Flexibility for tangent and transverse immersions in Engel manifolds


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.

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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.

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Correspondence to Francisco Presas.

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del Pino, Á., Presas, F. Flexibility for tangent and transverse immersions in Engel manifolds. Rev Mat Complut 32, 215–238 (2019). https://doi.org/10.1007/s13163-018-0277-2

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  • Engel structure
  • h-principle
  • Horizontal curve

Mathematics Subject Classification

  • Primary 58A30