Skip to main content

An approximation theorem for immersions with stable configurations of lines of principal curvature

Part of the Lecture Notes in Mathematics book series (LNM,volume 1007)

Abstract

It is proved that every immersion of a compact oriented two-dimensional smooth manifold into R3 can be arbitrarily C2-approximated by smooth immersions β whose principal configurations Pβ = (Uβ ,Fβ ,fβ) defined by umbilical points and families of lines of principal curvature, are stable under C3-sufficiently small perturbations of β. Actually, the elements β are found in the class S r, r≥4, of C3-principally structurally stable immersions, introduced in [3].

Examples of immersions with recurrent lines of principal curvature are also given.

Keywords

  • Principal Curvature
  • Minimal Cycle
  • Umbilical Point
  • Principal Line
  • Smooth Vector Field

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bruce, J.W., Giblin, P.J.-Generic curves and surfaces, J. London Math. Soc. (2) Vol. 24, 1981.

    Google Scholar 

  2. Darboux, G.-Sur la forme des lignes de courbure dans le voisinage d'un ombilic, Note VII; Leçons sur la théorie générale des surfaces, Vol. IV. Gauthier-Villars, 1896.

    Google Scholar 

  3. Gutiérrez, C., Sotomayor, J.-Structurally stable configurations of lines of principal curvature. To appear in Astérisque.

    Google Scholar 

  4. Gutiérrez, C.-Structural stability for flows on the torus with a cross-cap. Trans. AMS, Vol. 241, 1978.

    Google Scholar 

  5. Looijenga, E.-Structural stability of smooth families of C functions, Thesis, Univ. of Amsterdam, 1974.

    Google Scholar 

  6. Malgrange, B.-Ideals of Differentiable Functions, Oxford University Press, 1966.

    Google Scholar 

  7. Peixoto, M.-Structurally stable vector fields on two-dimensional manifolds, Topology, Vol. 1, 1962.

    Google Scholar 

  8. Porteous, I.R.-The normal singularities of a submanifold, J. Diff. Geom. Vol. 5, 1971.

    Google Scholar 

  9. Pugh, C.-The closing lemma. Am. J. Math. Vol. 89, 1967.

    Google Scholar 

  10. Struik, D.-Lectures on classical differential geometry, Addison Wesley, 1950.

    Google Scholar 

  11. Thom, R.-Stabilité Structurelle et Morphogénèse, Benjamin, 1972.

    Google Scholar 

  12. Wilson, L., Bleecker, D.-Stability of Gauss Maps, Ill. Journ. of Math. Vol. 22, 1978.

    Google Scholar 

  13. Whitney, H.-Differentiable manifolds, Annals of Math., Vol. 37, 1936.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Gutiérrez, C., Sotomayor, J. (1983). An approximation theorem for immersions with stable configurations of lines of principal curvature. In: Palis, J. (eds) Geometric Dynamics. Lecture Notes in Mathematics, vol 1007. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0061423

Download citation

  • DOI: https://doi.org/10.1007/BFb0061423

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12336-1

  • Online ISBN: 978-3-540-40969-4

  • eBook Packages: Springer Book Archive