The Engel-Lutz twist and overtwisted Engel structures

@article{Pino2017TheET,
  title={The Engel-Lutz twist and overtwisted Engel structures},
  author={'Alvaro del Pino and T. Vogel},
  journal={arXiv: Symplectic Geometry},
  year={2017}
}
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, and to prove a complete h-principle for overtwisted Engel structures with fixed overtwisted disc. 
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 areExpand
Engel structures on complex surfaces
We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes.Expand
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
Riemannian Properties of Engel Structures
This paper is about geometric and Riemannian properties of Engel structures, i.e. maximally non-integrable $2$-plane fields on $4$-manifolds. Two $1$-forms $\alpha$ and $\beta$ are called EngelExpand
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 thatExpand

References

SHOWING 1-10 OF 21 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
Existence and classification of overtwisted contact structures in all dimensions
We establish a parametric extension h-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the 3-dimensional result from [12]. ItExpand
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
On prolongations of contact manifolds
We apply spectral sequences to derive both an obstruction to the existence of $n$-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt toExpand
Loose Legendrian embeddings in high dimensional contact manifolds
We give an $h$--principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudo-holomorphicExpand
Legendrian and Transversal Knots
Publisher Summary Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi onExpand
Classification of overtwisted contact structures on 3-manifolds
A contac t s t ructure on a (2n + l ) -d imensional manifold is a cod imens ion 1 tangent d is t r ibut ion which can be defined (at least locally) by a 1-form 7 with 7/x (d~)" nowhere 0. In thisExpand
An introduction to contact topology
Foreword 1. Facets of Contact Geometry 2. Contact Manifolds 3. Knots in Contact 3-Manifolds 4. Contact Structures on 3-Manifolds 5. Symplectic Fillings and Convexity 6. Contact Surgery 7. FurtherExpand
Introduction to the h-Principle
Intrigue Holonomic approximation: Jets and holonomy Thom transversality theorem Holonomic approximation Applications Differential relations and Gromov's $h$-principle: Differential relations HomotopyExpand
...
1
2
3
...