Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes

TL;DR

This paper develops a method based on Cartan's theory to identify all symmetries in locally homogeneous spaces, applying it to 3D spaces embedded in 4D Lorentzian manifolds, with implications for Einstein's equations.

Contribution

It introduces a systematic approach to find symmetries in locally homogeneous spaces and applies it to derive solutions relevant to Einstein's field equations.

Findings

Identified all symmetries in 3D locally homogeneous spaces.

Recovered special solutions to Einstein's field equations.

Provided a local differential geometric framework for symmetry analysis.

Abstract

An answer to the question: Can, in general, the adoption of a given symmetry induce a further symmetry, which might be hidden at a first level? has been attempted in the context of differential geometry of locally homogeneous spaces. Based on E. Cartan's theory of moving frames, a methodology for finding all symmetries for any n dimensional locally homogeneous space is provided. The analysis is applied to 3 dimensional spaces, whereby the embedding of them into a 4 dimensional Lorentzian manifold is examined and special solutions to Einstein's field equations are recovered. The analysis is mainly of local character, since the interest is focused on local structures based on differential equations (and their symmetries), rather than on the implications of, e.g., the analytic continuation of their solution(s) and their dynamics in the large.