ETD

• characteristic set
• contact manifold
• Heisenberg group
• rectifiable set
• Riemmanian approximation.
• Rumin complex

Rumin complex is a complex of differential forms adapted to a contact geometry. It is defined from De Rham's one by a fairly complicated algebraic procedure. Despite having many good properties, Rumin complex does not emerge so naturally from the current literature.

Our goal is to define Rumin Complex in a natural way. This is not fully realized, but satisfactory partial results are obtained. The approach is the following: we approximate the Heisenberg Group with Riemannian manifolds, in order to consider the limit of De Rham complexes of the approximating manifolds. The resulting limit is a complex of forms strictly related to Rumin's one.

We also discuss the dual problem: the notion of regular surface and rectifiable set within Heisenberg group. One of the main result we present is that every rectifiable subset of high dimension of Heisenberg group is trivial. We also highlight a connection between this theorem and the theory of Characteristic Sets.