Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2000-11-07
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
6 pages, LaTex. (World Scientific style file: sprocl.sty needed) To appear in Proceedings of the Spanish Relativity Meeting (E
Scientific paper
In a recent paper Carot et al. considered the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this paper a similar result, valid under considerably weaker assumptions, is derived: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I, II, III, VII_0, VIII or IX. Some results on n-dimensional Lie groups are also derived and applied to show there are severe restrictions on the structure of the allowed four-dimensional Lie transformation groups compatible with cyclic symmetry.
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