Conformal structures and solitons in pseudo-Riemannian geometry

This thesis is divided into three distinct parts, each of which explores different aspects of mathematical structures. The first part focuses on the investigation of locally conformally flat structures on four-dimensional manifolds. More precisely, the study delves into the local conformal flatness of Kähler, para-Kähler, and null-Kähler manifolds, and provides a complete geometric description of four-dimensional para-Kähler Lie algebras. Moving on to the second part, the research focuses on the analysis of solitons associated to both the Bach and Ricci flows. This part offers a complete classification of four-dimensional left-invariant Lorentzian Ricci solitons and Riemannian algebraic Bach and Ricci solitons in dimension four.

Finally, the third part covers the description of four-dimensional homogeneous Riemannian manifolds that have half-harmonic Weyl curvature tensor and those homogenous manifolds which admit more than one homogeneous structure in dimension free.