Recovering Isometry Groups from Killing Vector Fields
Palavras-chave:
Killing vector fields, Lie groups, Riemannian manifolds, isometry groups, geometry, group theoryResumo
On pure geometric grounds, Killing vector fields play a central role in Riemannian manifolds. They serve as elements of tangent planes, generating local isometries and, in some approaches, could be used to construct global conserved structures on general relativistic spacetimes. Its arms extend to different areas, ranging from classical mechanics to the description of constant-curvature spaces of a homogeneous isotropic and static spacetimes passing through, recently, to their meaning in comprehending fluid flows on curved surfaces. According to several approaches, these vector fields on Lorentzian manifolds can be studied by employing proper actions of Lie groups. However, it’s not explicitly addressed how to recover the Lie group using the expression of Killing vector fields. Our investigation identifies a connection between Killing vector fields and Lie algebra elements by interpreting them as induced vector fields. This connection allows us to reconstruct the Lie groups that preserve the respective metrics, contributing to a deeper understanding of the interplay between geometry and group theory.
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