Conformal structures and solitons in pseudo-Riemannian geometry

Abstract

This thesis is divided into three distinct parts, each exploring different aspects of mathematical structures. The first part focuses on the investigation of locally conformally flat structures on fourdimensional manifolds. Specifically, the study delves into the local conformal flatness of Kähler, para-Kähler, and null-Kähler manifolds, and provides a complete description of the para-Kähler Lie algebras. Moving on to the second part, the research focuses on the analysis of solitons associated to geometric flows. This part offers a complete classification of four-dimensional Lorentzian Ricci solitons and Riemannian algebraic Bach solitons in dimension four. Finally, the third part covers the study of homogeneous manifolds that have half-harmonic Weyl curvature and those which admit more than one homogeneous structure.