Riemannian manifolds whose holonomy group lies inside the exceptional Lie group \(G_2\) are called \(G_2\)-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of \(G_2\)-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is \(G_2\). We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous spheres.