Pseudo-Riemannian structures on Lie groups, computation of reachable sets via geodesics vs optimal control problems

One of the famous results in Riemannian geometry is the Hopf-Rinow theorem stating that any two points on a compact Riemannian manifold can be joined by a geodesic segment. The latter is tantamount to saying that the exponential function of the corresponding affine connection is subjective, thus it provides a parametrization of the manifold. In the case of pseudo-Riemannian manifolds this need not be true any longer, the geodesic segments starting at a fixed point will always cover a neighbourhood of that point, but they may fail to cover the whole manifold. Motivated by applications in special relativity there is a great interest in Lorentzian manifolds and the set of points that are reachable along nonspace-like curves. We consider the case that the manifold is a Lie group and show how the Pontryagin maximum principle can be used to compute this reachable set. However, this need not yield a complete parametrization of the reachable set, but at least we obtain a parametrization of an open and dense subset whereas the geodesics may leave out big parts