Skip to main content
Log in

On the developable ruled surfaces of low smoothness

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

The classical description of the structure of developable surfaces of torse type is formally possible only starting with C 3-smoothness. We consider developable surfaces of class C 2 and show that the directions of their generators at the boundary points of a surface belong to the tangent cone of the boundary curve. In analytical terms we give a necessary and sufficient condition for C 1-smooth surfaces with locally Euclidean metric to belong to the class of the so-called normal developable surfaces introduced by Burago and Shefel’.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rashevskiĭ P. K., A Course in Differential Geometry [in Russian], Gostekhizdat, Moscow (1956).

    Google Scholar 

  2. Klingenberg W., A Course in Differential Geometry, Springer-Verlag, New York; Heidelberg; Berlin (1978).

    MATH  Google Scholar 

  3. Ushakov V., “Parametrization of developable surfaces by asymptotic lines,” Bull. Austral. Math. Soc., 54, 411–421 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  4. Hartman Ph. and Nirenberg L., “On spherical image maps whose Jacobians do not change sign,” Amer. J. Math., 81, No. 4, 901–920 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  5. Lebesgue H., “Intégrale, longueur, aire,” Ann. di Mat. Ser. III, 7, 231–359 (1902).

    Article  Google Scholar 

  6. Borisov Yu. F., “Irregular C 1,β-surfaces with an analytic metric,” Siberian Math. J., 45, No. 1, 19–52 (2004).

    Article  MathSciNet  Google Scholar 

  7. Pogorelov A. V., Extrinsic Geometry of Convex Surfaces Amer. Math. Soc., Providence (1973) (Transl. Math. Monogr.; 35).

  8. Burago Yu. D., “Surface geometry in Euclidean space,” in: Geometry III, Springer-Verlag, Berlin, 1992, pp. 1–85.

    Google Scholar 

  9. Shefel’ S. Z., “C 1-Smooth isometric immersions,” Sibirsk. Mat. Zh., 15, No. 6, 1372–1393 (1974).

    Google Scholar 

  10. Shefel’ S. Z., “C 1-Smooth surfaces of bounded positive extrinsic curvature,” Sibirsk. Mat. Zh., 16, No. 5, 1122–1123 (1975).

    Google Scholar 

  11. Ushakov V., “The explicit general solution of the trivial Monge-Ampère equation,” Comment. Math. Helv., 75, 125–133 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  12. Sabitov I. Kh., Isometric Immersions and Embeddings of Locally Euclidean Metrics, Cambridge Sci. Publ., Cambridge (2009).

    Google Scholar 

  13. Shtogrin M. I., “Piecewise smooth developable surfaces,” Trudy Mat. Inst. Steklov., 263, 227–250 (2008).

    Google Scholar 

  14. Zalgaller V. A., Theory of Envelopes [in Russian], Fizmatgiz, Moscow (1975).

    Google Scholar 

  15. Fikhtengol’ts G. M., A Course of Differential and Integral Calculus. Vol. 1 [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to I. Kh. Sabitov.

Additional information

To Yuriĭ Grigor’evich Reshetnyak in token of sincere respect.

Original Russian Text Copyright © 2009 Sabitov I. Kh.

The author was partially supported by the Ministry for Science and Education of the Russian Federation (Grant RNP 2.1.1.3704) and the Russian Foundation for Basic Research (Grant 01-09-00179).

__________

Moscow. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 5, pp. 1163–1175, September–October, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sabitov, I.K. On the developable ruled surfaces of low smoothness. Sib Math J 50, 919–928 (2009). https://doi.org/10.1007/s11202-009-0102-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-009-0102-8

Keywords

Navigation