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3. Engel dynamics

    1. Problem 3.1.

      What are the constraints on the dynamics of the characteristic line field $\mathcal{W}$ of an even contact structure imposed by an Engel structure? Otherwise said, which even contact structures support Engel structures?
        1. Remark. During the workshop some necessary conditions and some indications of possible non-trivial sufficient conditions were discussed.
            • Problem 3.2.

              Do Engel structures admit plugs in the sense of Wilson for $\mathcal{W}$?
                1. Remark. This is already interesting for even contact structures.
                    • Problem 3.4.

                      Does there exists an Engel structure on the 4-torus such that the characteristic foliation has exactly one closed leaf that is non-degenerate?
                        1. Remark. Consider on $T^3$ contact structure $\xi$ which is trivial as a bundle, and denote with $C_1,C_2$ a global trivialisation of $\xi$. Take the (oriented) Cartan prolongation $(T^3\times S^1,\mathcal{D}(\xi))=(T^4,\langle \partial/\partial t,\cos(t)C_1+\sin(t)C_2\rangle )$, where $t$ denotes the coordinate of $S^1$. The distribution \[ \mathcal{D}=\langle W=\partial/\partial t+\epsilon V,X=\cos(kt)C_1+\sin(kt)C_2\rangle \] is an Engel structure for $\epsilon>0$ small enough, $k\in\mathbb{N}$ big enough and $V$ contact vector field for $\xi$. Moreover the characteristic foliation associated to $\mathcal{D}$ is spanned by $W$ If we choose $V$ such that it has a unique singularity at a point $p\in T^3$ and such that its closed orbits are non-degenerate, then there exists an $\epsilon$ small enough and rationally independent with the periods of the orbits of $V$. This ensures that the only closed orbit of $W$ is the one passing through $(p,0)$. Notice that in general this construction only yields a degenerate closed orbit.
                            • Problem 3.5.

                              What can one say about the dynamics of the line field induced on a hypersurface on an Engel manifold? For instance, are there non-trivial constraints of Bennequin type on the monodromy of a hypersurface transverse to $\mathcal{D}$ but not necesseraly to $\mathcal{W}$?.
                                • Problem 3.6.

                                  Is there an Engel structure with trivial automorphism group?
                                    1. Remark. Montgomery [montg] gives an example with small automorphism group.

                                          Cite this as: AimPL: Engel structures, available at http://aimpl.org/engelstr.