Resumen de Engel structures and symplectic foliations: On their global topology : flexibility and rigidity
Two topics are studied in this thesis.
The first one refers to symplectic and contact foliations; these are simply foliations with a leafwise symplectic or contact structure. The results amount to showing that adding some additional restrictions (namely, asking for the symplectic foliation to be given by a global closed 2-form, which we call strongness, and asking for the contact foliation to have all leaves tight) leads to rigid geometrical theories. In particular, the theorems proven are the following: - Contact foliations with overtwisted leaves satisfy the h-principle - The Lefschetz hyperplane theorem holds for leaves in a strong symplectic foliation (where the divisors are given by asymptotically holomorphic techniques) - There is a complete h-principle for vector fields without closed orbits in dimension 3 onwards (and, in particular, this gives a complete h-principle for vector fields additionally tangent to foliations) - The Weinstein conjecture holds for contact foliations with overtwisted leaves and this result is sharp.
The second topic we address is that of Engel structures. These are maximally non--integrable 2-plane fields in dimension 4. This thesis conforms a first effort to initiate a systematic study of these structures (and potentially of other classes of distributions). The main results state: - Engel structures satisfy an existence h-principle.
- There is a distinguished subclass called the loose Engel structures that are purely classified by their formal invariants.
- The classic (orientable) examples of Engel structures (i.e. the prolongations) are loose.
This mimicks the developments in flexibility in contact topology. The main question left open is whether all Engel structures are actually loose. For this, we study immersed submanifolds that are either tangent or transverse to the Engel structure and we show that: - A complete h-principle holds for tangent and transverse immersions in Engel manifolds.
In particular, they cannot possibly produce invariants for Engel structures.
