Four topics in contact geometry from a sub-Riemannian view point

In the early 70’s J. Martinet proved that every three dimensional manifold admits a contact structure, i.e., a completely non-integrable plane field. Since then the study of contact manifolds grew into a rich and surprising chapter of differential topology. Beside being topologically interesting, contact manifolds play a central role in sub-Riemannian geometry, being the simplest instances of equiregular ge- ometries which are not Riemannian. A contact sub-Riemannian manifold is a contact manifold endowed with a sub-Riemannian metric. This thesis is a contribution to the study of the relations connecting the two subjects: sub-Riemannian and contact geometry. One can exploit the geometric tools of sub-Riemannian geometry to produce a quantitative study of contact manifolds. This approach is implemented in Chapter 1, where, through the use of sub- Riemannian metrics, we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of contact Jacobi curve, and prove sharp lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We also prove similar, but non-sharp, comparison theorems in terms of sub-Riemannian canonical cur- vature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan-Hadamard theorem. Conversely, one can study the geometric properties of a contact sub-Riemannian manifold through the techniques of contact topology: in Chapter 2, we consider smooth embedded surfaces in a contact sub-Riemannian manifold and we make use of techniques from convex surface theory to understand the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). We study closed surfaces of genus grater or equal to one, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the finiteness/not-finiteness of the distance. In Chapter 3 we make use of the functional metric invariants of contact sub- Riemannian geometry to define an index that determines the topological type of the fundamental cover of left-invariant contact structures on Lie groups. In Chapter 4 we consider a geometric procedure to construct Engel structures starting from contact ones. This can be seen as a geometric Cartan prolongation of the space of contact elements associated with a Riemannian manifold. The resulting Engel manifold comes with a natural sub-Riemannian metric, which is the object of study of the chapter.