Advertisement

Journal of Geometry

, Volume 108, Issue 1, pp 25–32 | Cite as

Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than \({{2\pi}}\)

  • Yannick MassonEmail author
  • Laurent Monasse
Article

Abstract

We prove the existence of a global smooth Chebyshev net on complete, simply connected surfaces when the total absolute curvature is bounded by \({2\pi}\). Following Samelson and Dayawansa, we look at Chebyshev nets given by a dual curve, splitting the surface into two connected half-surfaces, and a distribution of angles along it. An analogue to the Hazzidakis’ formula is used to control the angles of the net on each half-surface with the integral of the Gaussian curvature of this half-surface and the Cauchy boundary conditions. We can then prove the main result using a theorem about splitting the Gaussian curvature with a geodesic, obtained by Bonk and Lang.

Keywords

Chebyshev nets curvature bounds 

Mathematics Subject Classification

53C21 58G30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bakelman, I.Y.: Chebyshev nets in manifolds of bounded curvature. Trudy Math. Inst. Steklov. 76, 124–129 (1965). [English transl. Proc. Steklov. Inst. Math. 76, 154–160 (1967)]Google Scholar
  2. 2.
    Bianchi, L.: Lezione di geometria differenziale, vols. 1, 2, 2nd edn. Spoerri, Pisa (1902–1903)Google Scholar
  3. 3.
    Bonk M., Lang U.: Bi-lipschitz parameterization of surfaces. Mathe. Ann. 327(1), 135–169 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burago, Y.D., Ivanov, S.V., Malev, S.G.: Remarks on Chebyshev coordinates. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 329(5–13), 195 (2005). (Geom. i Topol. 9)Google Scholar
  5. 5.
    Douthe, C., Baverel, O., Caron, J-F.: Form-finding of a grid shell in composite materials. J. Int. Assoc. Shell Spat. Struct. 150 (53–62) (2006)Google Scholar
  6. 6.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. In: Universitext (Berlin. Print). U.S. Government Printing Office (2004)Google Scholar
  7. 7.
    Ghys É.: Sur la coupe des vêtements: variation autour d’un thème de Tchebychev. Enseign. Math. (2) 57(1–2), 165–208 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hazzidakis J.N.: Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass. J. Reine Angew. Math. 88, 68–73 (1880)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Samelson S.L., Dayawansa W.P.: On the existence of global Tchebychev nets. Trans. Am. Math. Soc. 347(2), 651–660 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Stoker, J.J.: Differential Geometry. In: Pure and Applied Mathematics, vol XX. Wiley, New York (1969)Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Université Paris-Est, CERMICS (ENPC)Marne la Vallée CedexFrance

Personalised recommendations