Loose Engel structures

@article{Casals2020LooseES,
  title={Loose Engel structures},
  author={Roger Casals and {\'A}lvaro del Pino and F. Presas},
  journal={Compositio Mathematica},
  year={2020},
  volume={156},
  pages={412 - 434}
}
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. 
A remark on the contactomorphism group of overtwisted contact spheres
We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some highExpand
Analytic torsion of generic rank two distributions in dimension five
We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finiteExpand
Universality of Euler flows and flexibility of Reeb embeddings
The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem forExpand

References

SHOWING 1-10 OF 50 REFERENCES
Existence of Engel structures
We develop a construction of Engel structures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an EngelExpand
Existence h-principle for Engel structures
In this article we prove that the inclusion of the space of Engel structures of a smooth 4-manifold into the space of full flags of its tangent bundle induces surjections in all homotopy groups. InExpand
On the classification of prolongations up to Engel homotopy
In [CPPP] it was shown that Engel structures satisfy an existence $h$-principle, and the question of whether a full $h$-principle holds was left open. In this note we address the classificationExpand
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 monodromyExpand
Engel structures and weakly hyperbolic flows on four-manifolds
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish aExpand
Geometry and dynamics of Engel structures
The aim of this paper is to extend basic understanding of Engel structures through developing geometric constructions which are canonical to a certain degree and the dynamics of CauchyExpand
Lagrangian Engel Structures
We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A \em{Lagrangian} EngelExpand
Foliated vector fields without periodic orbits
In this article parametric versions of Wilson’s plug and Kuperberg’s plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singularExpand
The Engel-Lutz twist and overtwisted Engel structures
We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is,Expand
Engel Manifolds and Contact 3-Orbifolds
In early study of Engel manifolds from R. Montgomery, the Cartan prolongation and the development map are central figures. However, the development map can be globally defined only if theExpand
...
1
2
3
4
5
...