Conformal structures and symmetries
The purpose of this thesis is to study methods by which conformal vector fields on pseudo-Riemannian manifolds can be simplified. A vector field on a manifold M with metric g is conformal if its local flows preserve the metric g up to a scaling and unlike Killing vector fields, which preserve g exactly, it cannot in general be linearised in a neighbourhood of any given point. The difference is that a Killing vector field is affine, that is it preserves a connection on the manifold. In this case the connection is the canonical (Levi-Civita) connection associated with g, but affine vector fields with respect to any connection are linearisable. The task is to find new connections with respect to which the set of conformal vector fields, or some subset of them, are affine. Suppose that we have a manifold M with a pseudo-Riemannian conformal structure and an orthogonal splitting of the tangent bundle. We construct, for a natural choice of torsion, a unique connection in the principal bundle of frames adapted to the splitting. Moreover this connection is preserved by any transformations which preserve the splitting of the tangent bundle. Thus any conformal vector field which preserves the splitting is affine. The splitting can be chosen to reflect the tangent to the orbits of a subalgebra of conformal vector fields, the principal null directions of the Weyl tensor or the flow of a perfect fluid. We also give a study of conformal vector fields in three-dimensional Lorentzian manifolds. An equivalent of the Cotton-York tensor is used to investigate the behaviour of these vector fields at their fixed points in the same spirit as the Weyl tensor is used in four dimensions.