# 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.

#### Figures from this paper

#### 6 Citations

Loose Engel structures

- Mathematics
- Compositio Mathematica
- 2020

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… Expand

Engel structures on complex surfaces

- Mathematics
- 2019

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

- Mathematics
- 2021

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 finite… Expand

Riemannian Properties of Engel Structures

- Mathematics
- 2019

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 Engel… Expand

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

- Mathematics
- 2019

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… Expand

#### References

SHOWING 1-10 OF 21 REFERENCES

Existence of Engel structures

- Mathematics
- 2004

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 Engel… Expand

Existence h-principle for Engel structures

- Mathematics
- 2015

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. In… Expand

Existence and classification of overtwisted contact structures in all dimensions

- Mathematics
- 2014

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]. It… Expand

On the classification of prolongations up to Engel homotopy

- Mathematics
- 2017

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 classification… Expand

On prolongations of contact manifolds

- Mathematics
- 2011

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 to… Expand

Loose Legendrian embeddings in high dimensional contact manifolds

- Mathematics
- 2012

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-holomorphic… Expand

Legendrian and Transversal Knots

- Mathematics
- 2005

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 on… Expand

Classification of overtwisted contact structures on 3-manifolds

- Mathematics
- 1989

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 this… Expand

An introduction to contact topology

- Mathematics
- 2008

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. Further… Expand

Introduction to the h-Principle

- Mathematics
- 2002

Intrigue Holonomic approximation: Jets and holonomy Thom transversality theorem Holonomic approximation Applications Differential relations and Gromov's $h$-principle: Differential relations Homotopy… Expand