## Remarks on Chebyshev coordinates.(English. Russian original)Zbl 1151.53349

J. Math. Sci., New York 140, No. 4, 497-501 (2007); translation from Zap. Nauchn. Semin. POMI 329, 5-13 (2005).
Summary: Some results on the existence of global Chebyshev coordinates on a Riemannian two-manifold or, more generally, on an Aleksandrov surface $$M$$ are proved. For instance, if the positive and the negative part of the integral curvature of $$M$$ are less than $$2\pi$$, then there exist global Chebyshev coordinates on $$M$$. Such coordinates help one to construct bi-Lipschitz maps between surfaces.

### MSC:

 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 53B21 Methods of local Riemannian geometry
Full Text:

### References:

 [1] A. D. Aleksandrov (Alexandrov) and V. A. Zalgaller, Intrinsic Geometry of Surfaces (Transl. Math. Monographs, 15), Amer. Math. Soc., Providence (1967). [2] I. Ya. Bakelman, ”Chebyshev nets on manifolds of bounded curvature,” Trudy MIAN, 76, 124–129 (1965). [3] A. Belenkii and Yu. D. Burago, ”Bi-Lipschitz equivalent Aleksandrov surfaces. I,” Algebra Analiz, 16, No. 4, 24–40 (2004). [4] M. Bonk and U. Lang, ”Bi-Lipschitz parametrization of surfaces,” Math. Ann., 327, 135–169 (2003). · Zbl 1042.53044 [5] Yu. D. Burago, ”Bi-Lipschitz equivalent Aleksandrov surfaces. II,” Algebra Analiz, 16, No. 6, 28–52 (2004). [6] H. Hopf, ”Differential geometry in the large,” Lect. Notes Math., 1000 (1983). · Zbl 0526.53002 [7] O. A. Ladyzhenskaya and V. I. Shubov, ”Unique solvability of the Cauchy problem of equations of the two dimensional chiral fields with values in a Riemannian manifold,” Zap. Nauchn. Semin. LOMI, 110, 81–94 (1981). · Zbl 0482.58011 [8] Yu. G. Reshetnyak, ”Two-dimensional manifolds of bounded curvature,” in: Geometry IV. Non-Regular Riemannian Geometry, Encyclopaedia Math. Sci., Springer Verlag, Berlin (1999), pp. 3–163. [9] S. L. Samelson and W. P. Dayawansa, ”On the existence of global Tchebyshev nets,” Trans. Amer. Math. Soc., 347, 651–660 (1995). · Zbl 0820.58055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.