
SIGMA 5 (2009), 066, 23 pages arXiv:0906.5227
https://doi.org/10.3842/SIGMA.2009.066
Contribution to the Special Issue “Élie Cartan and Differential Geometry”
Holonomy and Projective Equivalence in 4Dimensional 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 4dimensional Lorentz manifolds which
are projectively related, that is, whose LeviCivita 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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