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Finitely determined singularities of ruled surfaces in 3

Published online by Cambridge University Press:  22 May 2009

R. MARTINS  and
J. J. NUÑO–BALLESTEROS
R. MARTINS
Affiliation:
Universidade Estadual de Maringá, Av. Colombo, 5.790 Jd. Universitário, Maringá – Paraná – CEP 87020-900, Brazil. e-mail: rmartins@uem.br
J. J. NUÑO–BALLESTEROS
Affiliation:
Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot, 46100 Burjassot, Spain. e-mail: juan.nuno@uv.es
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Abstract

We study local singularities of ruled surfaces in 3. We show that any map germ f : (2, 0) → (3, 0) with a simple singularity is -equivalent to a ruled surface. Moreover, we give a topological classification of -finitely determined singularities of ruled surfaces and show that there are just eleven topological classes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 701 - 733
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Carter, J. S.How Surfaces Intersect in Space: An Introdution to Topology (Second Edition). (World Scientific, 1995).CrossRefGoogle Scholar
[2]Damon, J.Finite determinacy and topological triviality I. Invent. Math. 62 (1980/81), 299–324.CrossRefGoogle Scholar
[3]Fukuda, T.Local topological properties of differentiable mappings I. Invent. Math. 65 (1981/82), 227–250.CrossRefGoogle Scholar
[4]Gauss, C. F.Werke VIII (Teubner, Leipzig. 1900), 271–286.Google Scholar
[5]Ishikawa, G.Topological classification of the tangent developables of space curves. J. London Math. Soc. 62 (2000), 583–598.CrossRefGoogle Scholar
[6]Izumiya, S. and Takeuchi, N.Singularities of ruled surfaces in 3. Math. Proc. Camb. Phil. Soc. 130. 1 (2001), 1–11.CrossRefGoogle Scholar
[7]Kuiper, N. H.C 1-equivalence of functions near isolated critical points. Annals Math. Stud. 69 (1972), 199–218 (Symposium in infinite dimensional Topology, Baton Rouge, 1967).Google Scholar
[8]Marar, W. L. and Mond, D.Multiple point schemes for corank 1 maps. J. London Math. Soc. 39 (1989), 553–567.CrossRefGoogle Scholar
[9]Marar, W. and Nuño-Ballesteros, J. J.The doodle of a finitely determined map germ from 2 to 3. Adv. Math. 221 (2009), 1281–1301.CrossRefGoogle Scholar
[10]Mond, D.On the classification of germs of maps from 2 to 3. Proc. London Math. Soc. 50 (1985), 333–369.CrossRefGoogle Scholar
[11]Mond, D.Some remarks on the geometry and classification of germs of maps from surfaces to 3-space. Topology 26 (1987), 361–383.CrossRefGoogle Scholar
[12]Montesinos–Amilibia, A.SphereXSurface, software available at http://www.uv.es/montesinGoogle Scholar
[13]Wall, C. T. C.Finite determinacy of smoth map-germs. Bull. London. Math. Soc. 13 (1981), 481–539.CrossRefGoogle Scholar