Locally homogeneous spaces, induced Killing vector fields and applications to Bianchi prototypes

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 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 three-dimensional spaces, whereby the embedding of them into a four-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. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4732119]