Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T16:28:41.064Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
J. J. NUÑO–BALLESTEROS
Affiliation:
Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot, 46100 Burjassot, Spain. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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