Surjectivity properties of the exponential function of an ordered manifold with affine connection

It is a classical theorem that the exponential function of a Lie group with compact Lie algebra is surjective. We recall that a Lie algebra is said to be compact, if there exists a positive definite bilinear form which is invariant under the adjoint action. One can prove this theorem by means of the Pontryagin maximum principle (PMP). The idea is to consider a certain optimal control problem and prove that the solutions are one-parameter semigroups. The latter is equivalent to the statement that the optimal controls are constant. Another theorem from Lorentzian geometry states that two points p and q of a Lorentzian manifold may be joined by a geodesic segment, provided that the order interval [p, q] is compact. These two results are not as unrelated as one might expect, for they can be deduced from a more general theorem on the exponential function of an ordered manifold with affine connection