The investigation of rigidity phenomena is a central and wide topic in pseudo- Riemannian geometry. Rigidity results may appear at the metric level, like splitting theorems, or at the topological level, being compactness theorems or results involving the rst fundamental group classical examples. Moreover, if the manifold is equipped with some additional structure, one analyzes its behavior as it often gives rise to restrictions at both levels. In this thesis we consider Lorentzian manifolds equipped with an additional structure given by certain di erential equations: the Ricci soliton and the quasi- Einstein equations. Traditionally, in Analysis, one is mostly interested in the existence of a nontrivial solution to a di erential equation on a certain domain. However, from a more geometric point of view, one can also argue the existence of a domain manifold or structure for a di erential equation to provide a nontrivial solution, and this leads to a rigidity result for the corresponding structure.