Some Restrictions on Symmetry Groups of Axially Symmetric Spacetimes

Lie transformation groups containing a one-dimensional subgroup acting cyclically on a manifold are considered. The structure of the group is found to be considerably restricted by the existence of a one-dimensional subgroup whose orbits are circles. The results proved do not depend on the dimension of the manifold nor on the existence of a metric, but merely on the fact that the Lie group acts globally on the manifold. Firstly some results for the general case of an $m+1$-dimensional Lie group are derived: those commutators of the associated Lie algebra involving the generator of the cyclic subgroup, $X_0$ say, are severely restricted and, in a suitably chosen basis, take a simple form. The Jacobi identities involving $X_0$ are then applied to show there are further restrictions on the structure of the Lie algebra. All Lie algebras of dimensions 2 and 3 compatible with cyclic symmetry are obtained. In the two-dimensional case the group must be Abelian. For the three-dimensional case, the Bianchi type of the Lie algebra must be I, II, III, VII$_0$, VIII or IX and furthermore in all cases the vector $X_0$ forms part of a basis in which the algebra takes its canonical form. Finally four-dimensional Lie algebras compatible with cyclic symmetry are considered and the results are related to the Petrov-Kruchkovich classification of all four-dimensional Lie algebras.