Projective structure in 4-dimensional manifolds with metric signature (+,+,−,−)

This paper considers the situation for a 4-dimensional manifold admitting two metric connections, one of which is compatible with a metric of signature (+,+,−,−)(+,+,−,−), and which have the same unparametrised geodesics. It shows how, in many cases (decided on holonomy type), the relationship between these connections and metrics can be found. In many of these cases, the connections are found to be necessarily equal. The general techniques used are based on holonomy theory.