Revista Matemática Complutense

, Volume 32, Issue 1, pp 215–238 | Cite as

Flexibility for tangent and transverse immersions in Engel manifolds

  • Álvaro del Pino
  • Francisco PresasEmail author


We study the space of immersions of \({\mathbb {S}}^1\) that are tangent to an Engel structure \({\mathcal {D}}\). We show that the full h-principle holds as soon as one excludes the closed orbits of \({\mathcal {W}}\), the characteristic foliation of \({\mathcal {D}}\). This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to \({\mathcal {W}}\) sometimes form additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if \({\mathcal {D}}\) is \(C^\infty \)-generic, curves tangent to \({\mathcal {W}}\) are not isolated anymore. These results, in conjunction with an argument due to M. Gromov, prove that a full h-principle holds for immersions transverse to the Engel structure.


Engel structure h-principle Horizontal curve 

Mathematics Subject Classification

Primary 58A30 



The authors are grateful to T. Vogel for bringing the problem of transverse submanifolds to their attention, and to V. Ginzburg for the many conversations that gave birth to this Project. They are also thankful to R. Casals, J.L. Pérez, and F.J. Martínez for reading a preliminary version of the paper. Lastly, we thank the referees for their comments.


  1. 1.
    Adachi, J.: Classification of horizontal loops in standard Engel space. Int. Math. Res. Not. 2007, Article ID rnm008 (2007).
  2. 2.
    Bryant, R.L., Hsu, L.: Rigidity of integral curves of rank 2 distributions. Invent. Math. 114(2), 435–461 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Casals, R., del Pino, A.: Classification of Engel knots. Math. Ann. 371, 391–404 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casals, R., Pérez, J.L., del Pino, A., Presas, F.: Existence h-principle for Engel structures. Invent. Math. 210, 417–451 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eliashberg, Y., Mishachev, N.: Introduction to the h-Principle. Graduate Studies in Mathematics, vol. 48. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  6. 6.
    Engel, F.: Zur Invariantentheorie der systeme Pfaff’scher Gleichungen. Leipz. Ber. Band. 41, 157176 (1889)Google Scholar
  7. 7.
    Geiges, H.: Horizontal loops in Engel space. Math. Ann. 342(2), 291–296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gromov, M.: Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 9. Springer, Berlin (1986)Google Scholar
  9. 9.
    Hsu, L.: Calculus of variations via the Griffiths formalism. J. Differ. Geom. 36(3), 551–589 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Montgomery, R.: Engel deformations and contact structures. In: Northern California symplectic geometry seminar. American Mathematical Society. Translations Series 2, vol. 196, no. 45, p. 103–117. American Mathematical Society, Providence, RI (1999)Google Scholar
  11. 11.
    Peixoto, M.M.: On an approximation theorem of Kupka and Smale. J. Differ. Equ. 3, 214–227 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Thurston, W.: The theory of foliations of codimension greater than one. Comment. Math. Helv. 49, 214–231 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vogel, T.: Existence of Engel structures. Ann. Math. (2) 169(1), 79–137 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCMMadridSpain

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