Holonomy Theory and 4-dimensional Lorentz Manifolds
by
Graham Hall
University of Aberdeen, Scotland UK

Let M be a smooth, 4-dimensional, connected, Hausdorff manifold with smooth Lorentz metric g. Let \Phi be the holonomy group of M associated with the Levi-Civita connection from g with holonomy algebra \phi. The purpose of this talk is to give some results regarding the possibilities for \Phi. Let L be the Lorentz group with connected component L₀ and with Lie algebra L. Clearly, \Phi is a subgroup of L and the well-known classification of L is thus important in this work. \Phi is a Lie group and, if M is simply connected, \Phi is a connected subgroup of L₀. In this case, since \phi is a subalgebra of L, there exists a one-to-one correspondence between the possibilities for \Phi and the subalgebras of L. This gives a classification up to isomorphism of the possible holonomy groups of M. A coarser classification can (and will) be given in terms of covariantly constant and recurrent vector fields on M. Another useful approach (thinking of the pair (M, g) as the space-time of Einstein's general relativity) is to seek the possibilities for \Phi when the energy-momentum tensor representing the physics of space-time is given. This will be discussed for the more commonly used energy-momentum tensors. Some remarks will also be made concerning the use of holonomy theory in the study of space-time curvature structure and space-time symmetries and in the theory of the Petrov classification of the Weyl tensor for gravitational fields.

Date received: March 26, 2002