HOLONOMY THEORY AND 4–DIMENSIONAL LORENTZ MANIFOLDS

GRAHAM S. HALL

Abstract

Let M be a smooth, 4–dimensional, connected, Hausdorff manifold with smooth Lorentz metric g. Let Φ be the holonomy group of M associated with the Levi–Civita connection from g with holonomy algebra ϕ. The purpose of this talk is to give some results regarding the possibilities for Φ. Let L be the Lorentz group with connected component L_{′} and with Lie algebra L. Clearly, Φ is a subgroup of L and the well-known classification of L is thus important in this work. Φ is a Lie group and, if M is simply connected, Φ is a connected subgroup of L_{′}. In this case, since ϕ is a subalgebra of L, there exists a one–to–one correspondence between the possibilities for Φ 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 Φ 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.

Department of Mathematical Sciences

University of Aberdeen

Aberdeen AB24 3UE

Scotland

UNITED KINGDOM

e-mail: g.hall@maths.abdn.ac.uk