flat fronts in hyperbolic space. Pac. J. Math. 221–2, 303–351 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kreyszig, E.: Differential Geometry. University of Toronto Press, Toronto (1959)Google Scholar
  24. 24.
    Lawrence S.: Developable surfaces, their history and application. Nexus Netw. J. 13–3, 701–714 (2011)CrossRefGoogle Scholar
  25. 25.
    Mond D.: On the tangent developable of a space curve. Math. Proc. Camb. Philos. Soc. 91, 351–355 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mond D.: Singularities of the tangent developable surface of a space curve. Q. J. Math. Oxford 40, 79–91 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nuño Ballesteros J.J.: On the number of triple points of the tangent developable. Geom. Dedicata 47–3, 241–254 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nuñso Ballesteros, J.J., Saeki, O.: Singular surfaces in 3-manifolds, the tangent developable of a space curve and the dual of an immersed surface in 3-space, Real and complex singularities (São Carlos, 1994), 49–64, Pitman Res. Notes Math. Ser., 333, Longman, Harlow (1995)Google Scholar
  29. 29.
    Porteous I.R.: Geomric Differentiation, for the Intelligence of Curves and Surfaces. Cambridge Univ. Press, Cambridge (1994)Google Scholar
  30. 30.
    Saji K.: Criteria for singularities of smooth maps from the plane into the plane and their applications. Hiroshima Math. J. 40, 229–239 (2010)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Saji K., Umehara M., Yamada K.: A k singularities of wave fronts. Math. Proc. Camb. Philos. Soc. 146–3, 731–746 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sharpe, R.W.: Differential Geometry, Cartan’s Generalization of Klein’s Erlangen Program Graduate Texts in Math., 166, Springer, New York (1997)Google Scholar
  33. 33.
    Shcherbak O.P.: Projectively dual space curves and Legendre singularities. Trudy Tbiliss. Univ. 232–233, 280–336 (1982)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Shcherbak O.P.: Projectively dual space curves and Legendre singularities. Sel. Math. Sov. 5–4, 391–421 (1986)zbMATHGoogle Scholar
  35. 35.
    Shcherbak O.P.: Wavefront and reflection groups. Russian Math. Surv. 43–3, 149–194 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Whitney H.: On singularities of mappings of Euclidean spaces I, Mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations