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arXiv · 1609.09306

Flexibility for tangent and transverse immersions in Engel manifolds

Alvaro del Pino·Francisco Presas
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Abstract

In this article we study immersions of the circle that are tangent to an Engel structure $\mathcal{D}$. We show that a full $h$-principle does exist as soon as one excludes the closed orbits of $\mathcal{W}$, the kernel of $\mathcal{D}$. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to $\mathcal{W}$ often conform additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if $\mathcal{D}$ is generic, curves tangent to $\mathcal{W}$ are not isolated anymore. We then go on to show that a full $h$-principle holds for immersions transverse to the Engel structure.

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Alvaro del Pino, Francisco Presas. 2016-09-29. Flexibility for tangent and transverse immersions in Engel manifolds. https://arxiv.org/abs/1609.09306

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