These last two decades the connected sum techniques, essentially based on analytical tools, are revealed to be a powerful instrument to understand solutions of several nonlinear problem issued from the geometry (constant scalar curvature metrics in Riemannian geometry, self-dual metrics, metrics with special holonomy group, extremal Kaehler metrics, Yang-Mills equations, minimal and constant mean curvature surfaces, Einstein metrics, etc.). Even tough the techniques which allows one to consider the connected sum at points for solutions of nonlinear PDE's are frequently used and deeply understood, the analogous techniques for connected sums along sub-manifolds have not been mastered yet. The main purpose of this thesis is to (partially) plug this gap by developing such techniques in the context of the constant scalar curvature metrics and the Einstein constraint equations in general relativity
