For the last 3 years we have set up the foundations for the global study of Engel structures. This is a radiography of the project. It was started by the fruitful interaction of classical differential geometry techniques with the Gromov's h--principle. The theory has been very much related to the study of families of convex curves in the round 2-sphere.
The research has been funded by a National Spanish research fund EXPLORA for highly risky and innovative research ideas. Most of the activities of the group have been partially financed by this fund.
Task:
Mimic the development of Contact Topology. Prove results that create a dychotomy between flexible and rigid aspects of Engel structures. We have been focused for the first 3 years in showing that Engel geometry has a flexible side. Now, we are trying to change the focus and find some rigidity. Moreover, we have found that flexibility is our friend. We have found that computing the spaces of horizontal embeddings is usually easier than to compute the smooth analogues, but since the horizontal embedding spaces are flexible, the computations provide the answer in the smooth case.
Papers:
Overview:
Presas, F., Non-integrable distributions and the h-principle. Eur. Math. Soc. Newsl. 99, 18--26 (2016).
Flexibility of Engel structures
R. Casals, J. L. Pérez, A. del Pino, F. Presas, Existence h-principle for Engel structures, Invent. Math. 210 (2017), no. 2, 417-451.
R. Casals, A. del Pino, F. Presas, Loose Engel structures.
A. del Pino, T. Vogel, The Engel-Lutz twist and overtwisted Engel estructures.
Horizontal knots
A. del Pino, F. Presas, Flexibility for horizontal and transverse immersions in Engel manifolds.
E. Fernández, J. Martínez-Aguinaga, F. Presas, Fundamental groups of formal Legendrian and horizontal embedding spaces
A- del Pino, R. Casals, Classification of Engel knots
General constructions
V. Colin, F. Presas, T. Vogel, Notes on open book decompositions for Engel structures.
Members of the group
Eduardo Fernández
Francisco Javier Martínez
Conferences related to the group (partially TOPE funded)
AIM workshop: Engel structures
Summer school: Non-integrable distributions and the h--principle. Notes by Alvin Jin
Workshop on Topological Aspects of Symplectic Foliations
Workshop for the Chaire d'excellence Eva Miranda: Geometry and Dynamics in Interaction