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4. Relations with other geometric structures


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      Problem 4.1.

      Is every compact holomorphic Engel manifold $X^4$ either a Cartan prolongation of a holomorphic contact 3 manifold or a Lorentz prolongation of a holomorphic conformal 3 manifold?
        1. Remark. Notice that these two types are classified in the projective case. Under some additional assumptions, the Problem has a positive answer in the projective case. [presas]


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              Problem 4.2.

              Describe Engel structures that are realizable as complex line fields on compact complex surfaces.


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                  Problem 4.3.

                  Describe Engel structures that are realizable as Lagrangian, respectively symplectic, distributions on symplectic manifolds.
                    1. Remark. [Bryant] Let $z,w$ be holomorphic coordinates on $\mathbb{C}^2$ and define \[ \omega=e^{i(z+\bar z)}dw-idz \] Let $\mathbb{Z}^4$ act on $\mathbb{C}^4$ via \[ (k_1,k_2,k_3,k_4)(z,w)=(z+\pi k_1+ik_2,w+k_3+ik_4) \] Then $\omega$ passes to the quotient $M=\mathbb{C}^2/\mathbb{Z}^4=T^4$, defining on it an Engel structure $\mathcal{D}=\ker\omega$.

                      Remark that $\mathcal{D}$ is a complex line field admitting a transverse foliation by elliptic curves defined by $\ker(dz)$. Moreover $\mathcal{D}$ is Lagrangian with respect to the symplectic form $\Omega_1=Re(dzdw)$ and symplectic with respect to $\Omega_2=dzd\bar z+dwd\bar w$.


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                          Problem 4.4.

                          Is there a useful notion of a Goursat or similar cobordism/con­cor­dance between Engel structures?
                            1. Remark. The cobordism definitely needs more structure than just a contact structure.


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                                  Problem 4.5.

                                  Let $N$ be a contact 3 manifold with connected boundary, fix an inner product on $\xi$ and let $\phi:N\rightarrow N$ be a contactomorphism that is the identity near $\partial N$. Let $\mathcal{L}$ be a Legendrian foliation on $N$ and define the angle function $\theta_\phi:N\rightarrow\mathbb{R}$ such that $\theta(p)$ is the angle between $\mathcal{L}_p$ and $\phi_*(\mathcal{L}_{\phi^{-1}(p)})$ in $\xi_p$.

                                  1. which functions $\theta:N\rightarrow\mathbb{R}$ are of the form $\theta=\theta_\phi$? Given $\theta:N\rightarrow\mathbb{R}$ does there exist $\phi:N\rightarrow N$ such that $\theta\le\theta_\phi$?
                                  2. Is it possible to use $\theta_\phi$ to define a kind of metric-like structure on the group of contactomorphism with compact support?
                                    1. Remark. In order to relate this problem to Engel structures consider the following extension problem. Let $M$ be a oriented $3$-manifold with non-empty boundary. Assume that near $\partial\left(M\times[0,1]\right)$ there is an Engel structure such that the characteristic foliation is tangent to the second factor and the induced contact structures on $M\times\{0\}\simeq M$ and $M\times\{1\}\simeq M$ coincide. We want to extend $\mathcal{D}$ to the interior.

                                      If the answer to the first part of the question is positive for all $\theta$, then one can find an extension $\mathcal{D}_{ext}$ such that all leaves of the characteristic foliation connect $M\times\{0\}$ to $M\times\{1\}$. Conversely, if this restricted extension problem is not always solvable one might hope to find metric like structure/ordering on the group of contact diffeomorphisms which coincide with the identity near the boundary (in the same spirit as positivity for contact diffeomorphisms).


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                                          Problem 4.6.

                                          Let $(N,\xi)$ be a contact 3 manifold and let $\Sigma_1,\Sigma_2\subset N$ be surfaces transversal to $\xi$. Which diffeomorphism $\psi:\Sigma_1\rightarrow\Sigma_2$ can be realized by the flow of a Legendrian field?


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                                              Problem 4.7.

                                              Are there canonical constructions on Engel structures that would yield geometric structures that obey a version of Moser’s Theorem, i.e. admit no deformation or only a finite parameter family of deformations (modulo equivalence) in the closed case?
                                                1. Remark. The existence of smooth families with varying dynamics of the characteristic line field means that there can be no general Gray stability result in the Engel case. The problem asks whether starting from Engel structures one can build other structures which have better stability properties.

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