Engel Structure Research Articles

On the dynamics of some vector fields tangent to non-integrable plane fields

Let $\mathcal{E}^3\subset TM^n$ be a smooth $3$-distribution on a smooth manifold of dimension $n$ and $\mathcal{W}\subset\mathcal{E}$ a line field such that $[\mathcal{W},\mathcal{E}]\subset\mathcal{E}$. Under some orientability hypothesis, we give a necessary condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W}\subset\mathcal{D}$ and $[\mathcal{D},\mathcal{D}]=\mathcal{E}$. Moreover we study the case where a section of $\mathcal{W}$ is non-singular Morse-Smale and we get a sufficient condition for the global existence of $\mathcal{D}$. As a corollary we get conditions for a non-singular vector field $W$ on a $3$-manifold to be Legendrian for a contact structure $\mathcal{D}$. Similarly with these techniques we can study when an even contact structure $\mathcal{E}\subset TM^4$ is induced by an Engel structure $\mathcal{D}$.

English

In this article we introduce a higher dimensional analogue of Engel structure, motivated by the Cartan prolongation of contact manifolds. We study the stability of such structure, generalizing the Gray-type stability for Engel manifolds.

The Geometry of Marked Contact Engel Structures

A contact twisted cubic structure({mathcal M},mathcal {C},{varvec{upgamma }}) is a 5-dimensional manifold {mathcal M} together with a contact distribution mathcal {C} and a bundle of twisted cubics {varvec{upgamma }}subset mathbb {P}(mathcal {C}) compatible with the conformal symplectic form on mathcal {C}. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group mathrm {G}_2. In the present paper we equip the contact Engel structure with a smooth section sigma : {mathcal M}rightarrow {varvec{upgamma }}, which “marks” a point in each fibre {varvec{upgamma }}_x. We study the local geometry of the resulting structures ({mathcal M},mathcal {C},{varvec{upgamma }}, sigma ), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of {mathcal M} by curves whose tangent directions are everywhere contained in {varvec{upgamma }}. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ge 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

Riemannian Properties of Engel Structures

Abstract This paper is about geometric and Riemannian properties of Engel structures. A choice of defining forms for an Engel structure $\mathcal{D}$ determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the Reeb distribution. We study conditions that ensure integrability of $\mathcal{R}$. For example, if we have a metric $g$ that makes the splitting $TM=\mathcal{D}\oplus \mathcal{R}$ orthogonal and such that $\mathcal{D}$ is totally geodesic then there exists another Reeb distribution, which is integrable. We introduce the notion of K-Engel structures in analogy with K-contact structures, and we classify the topology of K-Engel manifolds. As natural consequences of these methods, we provide a construction that is the analogue of the Boothby–Wang construction in the contact setting, and we give a notion of contact filling for an Engel structure.

Aerobatics of Flying Saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$$G_2$$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $$G_2$$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $$G_2$$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ geometries is used to construct a ‘$$G_2$$ joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ that transforms the difficult $$G_2$$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

Notes on open book decompositions for Engel structures

We relate open book decompositions of a 4-manifold M with its Engel structures. Our main result is, given an open book decomposition of M whose binding is a collection of 2-tori and whose monodromy preserves a framing of a page, the construction of an En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a contactomorphism. As a consequence, on a parallelizable closed 4-manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that amongst the supported Engel structures we construct, there is a class of loose Engel structures.

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.

Exotic Holomorphic Engel Structures on $$\mathbf {C}^4$$ C 4

A holomorphic Engel structure determines a flag of distributions $$\mathcal {W}\subset \mathcal {D}\subset {\mathcal {E}}$$ . We construct examples of Engel structures on $$\mathbf {C}^4$$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $$\mathbf {C}$$ . We also construct two infinite families of pairwise non-isomorphic Engel structures on $$\mathbf {C}^4$$ by controlling the curves $$f{:}\mathbf {C}\rightarrow \mathbf {C}^4$$ tangent to $$\mathcal {W}$$ . The first is characterised by the topology of the set of points in $$\mathbf {C}^4$$ admitting $$\mathcal {W}$$ -lines and the second by a finer geometric property of this set. A consequence of the second construction is the existence of uncountably many non-isomorphic holomorphic Engel structures on $$\mathbf {C}^4$$ .

Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4

We investigate the set Vf of horizontal critical points of a polynomial function f for the standard Engel structure defined by the 1-forms ?3 = dx3 ? x1dx2 and ?4 = dx4 ? x3dx2, endowed with the sub-Riemannian metric $g_{\text {SR}}=d{x_{1}^{2}}+d{x^{2}_{2}}$. For a generic polynomial, we show that the set Γf of points in Vf, where Vf is not transverse to the Engel distribution, does not have a connected component which is contained in a fiber of f. Then, we prove that each trajectory of the horizontal gradient of f approaching the set Vf has a limit.

On prolongations of contact manifolds

We apply spectral sequences to derive both an obstruction to the existence of n n -fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on M × S 1 M\times \mathbb {S}^1 with characteristic line field tangent to the fibers is determined by the contact structure induced on a cross section and the twisting of the Engel structure along the fibers. Our results show that this statement needs some modification: to classify the diffeomorphism type of the Engel structure, we additionally have to fix a class in the first cohomology of M M .

Generalized Pseudo-Kähler Structures

In the paper we consider pseudo bihermitian structures - a pair of complex structures compatible with a pseudo Riemannian metric. As in the positive definite case we establish its relations with generalized (pseudo) Kaehler geometry and holomorphic Poisson structures. We provide a list of compact complex surfaces which could admit such structure and also examples of bihermitian structures on some of them. We also consider a naturally defined null plane distribution on generalized pseudo Kaehler 4-manifold and show that under a mild restriction it determines an Engel structure.

Existence of Engel structures

We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, there is an Engel structure on M_1#M_2#(S^2xS^2) which is closely related to the original Engel structures.

Horizontal loops in Engel space

A simple proof is given of the following result first observed by Adachi: embedded circles tangent to the standard Engel structure on \({\mathbb {R}^4}\) are classified, up to isotopy via such embeddings, by their rotation number.

Semilocal rings with Engel conditions

The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group.

Germs of Engel structures along 3-manifolds

Engel structures along an embedded 3-manifold in a 4-manifold are studied in this article. It is shown that, among the Engel structures having the same oriented even-contact structure around the given embedded closed oriented 3-manifold as derived distributions with the induced orientations, the germ of Engel structure along the 3-manifold is determined by the singular line field on the 3-manifold traced by the Engel structure.

Open Engel manifolds admitting compact characteristic leaves

We give an example of an Engel structure on the 4-dimensional Euclidean space which admits a compact characteristic leaf. We also show that every Engel structure on an open 4-manifold can be modified so that the resulting structure has a compact characteristic leaf.

The engel structure of linear groups

The engel structure of linear groups