Projective structure and holonomy in four-dimensional Lorentz manifolds

This paper studies the situation when two four-dimensional Lorentz manifolds (that is, space–times) admit the same (unparametrised) geodesics, that is, when they are projectively related. A review of some known results is given and then the problem is considered further by treating each holonomy type in turn for the space–time connection. It transpires that all holonomy possibilities can be dealt with completely except the most general one and that the consequences of two space–times being projectively related leads, in many cases, to their associated Levi-Civita connections being identical.