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


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.


Chebyshev nets curvature bounds 

Mathematics Subject Classification

53C21 58G30 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

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

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