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Parameterisation of developable surfaces by asymptotic lines

Published online by Cambridge University Press:  17 April 2009

Vitaly Ushakov
Vitaly Ushakov
Affiliation:
Department of Mathematics, The University of Melbourne, Parkville, Vic 3052, Australia, e-mail: vit@mundoe.maths.mu.oz.au
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Abstract

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An example of a “non-developable” surface of vanishing Gaussian curvature from W. Klingenberg's textbook is considered and its place in the theory of 2-dimensional developable surfaces is pointed out. The surface is found in explicit form. Other examples of smooth developable surfaces not allowing smooth asymptotic parametrisation are analysed. In particular, Hartman and Nirenberg's example (1959) is incorrect.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 3 , December 1996 , pp. 411 - 421
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Hartman, Ph. and Nirenberg, L., ‘On spherical image maps whose Jacobians do not change sign’, Amer. J. Math. 81 (1959), 901–920.CrossRefGoogle Scholar
[2]Hartman, Ph. and Winter, A., ‘On the fundamental equations of differential geometry’, Amer. J. Math. 72 (1950), 757–774.CrossRefGoogle Scholar
[3]Hartman, Ph. and Winter, A., ‘On the asymptotic curves of a surface’, Amer. J. Math. 73 (1951), 149–172.CrossRefGoogle Scholar
[4]Klingenberg, W., A course in differential geometry (Springer-Verlag, Berlin, Heidelberg, New York, 1978), p. 178.CrossRefGoogle Scholar
[5]Massey, W., ‘Surfaces of Gaussian curvature zero in Euclidean 3-space’, Tohoku Math. J. (2) 14 (1962), 73–79.CrossRefGoogle Scholar
[6]Pogorelov, A.V., ‘On continuous mappings of bounded variation; An extension of Gauss' theorem on the spherical representation of surfaces of bounded exterior curvature’, (in Russian), Dokl. Akad. Nauk SSSR 111 (1956), 757–759. 945–947.Google Scholar
[7]Pogorelov, A.V., Surfaces of bounded exterior curvature, (in Russian) (Kharkov University, 1956).Google Scholar
[8]Pogorelov, A.V., Exterior geometry of convex surfaces, (in Russian) (‘Nauka’, Moscow, 1969), p. 759.Google Scholar
[9]Spivak, M., A comprehensive introduction to differential geometry III (Publish or Perish, Inc., Berkeley, 1979), p. 466.Google Scholar
[10]Encyclopaedia of Mathematics 9 (Kluwer Academic Publishers, 1993).Google Scholar