Projective structure in 4-dimensional manifolds with positive definite metrics

This paper considers the situation on a 4-dimensional manifold admitting two metric connections, one of which is compatible with a positive definite metric, and which have the same unparametrised geodesics. It shows how, in many cases, 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 technique used is that based on a certain classification of the curvature tensor together with holonomy theory.