# Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold

What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $$x''(t) + \nabla V(x,t) = 0$$) in a (complete) Riemann manifold?