The Geometry of Marked Contact Engel Structures

A contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$ ( M , C , γ ) is a 5-dimensional manifold $${\mathcal M}$$ M together with a contact distribution $$\mathcal {C}$$ C and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$ γ ⊂ P ( C ) compatible with the conformal symplectic form on $$\mathcal {C}$$ C . The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$ G 2 . In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$ σ : M → γ , which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$ γ x . We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$ ( M , C , γ , σ ) , which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$ M by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$ γ . We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$ ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.

Riemannian Properties of Engel Structures

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.

Loose Engel structures

This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$-principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.