Abstract
This work aims to explore a novel relation between Killing vector fields and Lie algebras using induced vector fields. On one hand, Killing vector fields on a (pseudo) Riemannian manifold are related to directions of local isometries along it. On the other hand, group elements are responsible for global isometries in physical systems, such as the elements of the groups \(SO(3)\) and \(SO(1,3)\) acting in space and space-time. In this sense, the main objective of this manuscript is to reconstruct such groups (together with translations), after establishing the connection between Killing vector fields and elements of the Lie algebra corresponding to such Lie groups, using mainly the definition of induced vector fields.