Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2000-04-06
Class.Quant.Grav. 17 (2000) 2605
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
6 pages, TeX
Scientific paper
10.1088/0264-9381/17/13/401
In a recent paper Carot et al. considered carefully 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 comment a similar result, valid under considerably weaker assumptions, is recalled: 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 to 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 (Abelian), II, III, VII_0, VIII or IX. The relationship between the present approach and that of the original paper is discussed.
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