Explorando a conexão entre grupos de Lie e campos vetoriais de Killing em espaços planos

Abstract

This work aims to study topics in differential geometry and group theory with applications in theoretical physics. On one hand, Killing vector fields on a (pseudo-)Riemannian manifold are related to directions of isometries along it. On the other hand, some elements belonging to groups are responsible for isometries in physical systems, such as the elements of the groups SO(3) and SO(1, 3). In this sense, the main objective of the work is to explore a connection between such Killing vector fields and elements of the Lie algebra of Lie groups relevant in the context of theoretical physics. Alongside our main goal of exploring this connection, we hope that the initial chapters of the work will be sufficient for readers seeking an introduction to differential geometry and Lie groups, through examples and figures distributed throughout the text, with the purpose of clarifying the advanced topics studied here. Regarding these topics, they will also serve as a starting point for more advanced studies in general relativity and, in the long term, quantum gravity.