From local isometries to global symmetries: Bridging Killing vectors and Lie algebras through induced vector fields

TL;DR

This paper establishes a novel connection between local isometries represented by Killing vectors and the global symmetry groups of maximally symmetric manifolds, with applications to Euclidean space and Minkowski spacetime.

Contribution

It introduces a new method to relate Killing vector fields to isometry Lie groups via induced vector fields, bridging local and global symmetry concepts.

Findings

Derived a framework linking Killing vectors to Lie group symmetries.

Applied the framework to Euclidean space and Minkowski spacetime.

Provided explicit examples demonstrating the theoretical connection.

Abstract

The study of symmetries in the realm of manifolds can be approached in two different ways. On one hand, Killing vector fields on a (pseudo-)Riemannian manifold correspond to the directions of local isometries within it. On the other hand, from an algebraic perspective, global symmetries of such manifolds are associated with group elements. Although the connection between these two concepts is well established in the literature, this work aims to build an unexplored bridge between the Killing vector fields of n-dimensional maximally symmetric spaces and their corresponding isometry Lie groups, anchoring primarily on the definition of induced vector fields. As an application of our main result, we explore two specific examples: the three-dimensional Euclidean space and Minkowski spacetime.