Abstract
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.
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References
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
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)
Geiges, H.: An Introduction to Contact Topology, Cambridge Stud. Adv. Math., vol. 109. Cambridge University Press, Cambridge (2008)
Geiges, H.: A contact geometric proof of the Whitney–Graustein theorem. Enseign. Math. (2) (to appear)
Montgomery, R.: A Tour of Subriemannian Geometries, their Geodesics and Applications, Math. Surveys Monogr., vol. 91. American Mathematical Society, Providence (2002)
Sarychev, A.V.: On homotopy properties of the space of trajectories of a completely nonholonomic differential system. Soviet Math. Dokl. 42, 674–678 (1991)
Smale, S.: Regular curves on Riemannian manifolds. Trans. Amer. Math. Soc. 87, 492–512 (1958)