Cauchy Problem in Convex Neighborhood

PLuz

While reading the reference

Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available here,

there is something that I don't quite understand. Eq.(16.6) is an evolution equation for de Green functional. Then in Eq.(16.7) Poisson et. al. look for a specific solution and they state that the separation of the Green functional is valid only in the convex neighborhood of a field point x. I assume that is because the Cauchy problem is valid only in that neighborhood... My question is why? Why is the Cauchy problem related to the imposition that the two points must be connected by a unique geodesic?

Vargo

From a glance, it looks like the calculation propagates the solution from one point to another by integrating along the geodesic that joins them. If they are connected by more than one geodesic, then the two calculations might not agree. Hence the restriction to a convex neighborhood.