The aim of this thesis is to give a method to produce families of complete Riemannian metrics with positive constant $\sigma_k$-curvature equal to $2^{-k}\binom{n}{k}$ on $M\setminus \Lambda^p$, where $M$ is a Riemannian manifold and $\Lambda^p$ is a smooth, closed, compact submanifold, whose dimension is $0 < p < \frac{n-2k}{2}$. To simplify the computations, we assume $k = 2$. However, we conjecture the method to work for $k > 2$. First, we give a geometrical motivation of the problem and then we establish a methodology to solve it using analysis tools. Most part of the analysis is concerned with the study of the mapping properties of the linearized operator around the approximate solutions.