Holonomy Theory and 4-dimensional Lorentz Manifolds

This lecture describes the holonomy group for a 4-dimensional Hausdorff, connected and simply connected manifold admitting a Lorentz metric and shows, briefly, some applications to Einstein’s space-time of general relativity. 1. Holonomy Theory on 4-Dimensional Lorentz Manifolds. Let M be a 4-dimensional, smooth, Hausdorff, connected, and simply connected manifold admitting a smooth Lorentz metric g (and thus is paracompact) and associated smooth Levi-Civita connection Γ and covariant derivative ∇ with curvature R. The curvature components are denoted by Rbcd and the corresponding Ricci tensor components by Rab ≡ Racb. Let Φ be the holonomy group of M derived in the usual way by parallel transport of the members of the tangent space TmM to M at m around smooth closed curves at m. (In fact Φ is independent of the differentiability class Ck (k ≥ 1) of these curves – for this and more details about holonomy theory see [6]). It follows that Φ is a connected Lie group and hence a connected Lie subgroup of the identity component L0 of the Lorentz group L. Thus Φ is determined by its Lie algebra φ which is then a subalgebra of the Lie algebra A of L0. The problem with the study of Φ arises from the Lorentz signature of g and from the resulting more diverse nature of the subgroup structure of L0. The Lorentz group and its Lie algebra can be represented by L = {B ∈ GL(4,R) : BηB = η}, A = {C ∈MnR : ηC + (ηC) = 0} where η = diag(−1, 1, 1, 1). The members of A are skew self-adjoint with respect to η and can be represented in Minkowski space (i.e. the manifold R4 with metric η) by skew symmetric matrices Fab = ηacCb for C ∈ A. If C 6= 0, F has (matrix) rank two or four and in the former case it can be