Abstract
Cauchy-Riemann geometry, CR for short, is the natural geometry of real pseudoconvex hypersurfaces of C^{n+1} for n≥1. We consider the generic case when CR manifolds are contact manifolds. CR geometry presents strong analogies with conformal geometry; hence, known invariants and techniques of conformal geometry can be transported to that context. We focus in this thesis on two such invariants. In a first part, using asymptotically complex hyperbolic geometry, we introduce a CR covariant differential operator on maps from a CR manifold to a Riemannian manifold, which coincides on functions with the CR Paneitz operator. In a second part, we propose a Yamabe invariant for contact manifolds which admit a CR structure, and we study its behaviour under connected sum.