Mathematische Annalen

, Volume 342, Issue 2, pp 291–296 | Cite as

Horizontal loops in Engel space

  • Hansjörg GeigesEmail author


A simple proof is given of the following result first observed by Adachi: embedded circles tangent to the standard Engel structure on \({\mathbb {R}^4}\) are classified, up to isotopy via such embeddings, by their rotation number.

Mathematics Subject Classification (2000)

57R40 57R42 57R15 53D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adachi, J.: Classification of horizontal loops in standard Engel space. Int. Math. Res. Not. 2007, vol. 2007: article ID rnm008, 29 p. doi: 10.1093/imrn/rnm008
  2. 2.
    Eliashberg, Ya., Fraser, M.: Classification of topologically trivial Legendrian knots. In: Geometry, Topology, and Dynamics (Montréal, 1995). CRM Proc. Lecture Notes, vol. 15, pp. 17–51. American Mathematical Society, Providence (1998)Google Scholar
  3. 3.
    Geiges, H.: An Introduction to Contact Topology, Cambridge Stud. Adv. Math., vol. 109. Cambridge University Press, Cambridge (2008)Google Scholar
  4. 4.
    Geiges, H.: A contact geometric proof of the Whitney–Graustein theorem. Enseign. Math. (2) (to appear)Google Scholar
  5. 5.
    Montgomery, R.: A Tour of Subriemannian Geometries, their Geodesics and Applications, Math. Surveys Monogr., vol. 91. American Mathematical Society, Providence (2002)Google Scholar
  6. 6.
    Sarychev, A.V.: On homotopy properties of the space of trajectories of a completely nonholonomic differential system. Soviet Math. Dokl. 42, 674–678 (1991)MathSciNetGoogle Scholar
  7. 7.
    Smale, S.: Regular curves on Riemannian manifolds. Trans. Amer. Math. Soc. 87, 492–512 (1958)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany

Personalised recommendations