Dynamical and Riemannian properties of Engel structures

The only families of distributions stable in the sense of Cartan are line fields, contact structures, even contact structures and Engel structures. This thesis studies various aspects of the theory of Engel structures, i.e. maximally non-integrable smooth 2-plane fields on smooth 4-manifolds. First of all we analyze the interplay between Engel structures and Riemannian metrics. A choice of a pair of Engel defining forms permits to define a distribution transverse to the Engel structure, called Reeb distribution. We furnish some sufficient conditions for the existence of integrable Reeb distributions in terms of metric properties of the Engel structure. This discussion is related to the study of Engel vector fields, i.e. vector fields whose flow preserves an Engel structure. We introduce K-Engel structures, i.e. Engel structures admitting transverse Killing Engel vector fields, and classified them topologically. Many of these structures admit interesting features, such as contact fillings. Every Engel structure induces an even contact structure, which in turn comes equipped with a line field called characteristic foliation. We study the dynamics of such line field and define a new invariant associated with closed orbits. This furnishes a necessary condition for an even contact structure to be induced by an Engel structure. This invariant is particularly useful with Morse-Smale line fields and remarkably the same techniques apply in a more general setting. For example, on a 3-manifold, they give a sufficient and necessary condition for a non-singular Morse-Smale field to be Legendrian with respect to a contact structure. Finally we study the holomorphic analogue of Engel structures and we produce an uncountable family of non-standard examples on C4. We do this by controlling the topology and the geometry of the space of points which admit holomorphic immersions of C tangent to the characteristic foliation.