The geometry of Lorentzian Ricci solitons

The investigation of rigidity phenomena is a central and broad 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 first 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 differential equations: the Ricci soliton and the quasi-Einstein equations.