Path-lifting properties of the exponential map with applications to geodesics

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work F. Browder R. Rheindboldt 1950s 1960s, apply their elegant theory to exponential maps. obtain thereby a number novel results existence multiplicity geodesics joining any two points connected affine manifold, well causal connecting causally related on Lorentzian manifold. These include generalization well-known Hadamard–Cartan theorem Riemannian geometry manifold context, new version so-called using weaker assumptions than global hyperbolicity timelike 1-connectedness required extant version. also general description pseudoconvexity disprisonment broad classes terms suitable restrictions map. The latter sheds further light relation between given such class one hand, geodesic connectedness by members that other.