Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 066, 23 pages      arXiv:0906.5227      http://dx.doi.org/10.3842/SIGMA.2009.066
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

Graham S. Hall a and David P. Lonie b
a) Department of Mathematical Sciences, University of Aberdeen, Meston Building, Aberdeen, AB24 3UE, Scotland, UK
b) 108e Anderson Drive, Aberdeen, AB15 6BW, Scotland, UK

Received March 18, 2009, in final form June 11, 2009; Published online June 29, 2009

Abstract
A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided.

Key words: projective structure; holonomy; Lorentz manifolds; geodesic equivalence.

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